Besov Regularity of Stochastic Partial Differential Equations on Bounded Lipschitz Domains

Transcription

1 Dissertation Besov Regularity of Stochastic Partial Differential Equations on Bounded Lischitz Domains Petru A. Cioica 23

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3 Besov Regularity of Stochastic Partial Differential Equations on Bounded Lischitz Domains Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. vorgelegt dem Fachbereich Mathematik und Informatik der Philis Universität Marburg von Petru A. Cioica geboren am 22. März 983 in Cluj-Naoca/Klausenburg/Kolozsvár Rumänien

4 Vom Fachbereich Mathematik und Informatik der Philis-Universität Marburg Hochschulkennziffer: 8 als Dissertation angenommen am: 2. Dezember 23 Erstgutachter: Prof. Dr. Stehan Dahlke, Philis-Universität Marburg Zweitgutachter: Prof. Dr. René L. Schilling, Technische Universität Dresden Drittgutachter: Prof. Dr. Stig Larsson, Chalmers University of Technology, Göteborg, Schweden Tag der mündlichen Prüfung: 7. Februar 24

5 For Christine. In memory of my mother, Carmen Luminiţa Cioica d2.

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7 i Contents Acknowledgement iii Introduction. Motivation Overview of the relevant regularity theory The thesis in a nutshell Outline Preliminaries 5 2. Some conventions Bounded oerators Domains in R d Measurable maings and L -saces Probabilistic setting Functions, distributions and the Fourier transform Miscellaneous notation Stochastic integration in UMD Banach saces Geometric roerties of Banach saces γ-radonifying oerators Stochastic integration for cylindrical Brownian motions Function saces Sobolev saces Saces of Bessel otentials Weighted Sobolev saces Besov saces Triebel-Lizorkin saces Semigrous of linear oerators Starting oint: Linear SPDEs in weighted Sobolev saces Stochastic arabolic weighted Sobolev saces H γ,q G, T An L -theory of linear SPDEs on bounded Lischitz domains Embeddings of weighted Sobolev saces into Besov saces Weighted Sobolev saces and Sobolev saces without weights Wavelet decomosition of Besov saces on R d Weighted Sobolev saces and the non-linear aroximation scale An alternative roof of Theorem

8 ii CONTENTS 5 Satial Besov regularity of SPDEs on bounded Lischitz domains Linear equations Semi-linear equations Sace time regularity of the inhomogeneous heat equation with additive noise7 6. Sace time regularity of elements from H γ,q O, T The saces H γ,q O, T and SPDEs The stochastic heat equation in H γ,q O, T A result on the L q L -regularity Sace time regularity Zusammenfassung 3 Notation 37 Bibliograhy 43 Index 5

9 iii Acknowledgement During my work on this thesis, I have been suorted by many eole and institutions, whom I want to exress my gratitude at this oint. I want to start with my advisor, Prof. Dr. Stehan Dahlke. I am deely grateful, Stehan, that you acceted me as your PhD student. Thank you for roviding an excellent work and study environment and for suorting me in all resects, in articular, when alying for fellowshis and grants, when demanding more background from different mathematical areas, or when any self-doubt came u. Next, I thank Prof. Dr. René L. Schilling for agreeing to overview this thesis and for the constant suort whenever needed. Thank you also for several invitations to Dresden, I enjoyed the atmoshere at the Institut für Mathematische Stochastik very much. My secial thanks goes to Junior-Prof. Dr. Felix Lindner. Thank you, Felix, for many fruitful discussions on SPDEs and related toics, for a lot of good advice, for your encouragement and your kind hositality during my visits to Dresden. Thank you also for reading very carefully arts of this manuscrit. Over the last years, I have had the leasure to articiate at several meetings of the roject Adative Wavelet Methods for SPDEs, which is funded by the German Research Council DFG and is art of the DFG-Priority Program 324 Mathematical methods for extracting quantifiable information from comlex systems DFG-SPP 324. I want to thank all the members of this roject and their affiliates for many fruitful discussions. Besides the already mentioned ersons, these are: Prof. Dr. Klaus Ritter and his assistants Nicolas Döhring and Dr. Tiange Xu from Kaiserslautern, Stefan Kinzel from Marburg and Junior-Prof. Dr. Thorsten Raasch from Mainz. During one of my visits to Dresden, I have had the leasure to meet Prof. Dr. Kyeong-Hun Kim and Prof. Dr. Kijung Lee, who have been in Dresden as Fellows of the DFG-SPP 324. A very fruitful collaboration started and I want to thank you both, Kyeong-Hun and Kijung, for teaching me many details on the regularity theory of SPDEs in weighted Sobolev saces and the techniques used in the analytic aroach. Thank you also for inviting me to Korea, where I have had the oortunity to attend a very well organized summer school on Stochastic Partial Differential Equations and Related Fields. I enjoyed the time at this summer school and at your deartment very much. I also want to thank Dr. Sonja G. Cox for acceting to visit Marburg two years ago and rovide an insight into recent results concerning numerical methods for SPDEs in Banach saces. Thank you, Sonja, for exlaining different asects from the semigrou aroach to SPDEs in Banach saces to me. Thank you also for a coy of your very nice thesis and the delicious Swiss chocolate. While working on this thesis, I have received great suort from my colleagues from the Workgrou Numerics at Philis-Universität Marburg. Thank you, guys, for daily lunch and coffee breaks, for heling me whenever needed, in articular, during the last year. I also want to thank Mrs Jutta Hael for being a erfectly organized and very kind secretary. My PhD studies have been mainly funded by a doctoral scholarshi of the Philis-Universität Marburg. In this context I want to thank the selection anel for their confidence, as well

10 iv Acknowledgement as Dr. Susanne Igler, Dr. Katja Heitmann and Dr. Ute Kämer for excellent assistance. Over the last years, I have had the oortunity to attend many workshos and conferences, and to intensify my collaborations while visiting different mathematical deartments. Financial suort by the DFG-SPP 324 for the travelling costs is gratefully acknowledged. At this oint, I also want to thank my father for his constant encouragement. I enjoy it very much, that you are almost always in good humour again. Last but not least, I want to thank you, Christine, for your love, your atience and for very recious moments after maths.

11 Chater Introduction This thesis is concerned with the regularity of semi-linear second order arabolic stochastic artial differential equations SPDEs, for short of Itô tye on bounded Lischitz domains. They have the following form: d d du = a ij u x i x j + b i u x i + cu + f + Lu dt u = i,j= + i= d σ ik u x i + µ k u + g k + Λu k dwt k on Ω [, T ] O, k= i= on Ω, T ] O, u = u on Ω O.. Here, and in the rest of this thesis, O is a bounded Lischitz domain in R d d 2 and T, denotes a finite time horizon. Moreover, w k t t [,T ], k N, is a sequence of indeendent real valued standard Brownian motions with resect to a normal filtration F t t [,T ] on a comlete robability sace Ω, F, P and du denotes Itô s stochastic differential with resect to the time t [, T ]. The coefficients a ij, b i, c, σ ik, and µ k with i, j {,..., d} and k N := {, 2,...}, are real valued functions on Ω [, T ] O and fulfil certain conditions which will be secified later on in Chater 3, see Assumtion 3.. The non-linearities L and Λ are assumed to be Lischitz continuous in suitable saces, see Chater 5, in articular Assumtion 5.9, for details. In this thesis we take a functional analytic oint of view, meaning that the solution u is not considered as a real valued function deending on ω, t, x Ω [, T ] O but as a function on Ω [, T ] taking values in the sace D O of real valued distributions on O. The most rominent equation of the tye. is the stochastic heat equation with additive or multilicative noise. More general equations of the form. with finitely many w k t t [,T ], k {,..., N}, aear in the context of non-linear filtering roblems, see, e.g., [8, Section 8.] and [7]. Choosing infinitely many Brownian motions w k t t [,T ], k N, allows us to consider equations driven by sace-time white noise, cf. [8, Section 8.3]. These equations are suggested, for instance, as mathematical models for reaction diffusion systems corruted by noise, see [32, Section.7] and the references therein, in articular, [9]. In general, the question whether a unique solution to Eq.. exits is well-studied. However, in the majority of cases, this solution can not be secified. Thus, in order to make equations of the form. ready to use as mathematical models in alications, the solution has to be constructively aroximated. Therefore, efficient numerical methods are needed. Usually, their erformance deends on the regularity or smoothness of the solutions to the considered SPDEs in secific scales of function

12 2 Introduction saces. As we will elaborate later on in detail, the scale B α τ,τ O, τ = α d +, α >, of Besov saces 2 being fixed lays an outstanding role in this context. We refer to Subsection for the definition of Besov saces. In this thesis we analyse the regularity of SPDEs of the form. in the scale. We will be mainly concerned with the following two tasks: T Satial regularity. We use the scale to measure the smoothness of the solution u with resect to the sace coordinates. That is, we ask for an α > as high as ossible, such that for all < α < α and /τ = α/d + /, the solution u is contained in the sace of equivalence classes of -integrable B α τ,τ O-valued stochastic rocesses. T2 Sace time regularity. Under the assumtion that the solution u is a B α τ,τ O-valued stochastic rocess with α and τ as in, we analyse the Hölder regularity of the aths of this rocess. Before we continue our exosition, we motivate our analysis by elaborating in detail the imortance of the toics T and T2. In articular, we will emhasize their link with the convergence analysis of certain numerical methods.. Motivation Our motivation to study the regularity of SPDEs in the scale of Besov saces is closely related to the theme of adative numerical wavelet methods. Since this toic is not a common rerequisite in the stochastic analysis community we give a rather detailed exosition aiming to oint out the significance of our results from the oint of view of numerical analysis. However, we will not be too rigorous in a formal sense, but rather try to emhasize some key rinciles and basic results from the theory of numerical methods and non-linear aroximation which motivate our analysis. For an in-deth treatment of these toics we refer to the monograh [27] on numerical wavelet methods and to the survey article [46] on non-linear aroximation theory, see also [37]. Usually, the term wavelet is used for the elements of a secific kind of basis for the sace L 2 O of quadratically Lebesgue-integrable functions on a domain O R d, which allows the decomosition of functions into comonents corresonding to different scales of resolution [33]. Such a basis is tyically constructed by means of a multiresolution analysis MRA, for short, i.e., a sequence S j j j of closed linear subsaces of L 2 O with S j S j+ for all j j, and j j S j L2 O = L 2 O. The latter means that the union of all S j, j j, is dense in L 2 O. The MRA is designed in such a way that for each j j, the sace S j is sanned by a Riesz basis {φ λ : λ j } of so-called scaling functions. Furthermore, the comlement of S j in S j+ is sanned by another Riesz basis {ψ λ : λ j } of so-called wavelets. Following the notation from [27] we write j := j and denote the scaling functions sanning S j also by ψ λ, λ j. Then, setting := j j j, we call {ψ λ : λ } := {ψ λ : λ j } j j

13 . Motivation 3 a wavelet Riesz basis of L 2 O. The index λ tyically encodes several tyes of information, namely the scale level j + j, if λ j, the satial location, and also the tye of the wavelet. For constructions of wavelet bases for diverse shaes of bounded domains including olygonal and olyhedral domains we refer to [42 44] or [9, 2], see also [27, Section.2] for a detailed discussion. Tyically, the elements of a wavelet basis are local in the sense that they have comact suorts and the size of the suorts decays exonentially with the scale. Furthermore, they fulfil aroriate smoothness assumtions and have vanishing moments u to a rescribed order. These roerties yield the following facts [33]: ˆ Weighted sequence norms of wavelet coefficients are equivalent to Lebesgue, Sobolev and Besov norms for a certain range of regularity and integrability arameters, deending in articular on the smoothness of the wavelets. ˆ ˆ The reresentation of a wide class of oerators in the wavelet basis is nearly diagonal. The vanishing moments of wavelets remove the smooth art of a function. Due to these features, wavelets become a owerful tool for solving oerator equations. Let us discuss this toic with the hel of a classical examle. We write W 2 O for the closure of the sace C O of infinitely differentiable functions with comact suort on O in the L 2O-Sobolev sace of order one, which we denote by W2 O; see Subsection 2.3. for a recise definition of Sobolev saces. Let a : W 2 O W 2 O R be a continuous, symmetric and ellitic bilinear form, so that, in articular, there exists a finite constant C >, such that It defines an isomorhism C u 2 W 2 O au, u C u 2 W 2 O, u W 2 O..2 A : W 2 O W 2 O u au,, where W2 O denotes the dual of W 2 O. Thus, for f W2 O, the equation Au = f,.3 has a unique solution u W 2 O, which is simultaneously the unique solution of the variational roblem au, v = fv, v W 2 O..4 However, in general this solution is not known exlicitly. Therefore, in order to use.3 as a mathematical model in real-life alications, the solution has to be constructively aroximated. To this end, Eq..4 is discretized. One classical way to discretize this equation is to emloy a Galerkin method. That is, we choose an increasing sequence V m m J with J N of subsaces of W 2 O and determine the solutions u m V m to the variational roblems au m, v m = fv m, v m V m,.5 successively for m J. The index m denotes the number of degrees of freedom here: scaling functions and wavelets sanning the subsace V m. We distinguish two kinds of numerical methods, deending on the way the refinement from a sace V m to its successor V m, m, m J, is erformed. In our context refinement means to add wavelets to the basis functions used to

14 4 Introduction aroximate the current aroximative solution u m. On the one hand, we can develo a uniform method which is based on the underlying MRA and set V mj := S j, j j, where mj = j i j i N for j j usually, on bounded domains the cardinality of j behaves like 2 jd. This method is called uniform, since when assing from V mj to V mj+ we add all the wavelets at the scale level j+, i.e., we choose a finer resolution uniformly on the entire domain. On the other hand, since the aroximation u m might be already sufficiently accurate in some regions of the domain, it is reasonable to look for a self-regulating udating strategy and try to refine the resolution only at that arts where the accuracy is not yet satisfactory. Such an adative method, executes the following stes successively for m J :. Solve Eq..5 in V m. 2. Estimate the local error of u u m in a suitable norm E. 3. Refine where it is necessary. Of course, for the second ste one needs a osteriori error estimators, since the solution u is not known exactly. These estimators should rely on local error indicators, so that they rovide information about the way the sace V m has to be refined in the subsequent ste. However, one is faced with at least three major difficulties on the way to a fully-fledged adative method. Firstly, the design of local error estimators as they are needed for adative strategies is not a trivial task. A second difficulty is the convergence roof for adative strategies and the estimation of their convergence rates. Thirdly, their imlementation turns out to be much more difficult than the imlementation of uniform counterarts. Thus, before we decide to work on the develoment of an adative method, we need to check whether adativity really ays, in the sense that there is any chance to obtain a higher convergence rate than by uniform alternatives. A numerical method is said to have convergence rate s > in the Banach sace E, E, if there exists a constant C,, which does not deend on the number of degrees of freedom m J needed to describe the aroximative solution u m V m, such that u u m E C m s, m J, where u E denotes the exact solution of the given roblem. The benchmark for any numerical aroximation method based on {ψ λ : λ } is the rate of the best m-term wavelet aroximation error σ m,e u := inf u u m E, u m Σ m where { Σ m := λ Λ c λ ψ λ : Λ, Λ } = m, c λ R, λ Λ is the sace of m-term aroximations from {ψ λ : λ }, m N. As it is easy to see, Σ m is not a linear sace: The sum of two functions, each of which uses m basis elements, might make use of u to 2m basis elements and is therefore usually not contained in Σ m. This is why m-term aroximation is referred to as a non-linear aroximation method. Obviously, the convergence rate of any numerical method based on {ψ λ : λ } is dominated by the decay rate of the best m-term aroximation error σ m,e u, m N. Since, in general, the solution u is not known, we will not be able to find aroximations u m, m N, reroducing the errors σ m,e u, m N. However, what we can aim for is to develo a numerical method which has the same convergence rate as the best m-term aroximation error.

15 . Motivation 5 If the convergence rate of a uniform method meets the benchmark, then working on the develoment of adative algorithms is suerfluously. However, if the converse is true, i.e., if the rate of best m-term aroximation is strictly higher than the convergence rate of uniform methods, the develoment of adative methods is comletely justified. Since the error of uniform numerical methods based on V mj = S j, with mj = j i j i N, j j, is dominated by e m,e u := inf u u m E, m = mj, j j, u m V m this means: Adativity ays, only if there exists an α > and a corresonding constant C, which does not deend on m N, such that σ m,e u C m α/d, m N,.6 holds for the solution u E, and, simultaneously, α > s max u := su { s : j N : e mj,e u C mj s/d, C indeendent of j }..7 The question whether.6 and.7 with E = L O are simultaneously fulfilled, where,, can be decided after a rigorous regularity analysis of the target function u. On the one hand, it is well-known that under certain technical assumtions on the wavelet basis, which can be found, e.g., in [27, Chater 3 and 4] the decay rate of e m u := e m,lou is linked with the L O-Sobolev regularity of the target function. That is, there exists an uer bound s N, deending on the smoothness and olynomial exactness of the wavelet basis, such that, for all s, s, u W s O imlies e m u C m s/d, m = mj, j j,.8 with a constant C, which does not deend on m. As mentioned in the introduction of [27, Chater 3], statements similar to.8 also hold for aroximation methods based on finite elements instead of wavelets of course, with adjusted saces V m, m J see also the standard literature on finite elements like [2] or []. One can also show the following converse of.8: The existence of a constant C, such that e m u C m s/d, m = mj, for all j j, imlies u W s O, s < s. In articular, if u / W s O for some s,, then s max u s with s max u as defined in.7. This yields s max u = s Sob maxu := su { s : u W s O }..9 On the other hand, the convergence rate of the best m-term wavelet aroximation error σ m u = σ m,lou, m N, is governed by the smoothness of u in the so-called L O-non-linear aroximation scale Bτ,τ α O, τ = α d +, α >, of Besov saces. That is, for all α, s, Therefore, if u B α τ,τ O, u B α τ,τ O, τ = α d + τ = α d + imlies σ m u C m α/d, m N. with α > s Sob maxu,. The factor /d in the exonent is just a useful convention in order to match with the results resented below.

16 6 Introduction then.6 and.7 are simultaneously satisfied with s max u = s Sob maxu. In this case, the decay rate of the best m-term wavelet aroximation error is higher than the convergence rate of the uniform wavelet method resented above. Thus, by our exositions above, if. is fulfilled, working on the develoment of adative wavelet methods is comletely justified. For deterministic ellitic equations it could be already shown that, indeed, adativity ays: The results from [34 36, 38, 4] together with [57, 58] show that solutions of ellitic equations on non-smooth domains generically behave like described by.. Simultaneously, for this class of equations, there exist adative wavelet methods which realise the convergence rate of the best m-term aroximation error in a Hilbert sace setting = 2, see, e.g., [28,39]. The error is measured in the energy norm induced by the equation, which is, in general, equivalent to a suitable Sobolev norm. In our examle from above, the energy norm is given by a := a, and it is equivalent to the L 2 O-Sobolev norm of order one in W 2 O by.2. There also exist otimal adative wavelet algorithms for more general deterministic equations, see, e.g., [29,2], this list being by no means comlete. Our analysis is motivated by the question whether these results can be extended to solutions of SPDEs of the form.. We tackle and solve the tasks T and T2 with the following scoes: ad T. Satial regularity. By analysing the satial regularity of the solution rocess u in the scale of Besov saces we aim to clarify whether u = uω, t, fulfils u L Ω [, T ]; B α τ,τ O, τ = α d +, with α > ssob maxu,. where { } s Sob maxu := su s : u L Ω [, T ]; W s O..2 If so, the decay rate of the best m-term wavelet aroximation error for the solution to the considered SPDE with resect to the sace variables is higher than the convergence rates of uniform wavelet based alternatives. In this case, the attemt to develo numerical wavelet methods for SPDEs working adatively in sace direction is comletely justified. ad T2. Sace time regularity. If our analysis of the satial regularity shows that, indeed, adativity with resect to the sace coordinates ays, the next reasonable ste is to develo a sace time scheme for the athwise aroximation of the solutions to SPDEs of the tye., which works adatively in sace direction. To this end, variants of Rothe s method suggest themselves. That is, the equations is first dicretized in time. Then, since for stability reasons one has to take an imlicit scheme, in each time ste an ellitic subroblem has to be solved. To this end, otimal adative solvers of the tye mentioned above have to be emloyed. At the end, we need to estimate the overall error of such a scheme. We conjecture that our analysis of the Hölder regularity of the aths of the solution, considered as a stochastic rocess taking values in the Besov saces from the non-linear aroximation scale, can be used for estimating the overall error of satially adative variants of Rothe s method. Such an analysis has been started in [23], see also [77], but is still in its infancy..2 Overview of the relevant regularity theory In order to relate our results to the current state of research, we give a brief overview of the regularity theory which is relevant for our analysis. We begin with the significant achievements

17 .2 Overview of the relevant regularity theory 7 obtained from the analytic and from the semigrou aroach to SPDEs. Then we discuss what is known about the regularity of deterministic and stochastic equations in the non-linear aroximation scale. In contrast to the rest of this thesis, in this section, we do not assume that O R d is bounded or Lischitz. The analytic aroach of N.V. Krylov rovides a quite comlete and satisfactory L -theory 2 for semi-linear arabolic SPDEs of second order on the whole sace R d, see in articular [79, 8]. Roughly seaking, the main results concerning the satial regularity are of the form: If the free term f in Eq.. with O = R d and without the boundary condition takes values in the sace H γ R d of Bessel otentials, and the g = g k k N take values in the corresonding sace H γ+ R d ; l 2, then there exists a unique solution of this equation with values in H γ+2 R d. Thus, the saces of Bessel otentials are suitable for the regularity analysis of SPDEs on the whole sace R d. Recall that, for γ N, H γ R d coincides with W γ R d, the L -Sobolev sace of order γ, see, e.g., [84, Theorem 3.3.2]. A recise definition of the saces of Bessel otentials and their counterarts H γ R d ; l 2 for l 2 -valued functions can be found in Subsection On domains O R d with non-emty boundary O one is faced with at least two additional difficulties in order to obtain a similar theory. On the one hand, because of the behaviour of the infinitesimal differences of the driving noise, the second derivatives of the solution to Eq.. may blow u near the boundary. Then, the solution rocess fails to take values in W2 2 O. This may haen, even if the domain and the data of the equation are smooth, see, e.g., [78]. On the other hand, if the boundary of the domain is not very smooth, the singularities may become even worse caused by the influence of the shae of the boundary, see [9]. A natural way to deal with these difficulties is to consider the solution ut t [,T ] as a stochastic rocess taking values in suitable weighted Sobolev saces. These saces allow to include solutions for which the higher-order derivatives might exlode near the boundary, since this behaviour is comensated by the weight. This aroach has been initiated and develoed by Krylov and collaborators: first as an L 2 -theory for general smooth domains [78], then as an L -theory 2 for the half sace [85,86] and subsequently also for general smooth domains [72,76]. Recently, an L -theory 2 for SPDEs on more general bounded domains admitting Hardy s inequality, such as bounded Lischitz domains, has been established by K.-H. Kim in [75]. The results in those ublications are roven for linear equations of the form. with L = Λ =. The weighted Sobolev saces H γ O D O used in the theory described above are of the following form: For integer γ N and θ R, they consist of all measurable functions having finite norm u α γ O ρ O x α D α ux ρ O x θ d dx /, where ρ O x denotes the distance of a oint x O to the boundary O of the domain. For noninteger γ > they can be characterized as comlex interolation saces and for γ < the usual duality relation holds. A recise definition can be found in Subsection It turns out that this is a suitable scale to study the regularity of second-order semi-linear arabolic SPDEs on domains in the following sense: If the free terms f and g = g k k N in the equation have satial weighted Sobolev regularity γ and γ +, resectively, and the initial condition u is smooth enough, then the solution has satial weighted Sobolev regularity γ + 2 with roerly chosen weight arameters θ R on the different arts of the equation. Hence, the satial regularity of the solution in the scale H γ O, γ >, increases with the weighted Sobolev regularity of the free terms f and g of the equation. Furthermore, the weighted Sobolev norm of the solution rocess can be estimated from above by the corresonding weighted Sobolev norms of f, g and u.

18 8 Introduction Another way to analyse the regularity of solutions of Eq.. is the semigrou aroach. Develoed mainly by G. Da Prato and J. Zabczyk in a Hilbert sace framework [32], it has been generalized by Brzeźniak to M-tye 2 Banach saces [5, 6] and by J.M.A.M. van Neerven, M.C. Veraar and L. Weis to umd Banach saces [2, 22] umd stands for unconditional martingale differences. In this aroach, infinite dimensional ordinary stochastic differential equations SDEs, for short of the form } dut + AUt dt = F t, Ut dt + Σt, Ut dw H t, t [, T ],.3 U = u, are considered. The oerator A is the infinitesimal generator of a strongly continuous analytic semigrou on a suitable Banach sace E usually L O with 2, and Eq..3 is interreted as an abstract Cauchy roblem. Roughly seaking, tyical results are of the following form: If A has a good H -functional calculus in the sense of McIntosh, see Section 2.4 for details and the coefficients and non-linearities of the equations are smooth enough where the smoothness is measured in domains of fractional owers of the leading oerator, then there exists a unique strong solution in the sace L q Ω, T ; DA L q Ω; C[, T ]; E, DA q,q. Here, DA denotes the domain of the oerator A in the Banach sace E, whereas E, DA q,q is a real interolation sace. For many rominent examles the domain of the leading oerator A can be characterized in terms of well-studied function saces, so that the abstract results of [2, 22] ave the way to a owerful regularity theory for SPDEs. In contrast to the theory of Krylov and collaborators, which relies mainly on hard PDE techniques, in this aroach almost everything stands and falls with the good H -functional calculus of A. To mention an examle, the Dirichlet-Lalacian D,w on L O 2 has an H -calculus which is good enough, rovided the boundary O of the domain is sufficiently regular in general, C 2 is assumed. In this case, D D,w = W 2 O W O, where W O denotes the closure of C O in W O. Using these facts and the abstract theory from [2], one obtains an L q L -theory for the heat equation on bounded smooth domains. It is worth noting that similar results hold also for more general second order ellitic oerators, if the boundary of the domain O is smooth enough. Hence, equations of the form., which are analysed in the analytic aroach, also fit into this framework. However, we would like to mention that in the semigrou aroach certain comatibility conditions between the noise term and the leading oerator A have to be fulfilled. This makes the admissible class of noises smaller comared to those that can be treated with the analytic aroach, see, e.g., the discussion in [2, Section 7.4]. On the lus side, one obtains L q L -regularity results with different integrability arameters q and in time and sace even the case q < is ossible. With the techniques used by Krylov and collaborators, such results could not yet been roven. Also, in the semigrou framework one can treat more general second 2m-th order arabolic equations with Dirichlet and Neumann boundary conditions, stochastic Navier-Stokes equations and other imortant classes of equations see, e.g., the examles resented in [2]. In this thesis, we are exlicitly interested in domains with non-smooth boundary, in articular, we focus on general bounded Lischitz domains O R d. This covers nearly all domains of ractical interest. However, the characterization of the domain of the Dirichlet-Lalacian in terms of Sobolev saces resented above, fails to be true if the boundary of the domain O is assumed to be only Lischitz. Indeed, it has been roven in [57, 58] for olygonal and olyhedral domains, and in [67] for general bounded Lischitz domains, that W 2 2 O W 2 O D D,w. Moreover, to the best of our knowledge, in the case of general bounded Lischitz domains,

19 .3 The thesis in a nutshell 9 a characterization of D D,w in terms of function saces is not yet available. Thus, a direct alication of the results from [2] does not lead to otimal regularity results. To the best of our knowledge, so far there does not exist any analysis of the regularity of SPDEs in the non-linear aroximation scale of Besov saces excet the recent results in [22, 25, 26] by the author and collaborators, which are essential arts of this thesis. It is worth noting that a direct alication of the semigrou aroach does not immediately lead to regularity results in the scale. As already mentioned above, the semigrou framework has been used in [2] to derive regularity results in L -Sobolev and L -Besov saces 2 on sufficiently smooth domains O R d. The cornerstone for this theory is a generalization of Itô s stochastic integration theory to umd Banach saces, see Section 2.2 for details. However, for α > d /, the scale does not consist of Banach saces, but of quasi-banach saces. Thus, a direct alication of the semigrou aroach in order to obtain sufficiently high regularity in the scale requires at least! a fully-fledged theory of stochastic integration in roer classes of quasi-banach saces which is not yet available. We also want to mention that by the same reason, we can not exect direct results from the so called variational aroach for SPDEs initiated by E. Pradoux in []; we also refer to [4, Chater 4] and the literature therein for more details. This aroach has been designed articularly for the treatment of non-linear SPDEs and uses a Gelfand trile setting. In articular, the state sace of the solution rocess needs to be a reflexive Banach sace V which is continuously embedded into a Hilbert sace E. It is known that any Besov sace B α τ,τ O from the scale with = 2 is continuously embedded in the Hilbert sace L 2 O. However, as already mentioned, for α > d/2, B α τ,τ O is just a quasi-banach sace which is not reflexive. Since the reflexivity and the Banach sace roerty are essential in this framework, we can not obtain regularity results in the non-linear aroximation scale by a direct alication of the abstract results within this aroach. However, as already mentioned in Section., the non-linear aroximation scale has been already used for analysing the regularity of solutions to deterministic artial differential equations. First results on the regularity of the Dirichlet roblem for harmonic functions and of the Poisson equation on general bounded Lischitz domains in the Besov saces from have been obtained by S. Dahlke and R.A. DeVore in [38]. Several extensions followed: In [34] ellitic boundary value roblems with variable coefficients are analysed. The secial cases of olygonal and olyhedral domains have been considered in [35] and in [36], resectively. Also, equations on smooth and olyhedral cones have been considered, see [4]. Extensions to deterministic arabolic equations have been studied in [3 5]. Simultaneously, P. Grisvard shows in [57, 58] that the Sobolev regularity of solutions to ellitic and arabolic equations on non-smooth and non-convex domains is generically limited from above. Bringing those results together shows that, in general, solutions to deterministic artial differential equations on non-smooth and nonconvex domains have the behaviour described by.. Thus, in this case, the decay rate of the best m-term wavelet aroximation error is higher than the convergence rate of wavelet based uniform aroximation methods see Section. for details..3 The thesis in a nutshell Framework: the L -theory from the analytic aroach In the revious section, we exlained that the abstract results from the semigrou aroach and from the variational aroach can not be used directly to obtain regularity results for SPDEs in the non-linear aroximation scale. Therefore, we take an indirect way to rove regularity in of the solutions to SPDEs of the form.. Our analysis takes lace in the framework

20 Introduction of the analytic aroach. We borrow and exand the L -theory for linear SPDEs from [75], which gives us the existence and uniqueness of a solution to Eq.. on general bounded Lischitz domains O R d. Then, we analyse the satial Besov regularity, that is toic T, and the Hölder regularity of the aths, that is toic T2, of this solution. We start by roving a fundamental embedding of weighted Sobolev saces into Besov saces from the non-linear aroximation scale. Embeddings of weighted Sobolev saces into Besov saces The solutions to the linear SPDEs considered in [75] are elements of secial classes H γ O, T, consisting of certain redictable -Bochner integrable H γ O-valued stochastic rocesses. In articular, H γ O, T L Ω [, T ]; H γ O..4 means continuously linearly embedded. Hence, one way to extract regularity results in the non-linear aroximation scale from this theory, is to rove an embedding of weighted Sobolev saces into Besov saces from. This idea is underinned by the fact that, in the deterministic setting, weighted Sobolev estimates have been used to establish Besov regularity in the scale for the solutions of ellitic boundary value roblems, such as the Dirichlet roblem for harmonic functions and the Poisson equation, see, e.g., [38]. This has been erformed by estimating the wavelet coefficients of the unknown solution by means of weighted Sobolev semi-norms. Then, by using the equivalences of Besov norms and weighted sequence norms of wavelet coefficients, the desired Besov estimates were established. Using similar techniques, we can rove that for arbitrary bounded Lischitz domains O R d and arameters [2, and γ, ν,, H γ,d ν O Bα τ,τ O, τ = α d + {, for all < α < min γ, ν d d },.5 see Theorem 4.7. Our roof for integer γ N follows the line of the roof of [38, Theorem 3.2]. Additionally we use and rove the following embedding of weighted Sobolev saces into Sobolev saces without weights: H γ γ ν,d ν O W O,.6 which holds under the same requirements on the arameters and the shae of the domain Proosition 4.. By using comlex interolation we are able to rove Embedding.5 for arbitrary γ > Theorem 4.7. It is worth noting that this generalization has been roven in [26, Theorem 6.9] by the author and collaborators in a different more direct way without using interolation methods. The imact of.5 is obvious: U to a certain amount, the analysis of the satial regularity of SPDEs in the scale can be traced back to the analysis of the weighted Sobolev regularity of the solutions. In other words, every result on the weighted Sobolev regularity of SPDEs automatically encodes a statement about the Besov regularity in the scale. T Satial regularity in the non-linear aroximation scale As mentioned above, in this thesis, the solutions to SPDEs of the form. are elements of the classes H γ O, T with [2,, γ, θ R. Since θ = d + d θ,

21 .3 The thesis in a nutshell combining the embeddings.4 and.5 shows that H γ O, T L Ω [, T ]; Bτ,τ α O, τ = α d +, for all < α < γ + d θ d d..7 In Chater 5 we use this embedding to rove satial Besov regularity in the scale for linear and semi-linear SPDEs on general bounded Lischitz domains O R d. Linear equations The L -theory develoed in [75] rovides existence and uniqueness of solutions u H γ O, T, [2,, γ, θ R, for a wide class of linear second order stochastic arabolic differential equations of the form. with vanishing L and Λ. Alying Embedding.7 roves that u L Ω [, T ]; B α τ,τ O, see Theorem 5.2. Hence, we have found an { α := min γ, τ = α d +, for all < α < γ + d θ } d >, d + d θ d d,.8 such that for all < α < α and /τ = α/d+/, the solution u to the linear SPDEs as discussed in [75] is contained in the sace of equivalence classes of -integrable B α τ,τ O-valued stochastic rocesses. The recise conditions on the weight arameter θ R, for which.8 holds, can be found in the statement of our main result, Theorem 5.2. For examle, in the two-dimensional case, we can choose = 2, γ = 2 and θ = d = 2, which yields u L 2 Ω [, T ]; B α τ,τ O, τ = α 2 +, for all < α < 2. 2 Our result together with the analysis of the maximal Sobolev regularity of SPDEs in [92] shows that, in general, on bounded Lischitz domains O R d which are non-convex at the singularities of O, the solutions to the linear SPDEs considered in [75] behave as described in.. By our exosition in Section., this is a clear theoretical justification for the design of satially adative wavelet schemes for linear SPDEs. For the detailed analysis and several examles we refer to Section 5.. Semi-linear equations Many hysical or chemical systems are described by equations, which are rather non-linear. Thus, it is an immediate question whether the results resented above can be extended to nonlinear SPDEs. As a first ste in this direction we consider semi-linear equations. That is, we consider equations of the tye. with Lischitz continuous non-linearities L and Λ. As before, we use Embedding.7 to rove satial Besov regularity in the scale. Since there is no L -theory for semi-linear SPDEs on bounded Lischitz domains, we first rove existence and uniqueness of solutions in the classes H γ O, T, see Theorem 5.3. We assume that the non-linearities L and Λ in Eq.. fulfil suitable Lischitz conditions Assumtion 5.9, such that our equation can be interreted as a disturbed linear equation. Then, by using fixed oint arguments, see Lemma 5.6, we obtain existence and uniqueness of a solution u H γ O, T to Eq.., which by.7 automatically fulfils.8. In this way, satial regularity in the non-linear aroximation scale can be established also for semi-linear SPDEs, see our main result in Theorem 5.5.

22 2 Introduction T2 Sace time regularity H γ After we have roven that the solutions u H γ O, T to linear and semi-linear SPDEs of the form. can be considered as a Bτ,τ α O-valued stochastic rocesses for < α < α, where /τ = α/d + /, we can move on to the second main toic in this thesis: The analysis of the Hölder regularity of the aths of this rocess, which will be resented in Chater 6. The L -theory develoed in [75] already rovides Hölder estimates for elements of the classes O, T, considered as stochastic rocesses with values in weighted Sobolev saces. In articular, it has been shown therein that for u H γ O, T and 2/ < β < β, u C β/2 / [,T ];H γ β < P-a.s.,.9 O β where for any quasi-banach sace E, E, C κ [, T ]; E, C κ [,T ];E denotes the sace of κ-hölder continuous E-valued functions on [, T ], see Subsection 2..4 for a recise definition. An immediate idea is to use the embedding.5 and obtain Hölder regularity for the aths of the solutions u H γ O, T considered as stochastic rocesses taking values in the Besov saces from the scale. However, since the Hölder regularity in.9 deends on the summability arameter used to measure the regularity with resect to the sace variables and because of the restrictions on the weight arameters θ R needed in [75] to establish existence of solutions in the classes H γ O, T, this does not yield satisfactory results we refer to the introduction of Chater 6 for more details. We overcome these difficulties by using the following strategy. Instead of H γ O, T, we consider their counterarts H γ,q O, T, which consist of certain q-integrable Hγ O-valued stochastic rocesses, where the integrability arameter q in time direction and with resect to ω Ω is exlicitly allowed to be greater than the summability arameter used to measure the smoothness with resect to the sace variables. We first rove that for u H γ,q O, T with 2 q <, γ N and 2/q < β < β, u C β/2 /q [,T ];H γ β β O < P-a.s., see Theorem 6.. In articular, the Hölder regularity of the aths does not deend on the summability arameter with resect to the sace variables. Therefore, even if the restrictions from [75] on the weight arameter θ have to be imosed, satisfactory Hölder estimates for the aths of elements u H γ,q O, T, considered as stochastic rocesses with state saces from the scale, are ossible Theorem 6.2. However, if we want to aly these results in order to obtain imroved sace time regularity of the solutions to SPDEs, we have to rove that under suitable assumtions on the data of the considered equation the solution lies in H γ,q O, T where q and are exlicitly allowed to differ. In other words, we need to extend the L -theory from [75] to an L q L -theory for SPDEs with q. In this thesis we rove a first L q L -thoery result for the stochastic heat equation on general bounded Lischitz domains Theorem 6.. Our roofs rely on a combination of the semigrou aroach and the analytic aroach. From the semigrou aroach, we obtain the existence of a solution with low weighted Sobolev regularity Proosition 6.2. Using techniques from the analytic aroach we can lift this regularity, if we can increase the regularity of the free terms Theorem 6.7. At this oint, when merging results from the two different different aroaches, we will need the isomorhy between the saces H γ O; l 2, which are central within the analytic aroach, and the corresonding saces Γl 2, H γ O of γ-radonifying oerators from l 2 to H γ O. This will be roven in Subsection 2.3.3, see Theorem Finally, we can bring those results together roving Hölder regularity of the aths of the solution u H γ,q O, T to the stochastic heat equation, considered as a stochastic rocess

23 .4 Outline 3 with values in the Besov saces from the scale. In articular, we rove that, under suitable assumtions on the data of the equation, u C β/2 /q [,T ];B α τ,τ O < P-a.s., where 2 q < β <, τ = α d +, and < α < β d d. For the recise formulation of our main result on sace time regularity, which includes also estimates of the Hölder-Besov norm of the solution by the weighted Sobolev norms of the free terms, we refer to Theorem Outline This thesis starts with some reliminaries Chater 2. First we fix some notational and concetual conventions in Section 2.. Then, in Section 2.2, we give a brief inside into the theory of stochastic integration in umd Banach saces as develoed recently in [2]. In this context, we also discuss some geometric roerties of Banach saces, like tye and umd roerty, and the class of γ-radonifying oerators. Afterwards, in Section 2.3, we introduce and discuss some roerties of relevant function saces, ointing out several known relationshis between them. In articular, in Subsection 2.3.3, we focus on the weighted Sobolev saces H γ G and their counterarts H γ G; l 2 for l 2 -valued functions, which lay an imortant role within the analytic aroach G R d is an arbitrary domain with non-emty boundary. Section 2.4 deals with semigrous of linear oerators. We mainly focus on analytic semigrous and on the notion of H -calculus, which is relevant within the semigrou aroach for SPDEs. We also consider the class of variational oerators. Chater 3 is concerned with the L -theory for linear SPDEs in weighted Sobolev saces, recently develoed in [75] within the analytic aroach. The analysis therein takes lace in the stochastic arabolic weighted Sobolev saces H γ G, T, [2,, γ, θ R. In Section 3. we introduce and discuss some roerties of these saces and of their generalizations H γ,q G, T, q [2,. We also fix some other notation, which is common within the analytic aroach. Afterwards, in Section 3.2, we resent the main results from the aforementioned L -theory. We restrict ourselves to the case of bounded Lischitz domains. The solution concet borrowed from [75] is introduced in Definition 3. and it is related to the concet of weak solutions, as it is used within the semigrou aroach, in Proosition 3.8. In Chater 4 we leave the SPDE framework for a moment and rove Embedding.5 of weighted Sobolev saces on bounded Lischitz domains O R d into Besov saces from the non-linear aroximation scale, see Theorem 4.7. We also rove Embedding.6, see Proosition 4.. From the latter, we can conclude that the elements of weighted Sobolev saces are zero at the boundary in a well-defined sense, see Corollary 4.2 and Remark 4.3 for details. Chater 5 is devoted to the satial regularity of SPDEs in the scale of Besov saces, i.e., toic T. In Section 5., we state and rove our main result concerning linear equations, Theorem 5.2. We also resent several examles and discuss the results from the oint of view of aroximation theory and numerical analysis. In the subsequent Section 5.2 we consider semilinear equations. We first rove the existence of solutions in the classes H γ O, T, [2,, γ, θ R, under suitable assumtions on the non-linearities, see Theorem 5.3. Then, we rove our main result concerning the satial regularity of semi-linear SPDEs in the scale, see Theorem 5.5. The final Chater 6 is concerned with the sace time regularity of the solution to the stochastic heat equation on bounded Lischitz domains, i.e., with toic T2. In Section 6. we analyse

24 4 Introduction the Hölder regularity of the aths of elements from H γ,q O, T : first, considered as stochastic rocesses taking values in weighted Sobolev saces Theorem 6., and, subsequently, considered as stochastic rocesses with state saces from the non-linear aroximation scale Theorem 6.2. We are articularly interested in the case q. Afterwards, in Section 6.2 we show that the saces H γ,q O, T with q 2 are suitable for the analysis of SPDEs in the following sense: If we have a solution u H γ,q O, T with low regularity γ, but the free terms f and g have high L q L -regularity, then we can lift u the regularity of the solution Theorem 6.7. Finally, in Section 6.3 we rove the existence and uniqueness of a solution in the class H γ,q O, T to the stochastic heat equation Theorem 6.. Combined with the results mentioned above, this yields our main result on the sace time regularity of the stochastic heat equation, Theorem 6.7. A short German summary of this thesis starts on age 3. A list of notation can be found starting on age 37 and an index begins on age 5.

25 5 Chater 2 Preliminaries In this chater we resent definitions and results needed later on for our analysis. In the first section we fix some concetual and notational conventions from different mathematical areas. Afterwards, we give a brief inside into the theory of stochastic integration in Banach saces develoed mainly in [2] Section 2.2. In this context we will also discuss some geometric Banach sace roerties and the class of γ-radonifying oerators. In Section 2.3 we will introduce the function saces aearing in this thesis and discuss and rove some of their roerties which are relevant for the subsequent analysis. Finally, Section 2.4 is devoted to analytic semigrous and the concet of H -calculus, and to variational oerators. 2. Some conventions In order to guarantee concetual clarity, in this section we summarize the conventions made in this thesis. We give a fast overview of the notation and the basic concets we will use later on. We start with classes of bounded oerators. Then, we consider domains in R d and resent the definitions of different classes of domains. In articular, we substantiate the notion of a bounded Lischitz domain, which is central in this thesis. Afterwards, we recall the basics from quasi- Banach sace valued measure and integration theory. We continue with different asects from robability theory and the underlying robabilistic setting. Then, we strike the subject of real and comlex valued functions and distributions. At this oint, we want to emhasize that in this thesis, unless exlicitly stated otherwise, functions and distributions are meant to be real-valued. Finally, we resent some miscellaneous notation. In the course of this thesis, the reader is invited to use the list of notations on age 37 and the index on age 5 and come back to this section whenever more concetual clarity is needed. 2.. Bounded oerators Let E, E and F, F be two real normed saces. We write LE, F for the sace of all linear and bounded oerators from E to F, endowed with the classical norm R LE,F := su Rx F, R LE, F. x E, x E If F = E we use the common abbreviation LE := LE, E. E := LE, R denotes the dual sace of E. We will use the notation x, x := x, x E E := x x, x E, x E,

26 6 Preliminaries for the dual airing. The adjoint R determined by LF, E of an oerator R LE, F is uniquely R y, x E E = y, Rx F F, y F, x E. If H,, H is a real Hilbert sace, we usually identify H and H via the Riesz isometric isomorhism H h h, H H with h, H : H R g h, g H. If H is imlicitly given by the context, we write, :=, H for short. Assume that U,, U is a further real searable Hilbert sace. Following [32] and [4] we write L H, U for the sace of nuclear oerators and L 2 H, U for the sace of Hilbert-Schmidt oerators from H to U, see also 2.. We will also use the common abbreviations L H and L 2 H, resectively, if U = H. Recall that a quasi-normed sace E, E is a vector sace E endowed with a ma E : E [,, which is ositive definite and homogeneous as a norm but fails to fulfil the triangle inequality. Instead, there exists a constant C, which is allowed to be greater than one, such that x + y E C x E + y E, x, y E. Such a ma is called a quasi-norm. A quasi-banach sace is a quasi-normed sace which is comlete with resect to the quasi-metric dx, y := x y E, x, y E. We will use the notations from above also in the case of quasi-normed saces, whenever it makes sense Domains in R d Throughout this thesis, G will denote an arbitrary domain in R d, i.e., an oen and connected subset of the d-dimensional Euclidian sace R d d 2. If G has a non-emty boundary, we will denote it by G. In this case, we will write ρx := ρ G x := distx, G for the distance of a oint x G to the boundary G. Furthermore, in this thesis, O will always denote a bounded Lischitz domain in R d. Let us be more recise. Definition 2.. We call a bounded domain O R d a Lischitz domain if, and only if, for any x = x, x O, there exists a Lischitz continuous function µ : R d R such that, uon relabelling and reorienting the coordinate axes if necessary, we have i O B r x = {x = x, x B r x : x > µ x }, and ii µ x µ y K x y, for any x, y R d, where r, K are indeendent of x. Some results will be also formulated for domains of the following class. Definition 2.2. Let G be a domain in R d with non-emty boundary G. We say that G satisfies the outer ball condition if for each x G, there exists an r = rx > and a oint x = x x R d, such that B rx x R d \ G and x B rx x. 2. G satisfies a uniform outer ball condition if there exists an R >, such that for all x G, rx = R can be chosen in 2..

27 2. Some conventions 7 We will sometimes comare our results for SPDEs on bounded Lischitz domains with results which can be roven for bounded domains of class Cu. Equations on domains of this class have been analysed by N.V. Krylov and collaborators, see, e.g., [72,76]. We recall the definition given in [72, Assumtion 2.]. It is worth noting that the conditions imosed on the diffeomorhism Ψ therein and its inverse Ψ are not symmetric. We fix a function κ defined on [, such that κ ε for ε. Furthermore, Ψ i denotes the classical artial derivative of the i-th x j coordinate of a function Ψ : G R d R d with resect to the j-th variable x j, i, j {,..., d}. Definition 2.3. We call a domain G R d of class C u or simly a C u-domain, if there exist constants r, K > such that for any x G, there exists a one-to-one continuously differentiable Ψ from B r x onto a domain J R d such that i J + := ΨB r x G R d + := {y = y, y R d : y > } and Ψx = ; ii ΨB r x G = J {y R d : y = }; iii su x Br x Ψx + d i,j= Ψ i x K x j and Ψ y Ψ y 2 K y y 2 for any y, y 2 J; iv d i,j= Ψ i x x j Ψ i x x j 2 κ x x for any x, x 2 B r x Measurable maings and L -saces Let M, A, µ be a σ-finite measure sace and let E, E be a Banach sace. We call a function u : M E A-simle, if it has the form u = K k= A k x k with A k A and x k E for k K <. A function u : M E is called strongly A-measurable, if there exists a sequence f n n N of A-simle functions aroximating f ointwise in M. It is well-known that, if E is searable, a function u : M E is strongly A-measurable if, and only if, it is A/BE- measurable in the classical sense, i.e., if u B A for all B BE, where BE denotes the Borel σ-field on E. In this case, we also say u is A-measurable for short. Two strongly A- measurable functions which agree µ-almost everywhere on M are said to be µ-versions or simly versions of each other. For,, L M, A, µ; E denotes the sace of all µ-equivalence classes of strongly A-measurable functions u : M E such that u LM,A,µ;E := u E dµ <, 2.2 M the integral being understood as a Lebesgue integral see, e.g., []. As usual, we follow the convention that a µ-equivalence class [u] L M, A, µ; E contains all functions u : M\M E defined on M excet a µ-null set M A, µm =, such that u M\M [u]. We simly write u instead of the [u] L M, A, µ; E. We will sometimes use the common abbreviations L M; E and L M if E = R. u L M; E will be called -Bochner integrable or simly -integrable. A function u L M; E is also called Bochner integrable or simly integrable. In this case, u dµ = ux µdx = ux dµx E M M is well-defined as a Bochner integral, see, e.g., [8, Chater ] for details. L M, A, µ; E sometimes L M; E, for short denotes the Banach sace of all µ-equivalence classes of strongly A-measurable functions u : M E for which there exists a finite number r such that µ {x M : ux E > r} =. It is endowed with the norm M u L M;E := inf { r : µ {x M : ux E > r} = }, u L M; E.

28 8 Preliminaries For a countable set I, we write l I := L I, PI, i I δ i; R, where PI denotes the ower set of I and δ i is the common notation for the Dirac measure at the oint i I. For a l I we write a i := ai for the i-th coordinate. The function a, b l2 I := i I a i b i, a, b l 2 I, defines a scalar roduct on the Hilbert sace l 2 I and l2 I :=, l2 I is the corresonding norm. If I = N := {, 2, 3,...} we write l 2 := l 2 N, and denote by {e k : k N} the standard orthonormal basis of l 2, i.e., for i, k N, e i k =, if i k and ek k =. By λ d we denote the Lebesgue measure on BR d and its restriction to B BR d. When integrating with resect to λ d we will often write shorthand dx instead of λ d dx. If a measure µ has density g with resect to the Lebesgue measure λ d, we write µ = gλ d and gdx instead of gλ d dx. Moreover, we write L instead of L R d, BR d, λ d ; R. For f, g : G R, we will use the notation f, g := fg dx 2.3 whenever fg L G, BG, λ d ; R. We say a function f : G R is locally integrable in G, if it is BG/BR-measurable and fx dx <, K for every comact subset K of G. If E, E is just a quasi-banach sace and, we use the analogous notation L M, A, µ; E and the corresonding abbreviations to denote the set of all µ-equivalence classes of strongly A-measurable E-valued functions fulfilling Probabilistic setting Throughout this thesis Ω, F, P will denote a robability sace. Random variables A strongly F-measurable maing u from Ω into a quasi-banach sace E will be called E-valued random variable. If this function is F-simle, we will call it an F-simle random variable. If E is a Banach sace and u L Ω; E for some [,, we write E[u] for its exectation, i.e., E[u] := Ω u dp. If E[u] = we call the random variable centred. A random variable u : Ω E is called Gaussian if x, u is a real-valued Gaussian random variable for any x E. The ositive definite and symmetric oerator Q LE, E defined via E x Qx := E [ x, u E[u] u E[u] ] E is called covariance oerator of the Gaussian random variable u. Stochastic rocesses Let E be a quasi-banach sace and J an arbitrary set. A stochastic rocess u = uj j J on Ω, F, P with index set J is a maing u : Ω J E such that for any j J, the maing uj = u, j : Ω E is strongly F-measurable. We will sometimes use the notation u j j J instead of uj j J. For any ω Ω, the ma J j uω, E is called ath or trajectory of the rocess u. Two stochastic rocesses uj j J and vj j J on a common robability sace G

29 2. Some conventions 9 Ω, F, P are modifications of each other, if P{ω Ω : uω, j = vω, j} = for each j J. If P{ω Ω : uω, j = vω, j for all j J} =, the two rocesses are called indistinguishable. Assume that the index set J, is artially ordered e.g., J = [, T ] with T > or J is a searable subset of R. A filtration F j j J on Ω, F, P is an increasing family of sub-σ-fields of F. An E-valued stochastic rocess u = uj j J is called adated to the filtration F j j J F j j J -adated, for short, if for any j J the random variable uj is strongly F j -measurable. Let E be a Banach sace and J, be artially ordered. An F j j J -adated stochastic rocess u : Ω [, T ] E is called martingale with resect to F j j J if u j L Ω; E for any j J, and for any i, j J with i j, Eu j F i = u i P-a.s., where Eu j F i denotes the conditional exectation of u j with resect to F i. If, furthermore, for some [,, u j L Ω; E for all j J, the martingale u j j J is called an L -martingale. Regularity of aths Let ut t [,T ] be a stochastic rocess with index set J := [, T ] taking values in a quasi-banach sace E. We will measure the smoothness of the aths of u by means of their Hölder regularity. For κ, and a quasi-banach sace E, E we denote by C κ [, T ]; E the Hölder sace of continuous E-valued functions on [, T ] with finite norm C κ [,T ];E defined by [u] C κ [,T ];E := ut us E su s,t [,T ] t s κ, s t u C[,T ];E := su ut E, t [,T ] Oerator valued stochastic rocesses u C κ [,T ];E := u C[,T ];E + [u] C κ [,T ];E. Let E, E 2 be two Banach saces. An oerator valued function Φ : Ω [, T ] LE, E 2 is called an E -strongly measurable stochastic rocess if for any x E, the E 2 valued stochastic rocess Φx : Ω [, T ] E 2, ω, t Φxω, t := Φω, tx is strongly F B[, T ]-measurable. An E -strongly measurable stochastic rocess is called adated to a filtration F t t [,T ] on Ω, F, P or, simly F t t [,T ] -adated if for any x E, the rocess Φx is adated to F t t [,T ]. Let H,, H be a Hilbert sace and let E, E be a Banach sace. Let Φ : Ω [, T ] LH, E be an H-strongly measurable stochastic rocess. We write Φ : Ω [, T ] LE, H ω, t Φ ω, t := Φω, t, identifying H and its dual H via the Riesz isomorhism. Φ is said to belong to L 2 [, T ]; H scalarly almost surely if for all x E, Φ ω, x L 2 [, T ]; H for P-almost all ω Ω. Note that the excetional set may deend on x. For [2,, Φ is said to belong to L Ω; L 2 [, T ]; H scalarly if for all x E, Φ x L Ω; L 2 [, T ]; H.

30 2 Preliminaries A stochastic rocess Φ : Ω [, T ] LH, E which belongs to L 2 [, T ]; H scalarly almost surely is said to reresent a random variable X : Ω LL 2 [, T ]; H, E, if for all f L 2 [, T ]; H and x E we have x, Xωf E E = T Stochastic integration in Hilbert saces Φ ω, tx, ft H dt for P-almost all ω Ω. We assume that the reader is familiar with the issue of stochastic integration in the sense of Itô with resect to cylindrical Q-Wiener rocesses in the Hilbert sace setting, as described, e.g., in [32] or [4]. Let H,, H and U,, U be two real Hilbert saces. Furthermore, assume that W Q t t [,T ] is an H-valued Q-Wiener rocess with Q L H. We write t us dw Q s t [,T ] for the stochastic Itô integral of a rocess u : Ω T L 2 H, U which is stochastically integrable with resect to W Q. Here, H,, H := Q /2 H, Q /2, Q /2 H is the reroducing kernel Hilbert sace, Q /2 being the seudo-inverse of Q /2. In articular, if a real valued stochastic rocess g t t [,T ] is stochastically integrable with resect to a real valued Brownian motion w t t [,T ], we write t g s dw s t [,T ] for the stochastic integral rocess. A brief overview of the extension of this theory to certain classes of Banach saces will be given later on in Section 2.2. Miscellaneous conventions on the robabilistic setting In this thesis, T > will always denote a finite time horizon and w k t t [,T ], k N, will be a sequence of indeendent real-valued standard Brownian motions with resect to a normal filtration F t t [,T ] on a comlete robability sace Ω, F, P. By normal we mean that F t t [,T ] is right continuous and that F contains all P-null sets. We will use the common abbreviation Ω T := Ω [, T ] as well as P T := σ { F s s, t] : s < t T, F s F s } { F {} : F F } F B[, T ] for the F t t [,T ] -redictable σ-field. Furthermore, we will write P T for the roduct measure P dt on F B[, T ] and for its restriction to P T. The abbreviation L Ω T ; E := L Ω T, P T, P T ; E,, ], will often be used in this thesis to denote the set of redictable -integrable stochastic rocesses with values in a quasi-banach sace E Functions, distributions and the Fourier transform In this thesis, unless exlicitly stated otherwise, functions and distributions are meant to be real valued. For a domain G R d and r N, C r G denotes the sace of all r-times continuously differentiable functions, whereas CG is the sace of continuous functions. We will write C G for the set of test functions, i.e., the collection of all infinitely differentiable functions with

31 2. Some conventions 2 comact suort in G. For bounded domains G R d, the notation C r G is used for the set of all functions which are continuous on the closure G of G and ossess derivatives u to and including the order r N, which are continuous on G and can be extended by continuity to G. CG stands for the sace of continuous functions on G. For a multi-index α = α,..., α d N d with α := α α d = r and an r-times differentiable function u : G R, we write D α u = α x α x d αd u for the corresonding classical artial derivative. SR d denotes the Schwartz sace of raidly decreasing functions, see e.g. [8, Section 7.3]. The set of distributions on G will be denoted by D G, whereas S R d denotes the set of temered distributions on R d. The terms distribution and generalized function will be used synonymously. For the alication of a distribution u D G to a test function ϕ C G we write u, ϕ. The same notation will be used if u S R d and ϕ SR d. Let u D G. If there exists a locally integrable function f : G R such that u, ϕ = f, ϕ = fxϕx dx, for all ϕ C G, G we say that the distribution u is regular. Since such an f is uniquely determined, we do not distinguish between u and f. For u D G and a multi-index α = α,..., α d N d, we write D α u for the α-th generalized, weak or distributional derivative of u with resect to x = x,..., x d G, i.e., D α u is a distribution on G, uniquely determined by the formula D α u, ϕ := α u, D α ϕ, ϕ C G, see e.g. [8, Section 6.2]; D := Id. By making slight abuse of notation, for m N, we write D m u for any generalized m-th order derivative of u and for the vector of all m-th order derivatives of u. E.g. if we write D m u E, where E is a function sace on G, we mean D α u E for all α N d with α = m. We also use the notation u x i := De i u and u x i x j := De i D e j u, where for i {,..., d}, e i denotes the i-th unit vector in R d, i.e., e i i = and ek i = for i k. The notation u x resectively u xx is used synonymously for Du := D u resectively for D 2 u, whereas u x E := i u x i E resectively u xx E := i,j u x i x j E. We write u := d i= u x i x i whenever it makes sense. If we consider saces of comlex-valued functions and distributions we will indicate this exlicitly by writing e.g. S R d ; C or D G; C. The notation, is generalized to u, ϕ := uxϕx dx R d on S R d ; C SR d ; C. Here, ϕx denotes the comlex conjugate of ϕx, x R d. The analogous meaning is given to, on D G; C DG; C. In thesis, we denote by F : S R d ; C S R d ; C the Fourier transform on the sace of comlex valued temered distributions S R d ; C. For u S R d ; C, it is defined by Fu, ϕ := u, Fϕ, ϕ SR d ; C,

32 22 Preliminaries where Fϕ := 2π d/2 ϕxe i x,ξ dξ, R d ϕ SR d ; C. Its inverses is denoted by F. For more details on the Fourier transform of temered distributions we refer to [8, Chater 7] Miscellaneous notation For two quasi-normed saces E i, Ei, i =, 2, E E 2 means that E is continuously linearly embedded in E 2, i.e., there exists a linear continuous embedding j : E E 2. If we j want to secify the embedding, we write E E 2. If E and E 2 are normed and there exists a toological isomorhism between E and E 2, i.e., a bijective linear continuous maing from E to E 2 with bounded inverse, we write E E 2. The two saces are then called isomorhic. The notation E = E2 is used if there exists an isometric isomorhism between E and E 2, i.e., if there exists a norm reserving isomorhism from E to E 2. The sace E E 2 := {x, x 2 : x E, x 2 E 2 }, is endowed with the norm x, x 2 E E 2 := x E + x 2 E2, x, x 2 E E 2. For a comatible coule E, E 2 of Banach saces, [E, E 2 ] η denotes the interolation sace of exonent η, arising from the comlex interolation method, see, e.g., [3, Chater 4]. Furthermore, the intersection E E 2 is endowed with the norm x E E 2 := max { x E, x E2 }, x E E 2. Throughout this thesis, C denotes a ositive and finite constant which may change its value with every new aearance. If we have two terms deending on a arameter u, which might be a distribution or a distribution valued rocess or something else, and write f u C f 2 u we mean: There exists a constant C,, which does not deend on u, such that, if f 2 u makes sense and is finite, so does f u, and the inequality holds. We will also write f u f 2 u if simultaneously f u C f 2 u and f 2 u C f u. 2.2 Stochastic integration in UMD Banach saces The redominant art of this thesis is based on the generalization of Itô s stochastic integration theory for oerator valued stochastic rocesses with resect to cylindrical Wiener rocesses, where the oerators are maings from a real searable Hilbert sace into another. Even more, in the most arts we use just results from finite-dimensional stochastic analysis. However, in Chater 6, we will use regularity results in L -Sobolev saces 2 as derived by van Neerven, Veraar and Weis [2]. The cornerstone for this theory is a generalization of Itô s stochastic integration theory to certain classes of Banach saces, see [2] as well as [23] and [7]. It is the aim of this section to give a brief overview of this theory and to resent basic results which we will use later on.

33 2.2 Stochastic integration in UMD Banach saces Geometric roerties of Banach saces The generalization of Itô s stochastic integration theory does not work for arbitrary Banach saces. What we need are certain geometric roerties of these saces, which we resent now. Throughout this subsection, Ω, F, P denotes a robability sace and E, E is an arbitrary Banach sace. For further notation commonly used in robability theory, we refer to Subsection The first geometric roerty we introduce is the so-called umd roerty. Before we define this notion, we need to recall what is said to be a martingale difference sequence. Our definitions are taken from [8]. Definition 2.4. A sequence d k k N of E-valued Bochner integrable random variables is a martingale difference sequence relative to a filtration F k k N if for every k N, the random variable d k : Ω R is F k -measurable and Ed k+ F k =. 2.4 If, additionally, d k k N L Ω; E for some [,, then we call d k k N an L -martingale difference sequence. Definition 2.5. A Banach sace E, E is said to be a umd sace or to satisfy the umd roerty if for some equivalently, for all, there exists a constant C, deending only on and E, such that for any L -martingale difference sequence d k k N and any finite {, }-valued sequence ε k K k=, one has: [ K E ε k d k k= E ] [ K C E d k k= E ]. The abbreviation umd stands for unconditional martingale differences. The following wellknown facts about umd saces will be useful later on. Their roofs, as well as a roof of the -indeendence of the umd roerty, can be found in [8, Chater 2], see also the references therein. Lemma 2.6. i Banach saces isomorhic to closed subsaces of umd saces satisfy the umd roerty. ii Every Hilbert sace is a umd sace. iii Let E be a umd sace and M, A, µ be a σ-finite measure sace. Then L M, A, µ; E is a umd sace for any,. iv E is a umd sace if, and only if, the dual E is a umd sace. The second geometric roerty of Banach saces required later on is the so-called tye of a Banach sace. Before we resent its definition, we need to recall what is said to be a Rademacher sequence. Definition 2.7. A Rademacher sequence is a sequence {r k : k N} of indeendent and identically distributed {, }-valued random variables with Pr = = Pr = = /2. In what follows, {r k : k N} will always denote a Rademacher sequence.

34 24 Preliminaries Definition 2.8. Fix [, 2]. A Banach sace E, E is said to have tye, if there exists a constant C such that for all finite sequences x k K k= E the following estimate holds: [ K ] K E r k x k C x k E. 2.5 k= E Obviously, every Banach sace E has tye. If the tye of E is strictly greater than one, we say E has non-trivial tye. In the following lemma we collect some useful and well-known facts regarding the tye of Banach saces, see, e.g., [3] and the references therein. Lemma 2.9. i If a Banach sace E has tye for some [, 2], then E has tye for all [, ]. ii Every Hilbert sace has tye 2. iii Let M, A, µ be a σ-finite measure sace and let [,. Then, L M, A, µ; R has tye r := min{2, }. iv Let E and E 2 be isomorhic Banach saces and let [, 2]. Then, E has tye if, and only if, E 2 has tye. v Let E be a umd Banach sace. Then, the following assertions are equivalent: E has tye 2. 2 E has M-tye 2, i.e., there exists a constant C such that for every E-valued L 2 - martingale M k k N the following inequality holds: [ ] su E M k 2 C E k N with the usual convention M :=. k= [ E M k M k 2 The term on the left hand side of 2.5 deends on only u to a constant. This is due to the Kahane-Khintchine inequality, which is the content of the next lemma. A roof based on Lévy s inequality can be found e.g. in [8, Theorem 3.]. Lemma 2.. For all, q [,, there exists a constant C,q, deending only on and q, such that for all finite sequences x k K k= E we have [ K ] [ E r k x k C,q E K ] q q r k x k. k= E This result can be extended to the case where the Rademacher sequence is relaced by a Gaussian sequence. Let us first recall what we mean by that. Definition 2.. A Gaussian sequence is a sequence {γ k : k N} of indeendent real valued random variables, each of which is standard Gaussian. In what follows, {γ k : k N} will always denote a Gaussian sequence. A roof of the following generalization of Lemma 2., which is based on the central limit theorem, can be found in [8, Theorem 3.2]. We will refer to it as the Kahane-Khintchine inequality for Gaussian sums. Lemma 2.2. For all, q [,, there exists a constant C,q, deending only on and q, such that for all finite sequences x k K k= E we have [ K ] [ E γ k x k C,q E K ] q q γ k x k. k= E k= k= k= E E E ],

35 2.2 Stochastic integration in UMD Banach saces γ-radonifying oerators In this subsection we discuss the notion of γ-radonifying oerators from a real Hilbert sace H,, H into a Banach sace E, E. These oerators usually aear in the context of Banach sace valued Gaussian random variables. It is a well-known result that an oerator Q LE, E is the covariance oerator of a centred E-valued Gaussian random variable if, and only if, Q = RR with R being a γ-radonifying oerator from a Hilbert sace H into E, see e.g. [8, Theorem 5.2]. This class of oerators is also central in the develoment of the notion of stochastic integration in umd Banach saces with resect to H-cylindrical Brownian motions of van Neerven, Veraar and Weis [2], which we will discuss in the next subsection. Our exosition follows the lines of [8, Chater 5], see also the survey [9]. The notation used in the subsection before is still valid. Remember that {γ k : k N} is a Gaussian sequence, see Definition 2.. Definition 2.3. A linear oerator R : H E is called γ-summing if [ K 2 R Γ H,E := su E γ k Rh k k= E ] 2 <, 2.6 where the suremum is taken over all finite orthonormal systems {h,..., h K } H. The sace of γ-summing oerators from H to E will be denoted by Γ H, E and endowed with the norm Γ H,E introduced in 2.6. Remark 2.4. i Γ H, E LH, E, since for any R Γ H, E, R LH,E = su Rh E = h H = [ su E γ Rh ] 2 2 E h H = R Γ H,E. ii Let [,. Let Γ H, E be the set of all linear oerators R : H E such that R E[ Γ H,E := su K γ k Rh k k= E ] <, where the suremum is taken over all finite orthonormal systems {h,..., h K } H. Then, by the Kahane-Khintchine inequality for Gaussian sums, see Lemma 2.2, Γ H, E = Γ H, E. Moreover, the norms Γ H,E and Γ H,E are equivalent. iii Γ H, E, Γ H,E is a Banach sace, see, e.g., [8, Theorem 5.2]. For h H and x E, we sometimes write h x := h, H x LH, E for the rank one oerator H h h, h H x E. 2.7 Furthermore, we will use the notation L f H, E := { J } h j x j : h j H, x j E for j =,..., J LH, E J N j= for the subsace of linear and bounded finite rank oerators. Obviously, L f H, E Γ H, E.

36 26 Preliminaries Definition 2.5. The sace ΓH, E of γ-radonifying oerators is defined as the closure of the sace of finite rank oerators L f H, E in Γ H, E. I.e., ΓH, E := L f H, E Γ H,E Γ H, E. Remark 2.6. i By definition, ΓH, E, endowed with the norm Γ H,E inherited from Γ H, E is a Banach sace. We will use the abbreviations R ΓH,E := R Γ H,E, R ΓH, E, and, for [,, R ΓH,E := R Γ H,E, R ΓH, E. Note that for any [,, the norm equivalence holds, see also Remark 2.4ii. R ΓH,E R ΓH,E, R ΓH, E 2.8 ii In general, ΓH, E Γ H, E. An examle for a γ-summable oerator which is not γ-radonifying can be found in [9]. However, if the Banach sace E does not contain a closed subsace isomorhic to the sace c of sequences converging to zero endowed with the suremum norm, then ΓH, E = Γ H, E. This can be roven by using the results of Hoffmann-Jørgensen and Kwaień concerning sums of indeendent symmetric Banach sace valued random variables [65, 88], see [9, Theorem 4.3]. Now we collect some roerties and useful characterizations of ΓH, E. The roofs can be found in [8, Chater 5]. We start with the ideal roerty of γ-radonifying oerators. Theorem 2.7. Let R ΓH, E. Let H be another real Hilbert sace and E be another Banach sace. Then for all U LE, E and S LH, H we have URS ΓH, E and for all [,, URS ΓH,E U LE,E R ΓH,E S LH,H. Often, the Hilbert sace H is assumed to be searable and the following characterization of γ-radonifying oerators is taken as a definition. Theorem 2.8. Let H be a searable real Hilbert sace. Then for an oerator R LH, E the following assertions are equivalent: i R ΓH, E. ii For all orthonormal bases {h k : k N} of H and all [, the series k= γ krh k converges in L Ω; E. iii For some orthonormal basis {h k : k N} of H and some [, the series k= γ krh k converges in L Ω; E. In this situation, the sums in ii and iii converge almost surely and for all orthonormal bases {h k : k N} of H and [,, R [ ] Γ = E H,E γ k Rh k. 2.9 k= E

37 2.2 Stochastic integration in UMD Banach saces 27 Remark 2.9. Let H,, H be a searable real Hilbert sace and assume that E,, E is also a searable real Hilbert sace. Furthermore, let {h k : k N} be an orthonormal basis of H. Recall that L 2 H, E := {T LH, E : T 2 L2 H,E := } T h k 2 E < 2. is the sace of Hilbert-Schmidt oerators from H to E. Then, L 2 H, E = ΓH, E and for any R ΓH, E we have R ΓH,E = R L2 H,E. This is an immediate consequence of Theorem 2.8 above and Pythagoras theorem. If E is an L -sace on a σ-finite measure sace, ΓH, E can be also characterized as follows. The roof relies on the Kahane-Khintchine inequality for Gaussian sums, see Lemma 2.2. Theorem 2.2. Let H be a searable real Hilbert sace. Furthermore, let M, A, µ be a σ-finite measure sace and [,. For an oerator R LH, L M the following assertions are equivalent: i R ΓH, L M. k= ii For all orthonormal bases {h k L M. : k N} of H the function k= Rh k 2 2 belongs to iii For some orthonormal basis {h k : k N} of H the function k= Rh k 2 2 belongs to L M. In this case, there exists a constant C = C indeendent of R ΓH, L M such that C R ΓH,L M Rh k 2 2 C R ΓH,L LM M Stochastic integration for cylindrical Brownian motions k= Using the notions introduced in the revious subsections, we are now able to give a brief introduction to the theory of stochastic integration in Banach saces as develoed in [2, 23]. The integrator is an H-cylindrical Brownian motion. We collect the relevant definitions and results without roofs. We follow [8] and [3] in our exosition. Throughout this subsection, H,, H denotes a searable real Hilbert sace and F t t [,T ] is a normal filtration on a comlete robability sace Ω, F, P. E, E denotes an arbitrary real Banach sace. Definition 2.2. An H-cylindrical Brownian motion with resect to F t t [,T ] is a family W = W H t t [,T ] of linear oerators from H to L 2 Ω with the following roerties: [W] For every h H, W H th t [,T ] F t t [,T ]. is a real-valued Brownian motion with resect to [W2] For every t, t 2 [, T ] and h, h 2 H, we have E [W H t h W H t 2 h 2 ] = min{t, t 2 } h, h 2 H. From now on, W H = W H t t [,T ] denotes an H-cylindrical Brownian motion with resect to F t t [,T ]. We do not secify F t t [,T ] if it is clear from the context which filtration is meant.

38 28 Preliminaries Examle Let H be a searable real Hilbert sace with orthonormal basis {h k : k N}. Furthermore, let {wt k t [,T ] : k N} be a collection of indeendent real-valued standard Brownian motions with resect to F t t [,T ] on Ω, F, P. Then, H h W H th := wt k h k, h H L 2 Ω, t [, T ], 2.2 defines an H-cylindrical Brownian motion with resect to F t t [,T ]. k= Stochastic integration of functions: the Wiener integral We first define the stochastic integral of oerator valued functions having the following simle structure. Definition An oerator valued function Φ : [, T ] LH, E of the form Φt := J K tj,t j ]t h k, H x j,k, t [, T ], 2.3 j= k= with = t < t <... < t J = T, {h,..., h K } H orthonormal, and x j,k E, j J, k K, for some finite J, K N, is called finite rank ste function. The stochastic integral of a finite rank ste function is defined in the following natural way. Definition Let Φ be a finite rank ste function of the form 2.3. The stochastic integral of Φ on [, T ] with resect to W H is the E-valued random variable T Φt dw H t := J j= k= K W H t j h k W H t j h k x j,k. Note that the stochastic integral of a finite rank ste function is a centred E-valued Gaussian random variable. The class of stochastically integrable functions is introduced as follows. Definition A function Φ : [, T ] LH, E is said to be stochastically integrable with resect to W H if there exists a sequence Φ n n N of LH, E-valued finite rank ste functions on [, T ] such that: i for all h H we have lim n Φ n h = Φh in measure on [, T ], i.e., for any ε >, lim λ{ t [, T ] : Φ n th Φth E > ε } = ; n ii there exists an E-valued random variable X, such that lim n T Φ nt dw H t = X in robability. In this situation, the stochastic integral of Φ with resect to W H is defined as the limit in robability T T Φt dw H t := lim Φ n t dw H t. 2.4 n Since, as mentioned above, T Φ nt dw H t n N in 2.4 is a sequence of centred Gaussian random variables, it converges also in L Ω; E for any [,, and the stochastic integral T Φt dw Ht is again centred Gaussian, see, e.g., [8, Theorem 4.5].

39 2.2 Stochastic integration in UMD Banach saces 29 In [23], the following analogue of the Itô isometry has been roven for finite rank ste functions Φ : [, T ] LH, E: [ T E Φt dw H t where R Φ : L 2 [, T ]; H E is the oerator reresented by Φ, i.e., R Φ f := T 2 E ] 2 = RΦ ΓL2 [,T ];H,E, 2.5 Φtft dt, f L 2 [, T ]; H. 2.6 Furthermore, the following alternative characterization of the set of stochastically integrable functions holds: An H-strongly measurable function Φ : [, T ] LH, E is stochastically integrable on [, T ] with resect to W H if, and only if, t Φ x t := Φt x L 2 [, T ]; H for all x E and there exists an oerator R Φ ΓL 2 [, T ]; H, E, such that R Φx = Φ x in L 2 [, T ]; H for all x E. The isometry 2.5 extends to this situation. Remember that, unless exlicitly stated otherwise, we identify H with its dual sace H via the Riesz isometric isomorhism h h, H. Stochastic integration of stochastic rocesses The integral defined above has been extended to LH, E-valued stochastic rocesses for umd Banach saces E in [2]. The construction starts with the definition of the stochastic integral for so-called finite rank ste rocesses. Definition A finite rank F t t [,T ] -adated ste rocess is an LH, E-valued stochastic rocess Φ = Φt t [,T ] of the form J M K Φω, t := tj,t j ]t Aj,m ω h k, H x j,m,k, ω, t Ω T, 2.7 j= m= k= where = t < t <... < t J = T, and the sets {A j,,..., A j,m } F tj are disjoint for each j J with the convention t, t ] = {} and F = F, the vectors {h,..., h K } H are orthonormal, and x j,m,k E, j J, m M, k K, for some finite J, M, K N. In [2] such rocesses are called elementary adated to F t t [,T ]. Definition Let Φ be a finite rank F t t [,T ] -adated ste rocess of the form 2.7. The stochastic integral of Φ with resect to W H is defined as the E-valued random variable T Φt dw H t := J j= m= M K Aj,m W H t j h k W H t j h k x j,m,k. k= Note that the stochastic integral of a finite rank ste rocess is centred and -integrable for any [,. The class of stochastically integrable rocesses is defined as follows. Definition Let E be a Banach sace and fix,. Let W H be an H-cylindrical Brownian motion with resect to F t t [,T ]. An H-strongly measurable rocess Φ : Ω [, T ] LH, E is called L -stochastically integrable with resect to W H if there exists a sequence of finite rank F t t [,T ] -adated ste rocesses Φ n : Ω [, T ] LH, E, n N, such that: i for all h H we have lim n Φ n h = Φh in measure on Ω T, i.e., for any ε >, lim P { T ω, t ΩT : Φ n ω, th Φω, th E > ε } = ; n

40 3 Preliminaries ii there exists a random variable X L Ω; E such that T lim Φ n t dw H t = X n in L Ω; E. In this situation, the stochastic integral of Φ with resect to W H is defined as the L Ω; E-limit T T Φt dw H t := lim Φ n t dw H t. n It is easy to see that, if Φ : Ω [, T ] LH, E is L -stochastically integrable with resect to W H and S LE, F, F being another Banach sace, then, SΦ : Ω [, T ] LH, F is L -stochastically integrable with resect to W H. Furthermore, T T S Φt dw H t = SΦt dw H t in L Ω; F. 2.8 The aths of finite rank ste rocesses are finite rank ste functions. Thus, for any ω Ω, the ath t Φ ω t := Φω, t defines an oerator R Φω ΓL 2 [, T ]; H, E by 2.6. This leads to an F-simle random variable R Φ : Ω ΓL 2 [, T ]; H, E. If E is a umd Banach sace, one can use decouling and rove that for finite rank adated ste rocesses Φ, [ T E Φt dw H t E ] E [ R Φ ] ΓL 2 [,T ];H,E, where the constants involved do not deend on Φ. This extension of the analogue 2.5 of Itô s isometry can be generalized in the following way see [2, Theorem 3.6 and Remark 3.7]. Theorem Let E be a umd Banach sace and fix,. Furthermore, let Φ : Ω [, T ] LH, E be an H-strongly measurable F t t [,T ] -adated rocess. Assume that for any x E, the stochastic rocess Φ x : Ω [, T ] H ω, t Φω, t x belongs to L Ω; L 2 [, T ]; H, i.e., Φ is belongs to L Ω; L 2 [, T ]; H scalarly. Then, the following assertions are equivalent: [S] Φ is L -stochastically integrable with resect to W H. [S2] There exists a strongly measurable random variable X L Ω; E such that for all x E we have x, X = T Φ x t dw H t in L Ω. 2.9 [S3] There exists R Φ L Ω; ΓL 2 [, T ]; H, E such that for all x E we have R Φx = Φ x in L Ω; L 2 [, T ]; H. In this situation, X = T Φt dw Ht and R Φ in [S3] above is uniquely determined. Moreover, [ T E Φt dw H t where the constants involved do not deend on Φ and T. E ] [ E R Φ ΓL 2 [,T ];H,E ],

41 2.3 Function saces 3 Remark 2.3. In the setting of Theorem 2.29, [S3] means that Φ reresents an element R Φ L Ω; ΓL 2 [, T ]; H, E. By [2, Proositions 2. and 2.2], R Φ belongs to the closure of the finite rank ste rocesses in L Ω; ΓL 2 [, T ]; H, E, which, in analogy to the notation used in [2], will be denoted by L F Ω; ΓL 2 [, T ]; H, E. Later on, we will need the following series exansion of the stochastic integral which can be found in [2, Corollary 3.9]. Theorem 2.3. Let E be a umd Banach sace and fix,. Assume that the H-strongly measurable and F t t [,T ] -adated rocess Φ : Ω [, T ] LH, E is L -stochastically integrable with resect to W H. Then, for all h H the rocess Φh : Ω [, T ] E is L -stochastically integrable with resect to W H h. Moreover, if {h k : k N} is an orthonormal basis of H, then T Φt dw H t = T k= Φth k dw H th k convergence in L Ω; E. We have already mentioned in Lemma 2.9v, that a umd Banach sace has tye 2 if, and only if, it has M-tye 2. Stochastic integration of rocesses in M-tye 2 Banach saces has been studied by several authors, see the literature overview in [2,. 46] for a short list. As mentioned therein, according to [6, 24], if E has tye 2, then L 2 [, T ]; ΓH, E R ΓL 2 [, T ]; H, E, 2.2 with R : Φ R Φ given by 2.6. Consequently, the following result holds, see [2, Corollary 3.]. Theorem Let E be a umd Banach sace and fix,. If E has tye 2, then every H-strongly measurable and adated rocess Φ L Ω; L 2 [, T ]; ΓH, E belongs to L Ω; L 2 [, T ]; H scalarly, is L -stochastically integrable with resect to W H and we have [ T E Φt dw H t E where the constant C does not deend on Φ. ] [ C E Φ ] L 2, [,T ];ΓH,E 2.3 Function saces In this section we introduce some function saces, which will be used later on for the analysis of the regularity of SPDEs. We will also collect and rove some useful roerties of these saces, esecially of the weighted Sobolev saces in Subsection Sobolev saces The Sobolev saces on an arbitrary domain G R d are defined as follows. Definition Let,. i For m N, W m G := { } u L G : D α u L G for all α N d with α m

42 32 Preliminaries is called the Sobolev sace of smoothness order m with summability arameter. It is endowed with the norm / u W m G := D α u L G, u W m G. α m ii For s = m + σ with m N and σ,, we denote by W s G := { u W m G : u W s G := α =k G G D α ux D α uy x y σ+d dxdy < the Sobolev sace of fractional smoothness order s, \ N with summability arameter. It is endowed with the norm } u W s G := u W m G + u W s G /, u W s G. iii For s we denote by W s G the closure of the test functions C G in W s G, endowed with the norm W s G. iv For s <, we denote by W s G the dual sace of dual norm. W s G, endowed with the canonical Remark i Remember that in this thesis, D α u denotes the α-th generalized derivative of a distribution u see Subsection 2..5 for details. Thus, for m N and,, u W m G means that u L G and that for all α N d with α m, the α-th generalized derivative is regular and is interreted as a function an element of L G. ii For arbitrary domains G R d and arbitrary, and s, W s G is a Banach sace, see, e.g., [2, Theorem 3.3] for the case s N and [5, Theorem 6.3.3] for fractional s, \ N. Also, W s G is comlete, since it is a closed subsace of W s G. The corresonding duals W s G, s <, are consequently also Banach saces. iii For fractional s, \ N, the sace W s G is sometimes called Slobodeckij sace and the scale W s G, s, is referred to as the scale of Sobolev-Slobodeckij saces. Even if this terminology would be historically more correct, in this thesis we will call the sace W s G for any s R a Sobolev sace. For further details regarding the historical background, we refer to [5]. iv Let G R d be a bounded domain and,. For m N, the exression / [u] W m G := D α u L G, u W m G, α =m is an equivalent norm on W m G. This is a consequence of Poincaré s inequality, see, e.g. [53, Theorem 5.6/3].

43 2.3 Function saces Saces of Bessel otentials In this subsection we introduce the saces H γ R d of Bessel otentials and discuss some of their roerties which we will use in this thesis. Furthermore, we resent the definition of the saces H γ R d ; l 2. They have been used in [8] by N.V. Krylov for the develoment of the analytic aroach to SPDEs on the whole sace R d. We rove that the saces H γ R d ; l 2 are isomorhic to the corresonding saces Γl 2, H γ R d of γ-radonifying oerators from l 2 to H γ R d. Due to this fact, under suitable assumtions, the SPDEs from [8] can be rewritten as Banach sace valued stochastic differential equations in the sense of van Neerven, Veraar and Weis [2]. For γ R, we denote by γ/2 the seudo-differential oerator with symbol That is, γ/2 : S R d ; C S R d ; C R d ξ + ξ 2 γ/2 R +. u γ/2 u := F ξ + ξ 2 γ/2 Fuξ, where F denotes the Fourier transform on the sace of comlex valued temered distributions S R d ; C, see Subsection 2..5 for details. Definition Let, and γ R. Then H γ := H γ R d := γ/2 L R d = { } γ/2 f : f L R d is the sace of Bessel otentials of order γ with summability arameter. It is endowed with the norm u H γ R d := γ/2 u LR d, u Hγ R d. Remark i Recall that in Section 2. we have ostulated that in this thesis, unless exlicitly stated otherwise, functions and distributions are meant to be real valued. In articular, L R d stands for the sace of real valued -Bochner integrable functions on R d. What about the generalized functions in the saces H γ R d defined above? At first view, even if f is real valued, we can not guarantee that γ/2 f is real valued, since the seudo-differential oerator mas into the sace of comlex valued temered distributions S R d ; C. However, the following arguments show that γ/2 f is indeed a real valued temered distribution if f L R d, and, therefore, that H γ R d as defined above, consists of real valued generalized functions. Fix γ R and f L R d for some,. Then, γ/2 f S R d ; C is given by the formula γ/2 f, ϕ = f γ/2 ϕ dx, ϕ SR d ; C. 2.2 R d Assume that ϕ C Rd. Then, γ/2 ϕ can be exressed as γ/2 ϕx = Gyϕx y dy, x R d, 2.22 R d where G denotes the Green function of γ/2 and is given by Gy = 2π d lim R + ξ 2 γ/2 ei x,ξ dξ, y R d, { ξ R}

44 34 Preliminaries see [84, Chater 2] for details. Due to the symmetry roerties of the function R d ξ + ξ 2 γ/2 R +, G is real valued. Therefore and by 2.22, γ/2 ϕ is real valued too. Inserting this into 2.2, we obtain γ/2 f, ϕ = f γ/2 ϕ dx R, ϕ C R d. R d Hence, γ/2 f is a real valued generalized function. Consequently, the temered distribution γ/2 f S R d ; C is real valued, since for any ϕ SR d, there exists a sequence ϕ n n N C Rd converging to ϕ in SR d. All in all, we have shown that H γ R d = { } u S R d : u = γ/2 f for some f L R d. ii For, and γ R, the sace H γ R d of Bessel otentials is a Banach sace. For a roof see, e.g., [84, Theorem 3.3.3i] and use art i of this remark. The following result shows that the saces of Bessel otentials are generalizations of the Sobolev saces W m R d, m N. A roof can be found e.g. in [84, Theorem 3.3.2]. Lemma Let, and γ = m N. Then, H m R d = W m R d equivalent norms. It is well-known that for,, with / + / =, the maing Ψ : L R d L R d, Ψfg := R d fxgx dx, is an isometric isomorhism. In articular, L R d = L R d. This duality relation can be extended to the scale of Bessel otential saces, as the following result shows. The roof is left to the reader. Lemma Let γ R and,, with / + / =. Then, for any u H γ ϕ C Rd, u, ϕ u H γ ϕ H γ. Thus, for any u H γ, there exists a unique functional Ψ u H γ such that its restriction to the test functions C Rd coincides with u,, i.e., and Ψ u C R d = u, Moreover, the exression Ψ : H γ H γ u Ψu := Ψ u, with Ψ u from above, is an isometric isomorhism. In articular, H γ = H γ.

45 2.3 Function saces 35 Recall that the notation u, ϕ is used to denote the alication of a distribution u to a test function ϕ. In what follows we extend the meaning of, and write u, ϕ := Ψ u ϕ, for all u H γ and ϕ H γ, 2.24 with Ψ u as in Lemma This is justified by The following can be said about the geometric roerties of the saces of Bessel otentials. Remark For any, and γ R, H γ R d is a umd Banach sace, since γ/2 : H γ R d L R d is an isometric isomorhism and L R d is umd, see Lemma 2.6. The same argument, together with Lemma 2.9iii and iv, shows that H γ R d has tye r := min{2, }. Now we recall the definition of the saces H γ R d ; l 2 given in [8]. As already mentioned, these saces are used therein for the formulation of the SPDEs under consideration. Definition 2.4. For, and γ R, { H γ l 2 := H γ R d ; l 2 := g = g k k N H γ R d N : g H γ l 2 := γ/2 g k } L k N <. l2 Remark 2.4. Let,. Remember that in this thesis we write {e k : k N} for the standard orthonormal basis of l 2 = l 2 N. i The maing Φ : H R d ; l 2 L R d, BR d, λ d ; l 2 assigning the function equivalence class Φg : R d l 2 x g k x e k convergence in l 2 k= to each g H R d ; l 2 is an isometric isomorhism. In articular, we have H R d ; l 2 = L R d, BR d, λ d ; l 2. Consequently, since L R d ; l 2 is comlete, H R d ; l 2 is a Banach sace. ii Let γ R. Then, is an isometric maing. Furthermore, Φ γ : H γ R d ; l 2 H R d ; l 2 g = g k k N γ/2 g k k N Φ γ : H R d ; l 2 H γ R d ; l 2 g = g k k N γ/2 g k k N defines a right inverse for Φ γ. Thus, Φ γ is a surjective isometric maing, and therefore an isometric isomorhism. Consequently, H γ R d ; l 2 is a Banach sace, since H R d ; l 2 is comlete by art i of this remark. Finally, we rove that the sace H γ R d ; l 2 is isomorhic to Γl 2, H γ R d.

46 36 Preliminaries Theorem Let γ R and [2,. Then the oerator Φ : H γ R d ; l 2 Γl 2, H γ R d g k k N e k, l2 g k convergence in Γl 2, H γ R d k= is an isomorhism, and, therefore, H γ R d ; l 2 Γl 2, H γ R d. Proof. Fix γ R. First of all we have to show that the oerator Φ is well defined. To this end, since Γl 2, H γ R d is comlete, it is enough to rove that for any fixed g H γ R d ; l 2, the sequence n Rn n N := e k, l2 g k L f l 2, H γ R d k= n N is a Cauchy sequence in Γl 2, H γ R d. Fix g H γ R d ; l 2. Using the fact that γ/2 : HR d H γ R d is an isometric isomorhism together with the ideal roerty of γ-radonifying oerators, see Theorem 2.7 above, we obtain that for arbitrary m, n N, n R n R m Γl2,H γ R d = e k, l2 g k Γl2,H γ R d = k=m+ γ/2 n k=m+ γ/2 LH R d,h γ R d = n k=m+ e k, l2 γ/2 g k Γl2,H γ R d n k=m+ e k, l2 γ/2 g k Γl2,H Rd. Thus, since H R d = L R d, we obtain from 2., n R n R m Γl2,H γ R d C γ/2 g k 2 2. LR d k=m e k, l2 γ/2 g k Γl2,H Rd Therefore, R n n N converges in Γl 2, H γ R d and the series k= e k, l2 g k is well-defined. Moreover, using the same arguments, it follows that Φg Γl2,H γ R d = e k, l2 g k Γl2,H γ R d k= e k, l2 γ/2 g k Γl2,HR d k= C γ/2 g k 2 2 = C g LR d H γ R d ;l 2. k= Consequently, the obviously linear oerator Φ is bounded. Simultaneously, one easily checks that Φ : Γl 2, H γ R d H γ R d ; l 2 R Re k k N is the inverse of Φ, which is well-defined, linear and bounded by Theorem 2.2.

47 2.3 Function saces Weighted Sobolev saces In this subsection we recall the definition and some basic roerties of the weighted Sobolev saces H γ G as introduced e.g. in [93]. These saces serve as state saces for the solution rocesses u = ut t [,T ] of SPDEs on domains in the L -theory of N.V. Krylov and collaborators see, e.g., [72, 73, 75, 76, 85, 86]. Although in this thesis we consider only SPDEs on bounded Lischitz domains, we introduce and analyse the weighted Sobolev saces on arbitrary domains G in R d with non-emty boundary G. Among others, we will rove that H γ G ossesses the umd roerty and has tye r := min{, 2}, see Lemma 2.5. We will also consider the saces H γ G; l 2 used in the mentioned L -theory and rove that they are isomorhic to Γl 2, H γ G, see Theorem 2.54, which is a generalization of the just roven Theorem This will allow us to aly the stochastic integration theory from Section 2.2 to stochastic rocesses Φ : Ω [, T ] H γ G; l 2 in Chater 6. Let G R d be an arbitrary domain with non-emty boundary G. Remember that ρx = ρ G x denotes the distance of a oint x G to the boundary G. We fix an infinitely differentiable function ψ defined on G such that for all x G, ρx Cψx, ρx m D m ψx Cm < for all m N, 2.25 where C and Cm do not deend on x G. For a detailed construction of such a function see, e.g., [3, Chater VI, Section 2.]. Let ζ C R + be a non-negative function satisfying ζe n+t > c > for all t R n Z Note that any non-negative smooth function ζ C R + with ζ > on [e, e] satisfies Without loss of generality, wee assume that ζ takes values in the interval [, ]. For x G and n Z, define ζ n x := ζe n ψx Then, there exists k > such that, for all n Z, su ζ n G n := {x G : e n k < ρx < e n+k }, i.e., ζ n C G n. Moreover, D m ζ n x Cζ, me mn for all x G and m N, and n Z ζ nx δ > for all x G. Using this localisation sequence the weighted Sobolev saces H γ G can be introduced as follows. Definition Let G be an arbitrary domain in R d with non-emty boundary. Furthermore, let ζ n, n Z, be as above,,, and γ, θ R. Then H γ G := {u D G : u H γ G := e nθ ζ n e n ue n H γ n Z } <. It is called weighted Sobolev sace of smoothness order γ with summability arameter and weight arameter θ. Remark The reason why H γ G is called weighted Sobolev sace is the fact that L G := H G = L G, BG, ρ θ d λ d ; R, with equivalent norms, and that, if γ = m N is a ositive integer, } H {u m G = L G : ρ α D α u L G for all α N d with α m,

48 38 Preliminaries where u H k G := α N d α =k for k {,..., m}; see, e.g., [93, Proosition 2.2]. m u H m G u H k G 2.28 G k= ρx α D α ux ρx θ d dx, 2.29 Now we resent some useful roerties of the sace H γ G taken from [93], see also [8,8]. Lemma Let G R d be a domain with non-emty boundary G, γ, θ R, and,. i H γ G is a searable and reflexive Banach sace. ii The sace C G is dense in Hγ G. iii u H γ G if, and only if, u, ψu x H γ G and u H γ G C ψu x H γ G + C u H γ G C u H γ G. Also, u H γ G if, and only if, u, ψu x H γ G and u H γ G C ψu x H γ G + C u H γ G C u H γ G. iv For any ν, γ R, ψ ν H γ G = Hγ ν G and u H γ ν G C ψ ν u H γ G C u H γ ν G. v If < η <, γ = ην + ην, / = η/ + η/ and θ = ηθ + ηθ with ν, ν, θ, θ R and,,, then H γ G = [ H ν,θ G, H ν,θ G ] η equivalent norms. Consequently, if γ ν, ν then, for any ε >, there exists a constant C, deending on ν, ν, θ,, and ε, such that u H γ G ε u H ν G + Cν, ν, θ,, ε u H ν G. Also, if θ θ, θ then, for any ε >, there exists a constant C, deending on θ, θ, γ,, and ε, such that u H γ G ε u H γ G + Cθ, θ, γ,, ε u H γ G. vi There exists a constant c > deending on, θ, γ and the function ψ such that, for all c c, the oerator ψ 2 c is an isomorhism from H γ+ G to Hγ G. vii If G is bounded, then H γ G H γ 2 G for θ < θ 2.

49 2.3 Function saces 39 viii The dual of H γ G and the weighted Sobolev sace H γ,θ G with / + / = and θ/ + θ / = d are isomorhic. That is, H γ G H γ,θ G where + = and θ + θ = d. 2.3 Remark Assertions iv and vi in Lemma 2.45 imly the following: If u H γ G and u H γ + G, then u Hγ+2 G and there exists a constant C, which does not deend on u, such that u H γ+2 G C u H γ + G + C u H γ G. A roof of the following equivalent characterization of the weighted Sobolev saces H γ G can be found in [93, Proosition 2.2]. Lemma Let {ξ n : n Z} C G be such that for all n Z and m N, D m ξ n Cm c nm and su ξ n {x G : c n k < ρx < c n+k } 2.3 for some c > and k >, where the constant Cm does not deend on n Z and x G. Then, for any u H γ G, n Z c nθ ξ n c n uc n H γ C u H γ G. If in addition ξ n x δ > for all x G, 2.32 n Z then the converse inequality also holds. The following sequences {ξ n : n Z} will be useful, when we aly Lemma 2.47 in the roofs of Lemma 3.5i, Lemma 4.9 and Theorem 6.7, resectively. Remark i It can be shown that for any i, j {,..., d}, both { ξ n := e n ζ n x i : n Z } and { ξ 2 n := e 2n ζ n x i x j : n Z} satisfy 2.3 with c := e. Thus, for any, and θ R, e nθ e n ζ n x ie n ue n H γ + e2n ζ n x i x jen ue n H γ C u H γ G. n Z ii Let c > and k >. Fix a non-negative function ζ C R + with ζt = for all t [ ] C c k, C c k, where C and C are as in Then, the sequence {ξ n : n Z} C G defined by ξ n := ζc n ψ, n Z, fulfils the conditions 2.3 and 2.32 from Lemma 2.47 with c = c and a suitable k >. Furthermore, ξ n x = on { x G : c n k ρx c n+k }.

50 4 Preliminaries iii Let {ξ n : n Z} C G fulfil the conditions 2.3 and 2.32 from Lemma 2.47 for some fixed constants c > and k >. Consider the sequence {ξn : n Z} C G given by Obviously, ξ n := ξ n j Z ξ, n Z. j ξnx = for all x G n Z By standard calculations one can check that the sequence {ξn : n Z} C G also fulfils the condition 2.3 from Lemma The following fact might be useful: Any x G is contained in at most finitely many stries G n c, k := { x G : c n k < ρx < c n+k }, n Z Even more, there exists a finite constant C = Cc, k which does not deend on x G such that { n Z : x Gn c, k } C Let us also be a little bit more recise on the duality statement from Lemma 2.45viii. Remark Fix γ R,,, θ R and let and θ be as in 2.3. Fix {ξ n : n Z} C G with n Z ξ n = on G satisfying 2.3 from Lemma 2.47 for some fixed c > and k >. Simultaneously, assume that we have a sequence { ξ n : n Z} C G such that for every n Z, ξ n equals one on the suort of ξ n, i.e., suξn ξ n =, 2.36 and satisfying 2.3 with the same c > but ossibly different k > and 2.32 from Lemma By Remark 2.48ii and iii, it is clear that we can construct such sequences. The assertion of Lemma 2.45viii has been roven in [93, Proosition 2.4] by showing that the maing with Ψ : H γ,θ G H γ G u Ψu := [u, ] [, ] : H γ,θ G H γ G R, [u, v] := n Z c nd ξ n c n uc n, ξ n c n vc n 2.37 is an isomorhism; see 2.24 for the meaning of, on H γ R d H γ R d. From now on we will use this notation also on H γ,θ G H γ G and define, := [, ] on H γ,θ G H γ G, 2.38 with [, ] as in This is justified by the following calculation: Let u H γ,θ G D G and ϕ C G. Then, since n Z ξ n = and ξ n is constructed in such a way that 2.36 holds, we obtain u, ξ n ϕ = ξ n u, ξ n ϕ = [u, ϕ]. u, ϕ = u, n Z ξ n ϕ = n Z n Z

51 2.3 Function saces 4 After resenting and discussing these fundamental roerties of weighted Sobolev saces, we rove now that they satisfy the following geometric Banach sace roerties. Lemma 2.5. Let G be an arbitrary domain in R d with non-emty boundary. Let γ, θ R and,. Then H γ G is a umd sace with tye r := min{2, }. Proof. First we rove the umd roerty. Obviously, the linear oerator S : H γ G L Z, PZ, e nθ δ n ; H γ R d n Z u n ζ n e n ue n. is isometric. Therefore, and since H γ G is comlete, the range of S is a closed subsace of L Z, PZ, n Z enθ δ n ; H γ R d, which satisfies the umd roerty by Lemma 2.6iii. Thus, H γ G is isomorhic to a closed subsace of a umd Banach saces. Hence, due to Lemma 2.6i and Remark 2.39, the umd roerty of H γ G follows. In order to rove that H γ G has tye r = min{2, } we argue as follows: Fix an arbitrary Rademacher sequence r k k= and a finite set {u,..., u K } H γ G. Then, by the Kahane- Khintchine inequality, see Lemma 2., we have [ K r E r k u k k= H γ G ] r [ K C E r k u k By the definition of the weighted Sobolev norm, this yields [ K r E r k u k k= H γ G ] r k= H γ G ]. [ C E e nθ ζ n e n K r k u k e n n Z k= H γ R d Using Fubini s theorem and the fact that ζ n e n K k= r ku k en = K k= r kζ n e n u k e n, we obtain the estimate [ K r E r k u k k= H γ G ] r C n Z [ K e nθ E r k ζ n e n u k e n k= Thus, using again the Kahane-Khintchine inequality, we have [ K r E r k u k k= H γ G ] r C n Z [ e E K nθ r k ζ n e n u k e n k= Since H γ R d has tye r, see Remark 2.39, this leads to [ K r E r k u k k= H γ G ] r H γ R d r H γ R d K C e nθ ζ n e n u k e n r H γ R d n Z k= ] ].. ] r. r.

52 42 Preliminaries Therefore, alying the triangle inequality in L /r Z, PZ, n enθ δ n ; R yields [ K r E r k u k k= H γ G The assertion follows. ] r C K k= n ζ n e n u k e n r r H γ R d L /r Z,PZ, n enθ δ n;r K C n ζ n e n u k e n r H γ R d k= K = C e nθ ζ n e n u k e n r r H γ R d k= n Z K = C u k r H γ G k= r. L/r Z,PZ, n enθ δ n;r Remember that in this thesis we are mainly concerned with SPDEs on bounded Lischitz domains O R d. As already mentioned, the weighted Sobolev saces introduced above will serve as state saces for the solutions rocesses u = ut t [,T ] of the SPDEs under consideration. Therefore, since we are interested in solutions fulfilling a zero Dirichlet boundary condition, we need to check whether the elements of the the weighted Sobolev saces introduced above vanish at the boundary in an adequate way. In order to answer this question for the relevant range of arameters this will be done in Remark 4.3 we will need the following lemma. It is an immediate consequence of [87, Theorem 9.7]. Let us mention that this result holds for a wider class of domains. E.g. it stays true for bounded domains with Hölder continuous boundary, see [87, Remark 9.8ii] for details. However, in the course of this thesis, we will not need these generalizations. Lemma 2.5. For a bounded Lischitz domain O R d and k N, with equivalent norms. W k O = H k,d k O In order to formulate the stochastic equations under consideration, we will use the following saces H γ G; l 2. They are counterarts of the saces H γ R d ; l 2 introduced in the revious subsection. We define and discuss them for the general case of arbitrary domains with non-emty boundary, although later on we are mainly interested in the case of bounded Lischitz domains. Definition Let G be an arbitrary domain in R d with non-emty boundary. For γ R,, and θ R, we define H γ G; l 2 := { g = g k k N H γ G N : g H γ G;l 2 := e nθ ζ n e n g k e n k N }, < H γ l 2 n Z r with ζ n, n Z, from above, cf Remark For, and γ, θ R, H γ G; l 2 is a Banach sace. This can be roven by following the lines of the roof of the comleteness of H γ G resented in [93, Proosition 2.4.]. The details are left to the reader.

53 2.3 Function saces 43 In Chater 6 we will need the fact that H γ G is isomorhic to the corresonding class of γ-radonifying oerators from l 2 N to H γ G. We rove this now. As in Subsection 2.2.2, from now on {γ k : k N} denotes a Gaussian sequence, see Definition 2.. Remember that for h l 2 and u E, where E is a Banach sace, we use to write h u for the rank one oerator h, l2 u, see also 2.7. Theorem Let G be an arbitrary domain in R d with non-emty boundary. Furthermore, let γ, θ R and [2,. Then, the oerator Φ : H γ G; l 2 Γl 2, H γ G g k k N e k g k k= convergence in Γl 2, H γ G is an isomorhism, and therefore, H γ G; l 2 Γl 2, H γ G. Proof. First of all we show that Φ is well-defined and bounded. Fix g H γ G; l 2. Then, using the equality 2.9 from Theorem 2.8 together with the norm equivalence 2.8, for any m, m 2 N, we can estimate the norm of the finite rank oerator m 2 k=m e k g k as follows m 2 [ e k g k m 2 ] C E γ k g k. k=m Γl 2,H γ G k=m H γ G Since for every ω Ω, m 2 γ k ωg k = m 2 e nθ γ k ωζ n e n g k e n k=m H γ G n Z k=m H γ R d with {ζ n : n Z} as defined in 2.27, an alication of Beo-Levi s theorem yields m 2 e k g k C m 2 ] e E[ nθ γ k ζ n e n g k e n k=m Γl 2,H γ G n Z k=m H γ R d For every n Z, we can aly Equality 2.9 from Theorem 2.8 to the finite rank oerator m 2 e k ζ n e n g k e n L f l 2, H γ R d Γl 2, H γ R d, k=m followed by the norm equivalence 2.8, and obtain m 2 ] E[ γ k ζ n e n g k e n m 2 = C e k ζ n e n g k e n k=m H γ R d k=m m 2 C e k ζ n e n g k e n k=m, Γ l 2,H γ R d Γl 2,H γ R d. Thus, if we set { g nm k ζ n e n g k e n, } if k {m,..., m 2 }, m 2 :=,, else

54 44 Preliminaries obviously, g nm k, m 2 k N Hγ R d ; l 2, and an alication of Theorem 2.42 leads to m 2 ] E[ γ k ζ n e n g k e n C g nm k, m 2 k=m H γ k N, R d H γ R d ;l 2 the constant C being indeendent of n Z and g. Inserting this into the estimate 2.39, we obtain m 2 e k g k C e nθ k g n m, m 2 k=m Γl 2,H γ G k N. H γ R d ;l 2 n Z Since g H γ G; l 2, the right hand side converges to zero for m, m 2 by Lebesgue s dominated convergence theorem. Thus, the sequence m Rm m N := e k g k L f l 2, H γ G k= converges in Γl 2, H γ G and its limit k= e k g k is well-defined. The boundedness of Φ can now be roven by reeating the calculations above with m = and m 2 =. By the oen maing theorem, showing that m N Φ : Γl 2, H γ G Hγ G; l 2 R Re k k N is the inverse of Φ, would finish the roof. Let us check whether this oerator is well-defined. If so, then the fact that it is the inverse of Φ follows by simle calculations. Fix R Γl 2, H γ G. Since for every n Z, the oerator S n : H γ G Hγ R d u ζ n e n ue n is obviously bounded, the comosition S n R is γ-radonifying, i.e., S n R Γl 2, H γ R d, see Theorem 2.7. Furthermore, by Theorem 2.42, S n Re k k N = H γ ζ n e n Re k e n R d ;l 2 k N C S H γ n R R d ;l 2 Γl2,H γ R d with a constant C indeendent of n Z and R. Using this together with Equality 2.9 from Theorem 2.8 together with the norm equivalence 2.8, yields e nθ ζ n e n Re k e n k N C e nθ S n R H γ R d ;l 2 Γl 2,H γ R d n Z n Z C n Z e nθ E[ γ k S n Re k k= H γ R d Alying Beo-Levi s theorem and using the definitions of the norms we obtain e nθ ζ n e n Re k e n [ k N ] C E e nθ γ H γ k S n Re k R d ;l 2 n Z n Z k= H γ R d [ = C E e nθ γ k ζ n e n Re k e n n Z k= ] = C E[ γ k Re k. k= H γ G ]. H γ R d ]

55 2.3 Function saces 45 Therefore, another alication of Equality 2.9 from Theorem 2.8 followed by the norm equivalence 2.8, leads to e nθ ζ n e n Re k e n k N C H γ R d ;l 2 R Γl 2,H γ G. n Z Thus, Re k k N H γ G; l 2. We occasionally use the following roerties of the saces H γ G; l 2 in this thesis. In several ublications like [73, 75], these roerties are stated and used without roof. Since we did not find any roof in the literature, we sketch a roof based on the isomorhy from Theorem 2.54 above. The details are left to the reader. Lemma Let G be an arbitrary domain in R d with non-emty boundary,, and γ, θ R. i g = g k k N H γ G; l 2 if, and only if, g, ψgx k k N H γ G; l 2 and g H γ G;l 2 C ψgx k H k N γ G;l 2 + C g H γ G;l 2 C g H γ G;l 2. Also, g = g k k N H γ G; l 2 if, and only if, g, ψg k x k N H γ G; l 2 and g H γ G;l 2 C ψg k x H k N γ G;l 2 + C g H γ G;l 2 C g H γ G;l 2. ii For any ν, γ R, ψ ν H γ G; l 2 = H γ ν G; l 2 and g H γ ν G;l 2 C ψ ν g k k N H γ G;l 2 C g H γ ν G;l 2. iii There exists a constant c > deending on, θ, γ and the function ψ such that, for all c c, the oerator is an isomorhism. ψ 2 c : H γ+ G; l 2 H γ G; l 2 g = g k k N ψ 2 cg := ψ 2 cg k k N iv If G is bounded, then H γ G; l 2 H γ 2 G; l 2 for θ < θ 2. v If < η <, γ = ην + ην, / = η/ + η/ and θ = ηθ + ηθ with ν, ν, θ, θ R and,,, then H γ G; l 2 = [ H ν,θ G; l 2, H ν,θ G; l 2 ] η equivalent norms. Sketch of roof. The assertions can be roven by using the isomorhism from Theorem 2.54 together with the ideal roerty of γ-radonifying oerators Theorem 2.7 and the corresonding roerties of the saces H γ G, γ, θ R,,, from Lemma In order to rove the interolation statement v one additionally needs the fact that [ Γl2, H ν,θ G, Γl 2, H ν,θ G ] η = Γl 2, H γ G. This follows from [4, Theorem 2.] with H = H = l 2, X = H ν,θ G and X = H ν,θ G. Note that, by a result of G. Pisier, the B-convexity of X and X assumed in [4, Theorem 2.] is equivalent to the fact that the Banach saces have non-trivial tye, see, e.g., [49, Theorem 3.] for a roof.

56 46 Preliminaries Besov saces One of the main goals of this thesis is to analyse the satial regularity of the solutions to SPDEs on bounded Lischitz domains in the articular scale of Besov saces. In this subsection we recall the definition of Besov saces by means of the Fourier transform and resent an alternative intrinsic characterization via differences. In certain ublications, such as [27], [46] and [48], the latter is used as a definition of Besov saces for <, q < and s >. Denote by ϕ C Rd a comactly suorted, infinitely differentiable function having the roerties For k N define ϕ x = if x and ϕ x = if x 3/ ϕ k x := ϕ 2 k x ϕ 2 k+ x for x R d 2.4 to obtain a smooth dyadic resolution of unity on R d, i.e., ϕ k C Rd for all k N, and ϕ k x = for all x R d k= Definition Let {ϕ k } k N C Rd be a resolution of unity according to i Let <, q <, s R, and /q u B s,q R d := 2 ksq F [ϕ k Fu] q L R. d k= Then B s,qr d := { u S R d : u B s,q R d < } is the Besov sace of smoothness order s with summability arameter and fine tuning arameter q. ii Let G R d be an arbitrary domain. Then, for <, q < and s R, the Besov sace B s,qg of smoothness order s with summability arameter and fine tuning arameter q on G is defined as follows: B s,qg := { u D G : there exists g B s,qr d : g G = u }. It is endowed with the norm u B s,q G := inf g B s,qr d g G =u g B s,q R d, u B s,qg Remark For <, q < and s R, the Besov sace B s,qg, endowed with the quasinorm B s,q G, is a quasi-banach sace. If, q <, then B s,q G is a norm and therefore B s,qg, B s,q G is a Banach sace. A roof can be found in [5, Theorem 2.3.3i] for the case G = R d. For general domains we refer to the roof of [5, Proosition 3.2.3i]. Note that the assumed smoothness roerty for the boundary of the underlying domain therein does not have any relevance in the roof of the comleteness of the Besov saces.

57 2.3 Function saces 47 As already mentioned in the introduction to this subsection, besides the definition given above, Besov saces are frequently defined by means of differences. In general, it is not immediately clear whether the two definitions yield the same saces. However, if G = R d, for, q, and s > max{, d/ }, the two ways of defining Besov saces match. This follows from [5, Theorem 2.5.2] and the integral transformation formula for rotationally symmetric functions, see, e.g. [, Corollary 5.4]. In this thesis, we will also need this statement for the case that G is a bounded Lischitz domain in R d. It has been roven in [5, Theorem 3.8]. In the next theorem, we resent these results in detail. We use the following notation. Let G be an arbitrary domain in R d. For a function u : G R and a natural number n N let n n h [u]x := n h ux := G x + ih i= n j= n n j ux + jh j be the n-th difference of u with ste h R d. For,, the n-th order L -modulus of smoothness of u is given by ω n t, u, G := ω n t, u := su n h u, t >. LG h <t Theorem Let G be either R d or a bounded Lischitz domain in R d. Let, q,, s > max{, d/ } and n N with n > s. Then B s,qg is the collection of all functions u L G such that u B s,q G := /q q dt t s ω n t, u, G < t The function B s,qg u u LG + u B s,q G 2.45 is an equivalent quasi-norm for B s,q G on B s,qg. Remark In this thesis we will be mainly concerned with Besov saces B s,g with 2 and s >, and on the non-linear aroximation scale B α τ,τ G, /τ = α/d + /, α >, with 2, where either G = R d or G is a bounded Lischitz domain in R d. In both cases, the arameters fulfil the assumtions from Theorem Therefore, the definition based on the Fourier transform and the intrinsic characterization of Besov saces via differences are equivalent. In the next theorem we collect some arameter constellations, for which Besov and Sobolev saces coincide. As in the theorem before, we restrict ourselves to the cases which are relevant for this thesis and assume that the underlying domain is either the whole sace or a bounded Lischitz domain in R d. For G = R d the statements are taken from [6]: The first assertion can be found in [6, Section 2.5., esecially Remark 4] and the second follows from [6, Theorem 2.3.2d together with Theorem 2.3.3b]. For the case that G is a bounded Lischitz domain, the first assertion can be found in [6, Remark 4.4.2/2], whereas for the second statement we additionally need [6, Proosition together with Definition 4.2./ and Theorem 4.6.b]. It is worth noting that in the just mentioned references, the statements are roven for more general bounded domains. That is, the statements hold for bounded domains of cone-tye in the sense of [6, Definition 4.2.3]. However, bounded Lischitz domains are of cone-tye, see e.g. [2, Sections 4.8, 4.9 and 4.]. Theorem 2.6. Let G be either R d or a bounded Lischitz domain in R d.

58 48 Preliminaries i For, and s, \ N the following equality holds equivalent norms: W s G = B s,g. ii For s, the following equality holds equivalent norms: W s 2 G = B s 2,2G. We resent now three embeddings of Besov saces, which we will frequently use in this thesis. Theorem 2.6. i Let G be an arbitrary domain in R d. Then, for any, and s, s R with s > s the following embedding holds: B s,g B s,g ii Let G be either R d or a bounded Lischitz domain in R d and fix,. Furthermore, assume that α 2 > α > and let τ, τ 2 > fulfil Then the following embeddings hold: = α i τ i d +, i =, 2. B α 2 τ 2,τ 2 G B α τ,τ G L G iii Let O be a bounded Lischitz domain in R d and let < < q <. Then, for { } s > s ε > max, d, 2.48 the following embedding holds: B s q,qo B s ε, O Proof. i This is an immediate consequence of the definition of Besov saces given above. ii For G = R d, the first embedding in 2.47 is roved in [27, Corollary 3.7.], whereas the second embedding can be found in [7, Theorem.73i]. If G is a bounded Lischitz domain, the assertion follows from the case G = R d by using the existence and boundedness of the extension oerator introduced in [] for bounded Lischitz domains. iii Note that since 2.48 holds, we are in the setting of Theorem Thus, using the equivalent characterisation of Besov saces via differences, we immediately obtain B s q,qo B s,qo, by an alication of Hölder s inequality to the moduli of smoothness remember that O is bounded. In order to rove that simultaneously B s,qo B s ε, O holds, we can argue as follows. First one can check the equivalence of the quasi-semi-norm 2.44 and 2 js ω n 2 j, u, O lq, see also [27, Remark 3.2.]. Then, the arguments from the roof of [5, Proosition 2.3.2/2ii] yield the asserted embedding. j N

59 2.3 Function saces 49 r s B s,o ε B t 2,2 O = W t 2 O t { } q max, d q 2 q Figure 2.: Visualisation of Besov saces on bounded Lischitz domains O R d in a DeVore/Triebel diagram. Remark i In order to rove the embeddings 2.47 for bounded Lischitz domains, we only need the existence of a linear and bounded extension oerator as resented in []. Thus, the statement of Theorem 2.6ii stays true for any domain G R d, where a linear and bounded extension oerator from the roer Besov saces on G to the corresonding Besov saces on R d and from L G to L R d exist. ii The short roof of Theorem 2.6iii resented above reveals that, if the Besov saces are defined via differences, then the embedding 2.49 holds for any for any s > s ε > with O relaced by an arbitrary bounded domain G R d. Remark In Figure 2. we use a so-called DeVore/Triebel diagram for a visualisation of the results resented above for bounded Lischitz domains O R d. In this /q, r-diagram, a oint /, s in the first quadrant, 2 stands for the Besov sace B s,o as introduced in Definition The shaded area delimited by the coordinate axes and the ray with sloe d starting at the oint, reresents the range of arameters /q, r, 2 fulfilling { } r > max, d q. In articular, for any /q, r in this area, the alternative characterization of the corresonding Besov sace B r q,qo via differences from Theorem 2.58 holds. As we have seen in Theorem 2.6ii, for = 2, the Besov saces coincide with the Sobolev saces introduced in Subsection In our diagram these saces are reresented by the oints above /2,. The three arrows starting at /, s stand for the three embeddings from Theorem 2.6. In clockwise orientation: Firstly, the arrow ointing to the right stands for 2.49; the ε at the arrowhead indicates that the smoothness decreases by an arbitrarily small ε > in this case. Secondly, the trivial embedding 2.46 is reresented by the arrow ointing straight down. Finally, the third arrow starting at /, s and ointing to the south-west with a sloe d stands for the first embedding in This embedding is a generalization of the well-known Sobolev embedding. Therefore, a ray contained in the shaded area, starting at a oint /, with sloe d is usually

60 5 Preliminaries called a Sobolev embedding line. Note that for,, the non-linear aroximation scale of Besov saces is reresented by a Sobolev embedding line Triebel-Lizorkin saces In this subsection we resent a Fourier analytic definition of the Triebel-Lizorkin saces. They will be used when analysing the relationshi between Sobolev saces with and without weights, resectively, in Chater 4. Definition Let {ϕ k } k N C Rd be a resolution of unity according to i Let <, q <, s R, and Then u F s,q R d := 2 ksq F [ϕ k Fu] /q q LR. d k= F s,qr d := { u S R d : u F s,q R d < } is the Triebel-Lizorkin sace of smoothness order s with summability arameters, q. ii Let G R d be an arbitrary domain. Then, for <, q < and s R, the Triebel-Lizorkin sace F s,qg of order s on G is defined as follows: F s,qg := { u D G : there exists g F s,qr d : g G = u }. It is endowed with the norm u F s,q G := inf g F s,q Rd g G =u g F s,q R d, u F s,qg. Remark For <, q < and s R, the Triebel-Lizorkin sace F s,qg, endowed with the quasi-norm F s,q G, is a quasi-banach sace. If, q <, then F s,q G is a norm and therefore F s,qg, F s,q G is a Banach sace. A roof can be found in [5, Theorem 2.3.3ii] for the case G = R d. For general domains we refer to the roof of [5, Proosition 3.2.3iii]. Note that the assumed smoothness roerty for the boundary of the underlying domain therein does not have any relevance in the roof of the comleteness of the Triebel-Lizorkin saces. The following relationshis of Triebel-Lizorkin saces to Besov and Sobolev saces resectively will be used in this thesis. A roof of assertion i can be found in [6, Theorem 4.6.b]. The second art of the theorem is taken from [7, Proosition.22i] at least the case m N. A roof for the more general case of bounded domains of cone-tye in the sense of [6, Definition 4.2.3] can be found in [6, Theorem 4.2.4]. For m =, 2.5 holds for arbitrary domains G R d instead of O. This follows from the fact that F,2 Rd = L R d. A roof of the latter can be found in [5, Proosition 2.5.6]. Theorem i Let G be an arbitrary domain in R d. Then, for 2 and s R, F s,2g B s,g. ii Let O be a bounded Lischitz domain in R d. Then, for, and m N, F m,2o = W m O equivalent norms. 2.5

61 2.4 Semigrous of linear oerators 5 As a consequence, one obtains the following relationshi between Sobolev saces and Besov saces on bounded Lischitz domains, in the case that the smoothness arameter is a nonnegative integer. Corollary Let O be a bounded Lischitz domain in R d. Then, for 2 and m N, W m O B m,o. 2.4 Semigrous of linear oerators In the semigrou aroach to SPDEs, the equation is rewritten as a vector-valued ordinary stochastic differential equation of the form } dut + AUt dt = F Ut, t dt + ΣUt, t dw H t, t [, T ] U = u. The leading oerator A is usually unbounded and A generates a strongly continuous semigrou on a suitable Banach sace. In this section we recall the terminology from the semigrou theory, focusing first on analytic semigrous. Afterwards, we give a definition of what is sometimes called variational oerators and collect some roerties needed later on. Let E be a Banach sace real or comlex. In general, we call a linear maing B : DB E E, defined on a linear subsace DB of E, a linear oerator with domain DB. B, DB is said to be closed, if its grah {x, Bx : x DB} is a closed subset of E E. It is densely defined, if DB is dense in E, i.e., if DB E = E. The resolvent set of B is the set ρb consisting of all λ C for which there exists a bounded inverse λ B : E, E DB, E of λ B := λid B. The sectrum of B is its comlement σb := C \ ρb. If B is an oerator on a real Banach sace we ut ρb = ρb C and σb := σb C, where B C is a comlexification of B, see, e.g., [98] or [8,. 4ff.] for details. Recall that a family S = {St} t LE of bounded linear oerators is called a C - semigrou or, alternatively, a strongly continuous semigrou, if S = Id, StSs = St + s for any t, s, and lim t Stx x E = for every x E. S = {St} t is called a contraction semigrou, if additionally St LE for all t. The infinitesimal generator, or briefly the generator, of a C -semigrou S = {St} t is the unbounded linear oerator B : DB E E defined by { Stx x DB := x E : lim t t Stx x Bx := lim t t, x DB. } exists in E, By [2, Corollary 2.5], any generator B, DB of a C -semigrou is densely defined and closed. Therefore, the domain DB endowed with the grah norm x DB := x E + Bx E, x DB, becomes a Banach sace. Obviously, if ρb, the grah norm is equivalent to B E. A contraction semigrou S = {St} t is said to be of negative tye, if there exists an ω < such that St LE e ωt, t.

62 52 Preliminaries From [2, Theorem 5.3] one can deduce that, if B, DB is the generator of a semigrou S of negative tye, then A, DA := B, DB is ositive in the sense of [5, Definition.4.], i.e.,, ] ρa and there exists a constant C, such that A λ LE C, λ, ]. + λ A C -semigrou S = {St} t on the Banach sace E := L G with [, is called ositive if for each t, see [52,. 353]. Analytic semigrous f L G, f a.e. on G imlies Stf a.e. on G, Now we collect some definitions and results from the theory of analytic semigrous. We restrict ourselves to the toics we will need in this thesis and refer to the monograhs [2], [52], or [94] for an in-deth treatment of the theory. For σ, π, we write Σ σ := {z C \ {} : arg z < σ}. Definition Let σ, π/2. A C -semigrou S = {St} t LE acting on a Banach sace E is called analytic on Σ σ if A. S extends to an analytic function S : Σ σ LE, z Sz; A2. lim z,z Σσ Szx = x for every x E; A3. Sz Sz 2 = Sz + z 2 for z, z 2 Σ σ. We say that a C -semigrou S is analytic, if it is analytic on Σ σ for some σ, π/2. If, in addition, A4. z Sz LE is bounded in Σ σ for every < σ < σ, we call S a bounded analytic semigrou. Next, we introduce the notion H -calculus of a sectorial oerator. Originally develoed by McIntosh and collaborators [6,, 95], it has found various alications in the context of stochastic artial differential equations. Our definition is taken from [22]. Let A, D A be the generator of a bounded analytic semigrou on a Banach sace E. Then, see, e.g., [8, Proosition I..4.], σa Σ σ for some σ, π/2, and for all σ σ, π, su zz A LE <, z C\Σ σ i.e., in the terminology used e.g. in [6, Chater 2], A is a sectorial oerator. Let H Σ σ denote the Banach sace of all bounded analytic functions f : Σ σ C endowed with the suremum norm. Furthermore, H Σ σ denotes the subsace of H Σ σ, consisting of all functions satisfying z ε fz C + z 2 ε, z Σ σ, 2.5

63 2.4 Semigrous of linear oerators 53 for some ε >. For f H Σ σ and σ σ, σ, due to 2.5, the LE-valued Bochner integral fa := fzz A dz 2πi Σ σ converges absolutely. Furthermore, it is indeendent of σ. We say that the oerator A, DA admits a bounded H Σ σ -calculus if there exists a constant C, such that fa LE C f := C su z Σ σ fz, f H Σ σ. The infimum of all σ such that A, DA admits a bounded H Σ σ -calculus is called angle of the calculus. The following two results are mentioned in [22]. Theorem 2.69 [7, Corollary 5.2]. Let G R d be an arbitrary domain and let [,. If A, D A is the generator of a ositive analytic contraction semigrou on L G, then, A, DA admits a bounded H -calculus of angle less than π/2. The next result can be derived from [6, Corollary 3.5.7]. Theorem 2.7. If ρa and A, DA admits a bounded H -calculus of angle less than π/2, then, A has bounded imaginary owers and su t [,] A it LE <. Variatonal oerators Let V,, V be a searable real Hilbert sace. Furthermore, let a, : V V R be a continuous, symmetric and ellitic bilinear form. This means that there exist two constants δ ell, K ell >, such that for arbitrary u, v V, the bilinear form fulfils the following conditions: δ ell u 2 V au, u, au, v = av, u, au, v K ell u V v V Then, by the Lax-Milgram theorem, the oerator A : V V v Av := av, 2.53 is an isomorhism between V and its dual sace V. Let us now assume that V is densely embedded into a real Hilbert sace E,, E via a linear embedding j. Then, the adjoint ma j : E V of j embeds E densely into the dual V of V. If we identify the Hilbert sace E with its dual E via Riesz s isometric isomorhism E u Ψu := u, E E, we obtain a so called Gelfand trile V, E, V, We have V j E Ψ = E j V jv, jv 2 E = j Ψjv, v 2 V V for all v, v 2 V It is worth noting that, although V is a Hilbert sace, at this oint we do not identify V and its dual V via the Riesz isomorhism in V. This would not match with 2.54 and Here, the vector sace V is considered as a subsace of V by means of the embedding j Ψj, where Ψ is the Riesz isomorhism for E and not for V. In this setting, we can consider the oerator

64 54 Preliminaries A : V V as an in general unbounded oerator on the intermediate sace E. Therefore, we set DA := DA; E := {u V : Au j ΨE}, and define the oerator à : Dà := jda; E E E u Ãu := Ψ j A j u. The unbounded linear oerator à with domain D à := Dà is sometimes called variational. It is densely defined, since E is densely embedded in V and A is isomorhic. Furthermore, the symmetry of the bilinear form a, imlies that Ã, and therefore also Ã, is self-adjoint. That is, Ã, Dà = Ã, DÃ, where à : Dà E E denotes the adjoint oerator defined by Dà := { u E : u 2 E : Ãu, u E = u, u 2 E for all u DÃ}, à u := u 2, u DÃ, where u 2 E fulfils Ãu, u E = u, u 2 E for all u Dà and is unique by the density of Dà in E. At the same time, since A : V V is an isomorhism, the oerator à : E, E D Ã, E, defined by à := j A j Ψ is the bounded inverse of Ã. Thus, ρ à and, therefore, Ã, D à is a closed oerator on E. Moreover, by 2.52 and the definition of Ã, for arbitrary λ > and u D Ã, λid Ãu E λu, u E + au, u λ u E, i.e., à is dissiative, see [2, Theorem 4.2]. Therefore, by the Lumer-Philis Theorem, see in articular [2, Corollary 4.4], Ã, D à is the generator of a contraction semigrou on E. By making slight abuse of notation, we sometimes write A instead of Ã, esecially when j = Id.

65 55 Chater 3 Starting oint: Linear SPDEs in weighted Sobolev saces In this chater we resent and discuss the main results from the L -theory of SPDEs on nonsmooth domains as develoed in [75] within the analytic aroach. It is the starting oint for our regularity analysis, roviding existence and uniqueness of solutions for a wide class of linear SPDEs on general bounded Lischitz domains O R d. The solutions are elements of certain classes H γ O, T of redictable -Bochner integrable Hγ O-valued stochastic rocesses. Since in the next chater we will be able to rove a general embedding of weighted Sobolev saces into Besov saces from the scale, this L -theory turns out to be tailor-made for our regularity analysis in the non-linear aroximation scale. A combination of the existence results from this section with the aforedmentioned embedding will lead to a statement about the satial Besov regularity for linear SPDEs, as stated and roven in Section 5.. In order to obtain similar satial regularity results for semi-linear equations, in Section 5.2 we will also extend the L -theory from [75] to a class of semi-linear SPDEs. Furthermore, while analysing the Hölder regularity of the aths of the solution rocess in Chater 6, we resent an extension to an L q L -theory for the heat equation on bounded Lischitz domains. That is, we rove the existence of a solution in certain classes H γ,q O, T of redictable q-bochner integrable H γ O-valued stochastic rocesses, exlicitly allowing the summability arameter q in time and with resect to ω Ω to be greater than the summability arameter used to measure the smoothness in sace direction. We slit this chater in two arts: In Section 3. we discuss the saces H γ,q O, T, whereas Section 3.2 is concerned with those fragments from the L -regularity of SPDEs develoed in [75] which are relevant for our analysis. Before we start our exosition, we fix some notation and secify the class of equations considered in this chater. Let O be a bounded Lischitz domain in R d. Ω, F, P denotes a comlete robability sace and T > is a finite time horizon. Moreover, wt k t [,T ], k N, is a sequence of indeendent real-valued standard Brownian motions with resect to a normal filtration F t t [,T ] on Ω, F, P. We write shorthand Ω T instead of Ω [, T ]. We consider equations of the form du = a ij u x i x j + bi u x i + cu + f dt + σ ik u x i + µ k u + g k } dwt k on Ω T O, 3. u = u on Ω O, where the coefficients a ij, b i, c, σ i,k and µ k, for i, j {,..., d} and k N, are assumed to fulfil certain assumtions. We want to emhasize that in this thesis, for a better readability, we use

66 56 Starting oint: Linear SPDEs in weighted Sobolev saces the so-called summation convention on the reeated indices i, j, k when writing down equations, see also Remark 3.2i as well as Remark 5.2i. In order to state the assumtions on the coefficients, we need some additional notation. For x, y O we write ρx, y := ρx ρy, ρx being the distance of a oint x O to the boundary O, i.e., ρx = distx, O. For α R, δ, ] and m N we set [f] α m := su ρ m+α x D m fx, x O [f] α m+δ := su f α m := x,y O β =m m l= ρ m+α x, y Dβ fx D β fy x y δ, [f] α l and f α m+δ := f α m + [f] α m+δ, whenever it makes sense. We use the same notations for l 2 -valued functions just relace the absolute values in the above definitions by the l 2 -norms. Furthermore, let us fix an arbitrary function η : [, [,, vanishing only on the set of non-negative integers, i.e., ηj = if, and only if, j N. We set t + := t + ηt. These notations at hand, we resent the assumtions on the coefficients of Eq. 3., which are identical with the ones in [75, Assumtion 2.], see also [73, Assumtions 2.5 and 2.6]. The recise solution concet for equations of the tye 3. fulfilling these assumtions will be secified in Definition 3.. Assumtion 3.. i For any fixed x O, the coefficients a ij,, x, b i,, x, c,, x, σ ik,, x, µ k,, x : Ω T R are redictable rocesses with resect to the given normal filtration F t t [,T ]. ii Stochastic arabolicity There are constants δ, K,, such that for all ω, t, x Ω T O and λ R d, d δ λ 2 ã ij ω, t, xλ i λ j K λ 2, where ã ij := a ij 2 σi, σ j l2 iii For all ω, t Ω T, i,j= for i, j {,..., d}. a ij ω, t, γ + + b i ω, t, γ + + cω, t, 2 γ + + σ ik ω, t, k N γ+ + + µ k ω, t, k N γ+ + K. iv The coefficients a ij and σ i are uniformly continuous in x O, i.e., for any ε > there is a δ = δε >, such that a ij ω, t, x a ij ω, t, y + σ i ω, t, x σ i ω, t, y l2 ε, for all ω, t Ω T, whenever x, y O with x y δ.

67 3. Stochastic arabolic weighted Sobolev saces H γ,q G, T 57 v The behaviour of the coefficients b i, c and µ can be controlled near the boundary of O in the following way: } lim {ρx b i ω, t, x + ρ 2 x cω, t, x + ρx µω, t, x l2 =. ρx x O su ω Ω t [,T ] 3. Stochastic arabolic weighted Sobolev saces H γ,q G, T The analysis of SPDEs in the analytic aroach takes lace in the saces H γ,q G, T consisting of certain redictable q-bochner integrable H γ G-valued stochastic rocesses. In this section we resent the recise definition of these saces. Furthermore, we collect some of their roerties which are relevant for our analysis later on. We start with common notations for redictable rocesses taking values in weighted Sobolev saces, which are frequently used in the analytic aroach. In the sequel, we write P T for the F t t [,T ] -redictable σ-field on Ω T and P T := P dt. Unless exlicitly stated otherwise, G denotes an arbitrary domain in R d with non-emty boundary. Definition 3.2. Let G be a domain in R d with non-emty boundary. For, q, and γ, θ R we define H γ,q G, T := L qω T, P T, P T ; H γ G, H γ,q G, T ; l 2 := L q Ω T, P T, P T ; H γ G; l 2, U γ,q G := L qω, F, P; H γ 2/q 2/q G. If = q we also write H γ G, T, Hγ G, T ; l 2 and U γ G instead of Hγ, G, T, Hγ, G, T ; l 2 and U γ, G resectively. Unless exlicitly stated otherwise, from now on we assume that [2,, q [2,, γ R, θ R. Definition 3.3. Let G be a domain in R d with non-emty boundary,, q [2, and γ, θ R. We write u H γ,q γ,q G, T if, and only if, u Hγ,q G, T, u U G, and there exist some f H γ 2,q + G, T and g Hγ,q G, T ; l 2 such that du = f dt + g k dw k t in the sense of distributions. That is, for any ϕ C ut,, ϕ = u,, ϕ + t fs,, ϕ ds + G, with robability one, the equality k= t g k s,, ϕ dw k s 3.2 holds for all t [, T ], where the series is assumed to converge uniformly on [, T ] in robability. In this situation we write Du := f and Su := g for the deterministic and for the stochastic art of u, resectively. The norm in H γ,q G, T is defined as u H γ,q G,T := u H γ,q G,T + Du H γ 2,q If = q we also write H γ + G,T + Su H γ,q G, T instead of Hγ, G, T. G,T ;l 2 + u U γ,q G. 3.3

68 58 Starting oint: Linear SPDEs in weighted Sobolev saces Remark 3.4. i The hrase..., with robability one, the equality 3.2 holds for all t [, T ],... in the definition above, means: There exists a set Ω F with PΩ =, such that for any fixed ω Ω, Equality 3.2 is fulfilled for all t [, T ]. In articular, the P-null set where 3.2 might not hold, does not deend on t [, T ]. ii Relacing G by R d and omitting the weight arameters in the definitions above, we obtain the saces H γ,q T = H γ,q R d, T, H γ,q T ; l 2 = H γ,q R d, T ; l 2, U γ,q = U γ,q R d, and H γ,q T as introduced in [83, Definition 3.5]. The latter are denoted by H γ,q T in [82]; if q = they coincide with the saces HT γ introduced in [8, Definition 3.]. Lemma 3.5. Let G be a domain in R d with non-emty boundary,, q [2, and γ, θ R. i If g H γ,q G, T ; l 2 then, for any ϕ C G, the series g k s,, ϕ dws k 3.4 k= from 3.2 converges in L 2 Ω; C[, T ]; R. ii The air Du, Su H γ 2,q + G, T Hγ,q G, T ; l 2 in Definition 3.3 is uniquely determined by u H γ,q G, T. iii H γ,q G, T is a Banach sace. Proof. i The convergence of the sum 3.4 in L 2 Ω; C[, T ]; R has been roven in [9, Section 3.5]. However, we need to correct a minor mistake in the first equality in the last estimate on age 9 in [9]. Let {ξ n : n Z} C G with n Z ξ n = on G fulfil 2.3 for some c > and k >. Furthermore, fix a sequence { ξ n : n Z} C G, also fulfilling 2.3 with a ossibly different k >, such that su ξ n = for all n Z. {ξn} By Remark 2.48ii and iii, it is clear that we can construct such sequences. Now, by mimicking, the roof in [9, Section 3.5] with g κ n := ξ n c n g κ c n and φ n := ξ n c n φc n for n Z and κ N, the assertion follows. ii This assertion follows by using the arguments from [8, Remark 3.3]. iii By ii we know that the norm 3.3 is well-defined. The comleteness can be roven by following the lines of [83, Remark 3.8] with R d + relaced by G. Proosition 3.6. Let G be a domain in R d with non-emty boundary,, q [2, and γ, θ R. Fix g H γ,q G, T ; l 2 and let Φ : H γ G; l 2 Γl 2, H γ G be the isomorhism introduced in Theorem Then, the Γl 2, H γ G-valued stochastic rocess Φ g := Φ g 3.5 is L q -stochastically integrable with resect to the l 2 -cylindrical Brownian motion l 2 h W l2 th := wt k e k, h l2 L 2 Ω, t [, T ]. Moreover, in C[, T ]; R. k= k= g k s,, ϕ dw k s = Φ g s dw l2 s, ϕ P-a.s. 3.6

69 3. Stochastic arabolic weighted Sobolev saces H γ,q G, T 59 Proof. Fix g H γ,q G, T ; l 2 = L q Ω T, P T, P T ; H γ G; l 2. Then, since Φ is bounded from H γ G; l 2 to Γl 2, H γ G, we have Φ g L q Ω T, P T, P T ; Γl 2, H γ G. 3.7 In articular, Φ g is an l 2 -strongly measurable F t t [,T ] -adated rocess. Also, Φ g belongs to L q Ω; L 2 [, T ], H scalarly, and, since H γ G is a umd Banach sace with tye 2, comare Lemma 2.5, Φ g is stochastically integrable with resect to the l 2 -cylindrical Brownian motion W l2 t t [,T ], cf. Theorem Consequently, by Theorem 2.29, see also Remark 2.3, Φ g reresents a random variable R Φg L F q Ω; ΓL 2 [, T ]; H, E. In articular, by [2, Proosition 4.3], there exists a continuous F t t [,T ] -adated version of the H γ G-valued stochastic rocess t Φ g s dw l2 s, t [,T ] which, by the Burkholder-Davis-Gundy inequality roved in [2, Theorem 4.4], satisfies [ t ] q [ E su Φ g s dw l2 s C E R Φg q ΓL 2 [,T ];l 2 ].,H γ G t [,T ] H γ G Using Embedding 2.2 and the fact that Φ is an isomorhism between the saces H γ G; l 2 and Γl 2, H γ G, see Theorem 2.54, this leads to [ t ] q [ T ] E su Φ g s dw l2 s C E Φ g q Γl 2,H γ G dt C g q H γ,q G,T ;l 2. t [,T ] H γ G Fix two arbitrary ositive integers m n and set g m,n := gm,n k k N with { } g k gm,n k, if k {m,..., n} := H γ,q, else G, T ; l 2. Then, by the same arguments as above, the stochastic rocesses n t t g k s, dws k and Φ gm,n s dw l2 s k=m t [,T ] t [,T ] have continuous versions, which, by Theorem 2.3 coincide P-a.s. in C[, T ]; H γ G. Moreover, [ t ] q E su Φ gm,n s dw l2 s C g m,n q H γ,q G,T ;l 2. t [,T ] H γ G The right hand side converges to zero as m, n, since g H γ,q G, T ; l 2. Consequently, the series g k s, dws k k= converges in the Banach sace L q Ω; C[, T ]; H γ G, and, by another alication of Theorem 2.3, k= g k s, dw k s = Φ g s dw l2 s P-a.s.

70 6 Starting oint: Linear SPDEs in weighted Sobolev saces in C[, T ]; H γ G. For ϕ C G Hγ G H γ,θ G with / + / = and θ/ + θ / = d, see Lemma 2.45viii and ii together with Remark 2.49, the linear oerator, ϕ : H γ G R, v v, ϕ, is bounded. Thus, for any t [, T ], and, therefore, k= k= t g k s, dws k, ϕ = k= g k s,, ϕ dw k s = t g k s,, ϕ dw k s P-a.s., Φ g s dw l2 s, ϕ P-a.s. 3.8 in C[, T ]; R after ossibly changing to suitable versions of the rocesses. Remark 3.7. It is worth noting that similar arguments as in the roof of Proosition 3.6 yield an alternative roof of the convergence of the series 3.4 in L 2 Ω; C[, T ]; R, and even in L q Ω; C[, T ]; R. Besides the analysis of the satial regularity of solutions to SPDEs, in this thesis we are also interested in the Hölder regularity of the aths of the solution rocesses. Since our solutions will always be contained in H γ,q G, T, results on the Hölder regularity of the elements of these saces are of major interest. For = q [2, the following result concerning the regularity of the aths of an element of H γ G, T, considered as a stochastic rocess with values in weighted Sobolev saces, can be found in [75, Theorem 2.9]. Its roof strongly relies on [83, Corollary 4.2 and Remark 4.4], which are corresonding results on the whole sace R d. Note that the statement of [75, Theorem 2.9] is formulated only for a certain class of bounded nonsmooth domains. However, the arguments go through for arbitrary domains with non-emty boundary. Theorem 3.8. Let G R d be an arbitrary domain with non-emty boundary, γ R, and θ R. i Let 2/ < β < β. Then E [ u ] C T β β/2 u C β/2 / [,T ];H γ+2 β β G H γ+2 where C, is a constant indeendent of T and u. G,T, ii Let [2,. Then [ E su t [,T ] u H γ+ G ] C u H γ+2 G,T, where the constant C deends on d,, γ, θ, G, and T. The function T CT is non-decreasing. In articular, there exists a constant C,, such that for any u G, T and all t [, T ], H γ+2 u H γ+ G,t C t u H γ+2 G,sds. Remark 3.9. In Chater 6 we will need a generalization of Theorem 3.8i for the aths of elements of H γ,q O, T with q in order to obtain Hölder regularity of the solution to the stochastic heat equation, considered as a rocess taking values in the Besov saces from the scale ; see Theorem 6.. Its roof will require more involved arguments.

71 3.2 An L -theory of linear SPDEs on bounded Lischitz domains An L -theory of linear SPDEs on bounded Lischitz domains In this section we resent the main existence and uniqueness result of the L -theory of linear SPDEs on bounded non-smooth domains develoed recently in [75]. Since in this thesis we are only interested in SPDEs on bounded Lischitz domains O R d, we will restrict ourselves to this case, and consider equations of the form 3. with coefficients fulfilling Assumtion 3.. We first secify the solution concet which will be used in this thesis. For i, j {,..., d} and a D O-valued function u on Ω T we use the common notation u x i := u x iω, t := uω, t x i and u x i x j := u x i x jω, t := uω, t x i x j, ω, t Ω T, resectively. Definition 3.. Let O be a bounded Lischitz domain in R d. Given γ R, let a ij, b i, c, σ ik and µ k, where i, j {,..., d} and k N, satisfy Assumtions 3.. A stochastic rocess u H γ,q O, T is called a solution of Eq. 3. in the class Hγ,q O, T if, and only if, u O, T with H γ,q u, = u, Du = d d d a ij u x i x j + b i u x i + cu + f, and Su = σ ik u x i + µ k u + g k i,j= in the sense of Definition 3.3. i= Remark 3.. In this thesis, if we call an element u H γ,q O, T a solution of Eq. 3., we mean that u is a solution of Eq. 3. in the class H γ,q O, T. Remark 3.2. i As already mentioned, throughout this thesis, for a better readability, we omit the notation of the sums i,j and k when writing down equations and use the so-called summation convention on the reeated indices i, j, k. Thus the exression du = a ij u x i x j + bi u x i + cu + f dt + σ ik u x i + µ k u + g k dw k t is shorthand for d d d du = a ij u x i x j + b i u x i + cu + f dt + σ ik u x i + µ k u + g k dwt k i,j= in the sense of Definition 3.3. i= ii The solution concet resented in Definition 3. is a natural generalization of the definition given in [75]. Therein, only the case = q is considered. However, we will need this generalization later on in Chater 6. The main existence and uniqueness results for equations on bounded Lischitz domains roven in [75], see Theorem 2.2, Remark 2.3 as well as Theorem 2.5 therein, can be summarized as follows. Theorem 3.3. Let O be a bounded Lischitz domain in R d, and γ R. For i, j {,..., d} and k N, let a ij, b i, c, σ ik, and µ k be given coefficients satisfying Assumtion 3. with suitable constants δ and K. i For [2,, there exists a constant κ,, deending only on d,, δ, K and O, such that for any θ d + 2 κ, d κ, f H γ + O, T, g Hγ+ O, T ; l 2 and u U γ+2 O, Eq. 3. has a unique solution u in the class Hγ+2 O, T. For this solution u C f H γ+2 O,T H γ + O,T + g + u, 3.9 H γ+ O,T ;l 2 U γ+2 O i= i= k N

72 62 Starting oint: Linear SPDEs in weighted Sobolev saces where the constant C deends only on d,, γ, θ, δ, K, T and O. ii There exists > 2, such that the following statement holds: If [2,, then there exists a constant κ,, deending only on d,, δ, K and O, such that for any θ d κ, d + κ, f H γ + O, T, g Hγ+ O, T ; l 2 and u U γ+2 O, Eq. 3. has a unique solution u in the class H γ+2 O, T. For this solution, estimate 3.9 holds. Remark 3.4. i For = 2 there is no difference between i and ii in Theorem 3.3. In articular, existence of solutions in H γ 2,d O, T L 2Ω T ; W 2 O is guaranteed under suitable assumtions on the data of the equation. Things are different if > 2. Since we do not know the recise value of κ = κ d,, O, we can not exect that d d+ 2 κ, d+ 2 + κ if > 2. Thus, Theorem 3.3i does not yield the existence of a solution u H γ,d O, T, even if the data of the equation are assumed to be arbitrarily smooth. However, Theorem 3.3ii guarantees that at least for certain > 2, i.e., for [2, with some > 2, a solution u H γ,d O, T L Ω T ; W O exists under suitable assumtions on the data. In general, is not very high due to a counterexamle of N.V. Krylov, which can be found in [75, Examle 2.7]. It is shown therein that for any > 4, there exists a bounded Lischitz domain O R 2 and a function f L [, T ]; L O such that a solution of the deterministic heat equation du = u + f } dt on Ω T O, u = on Ω O, fails to be in L [, T ]; L,d O, see [75, Examle 2.7]. Thus, if we do not secify any further roerties of the domain O excet the fact that it is bounded and Lischitz, the assertion of Theorem 3.3ii holds only with 4. ii Assume that the bounded O is not only Lischitz but of class C u, see Definition 2.3. Then, if σ =, the statement of Theorem 3.3i holds for any [2, and θ R fulfilling d < θ < d + ; 3. see [72, Theorem 2.9 together with Remark 2.7]. That is: Let O be a bounded Cu-domain in R d, and assume that a ij, b i, c, and µ k with i, j {,..., d} and k N, are given coefficients satisfying Assumtion 3. for some γ R with σ = and suitable constants δ and K. Then for any [2, and any θ R fulfilling 3., Eq. 3. with u U γ+2 O, f H γ + O, T and g Hγ+ O, T ; l 2 has a unique solution u in the class H γ+2 O, T. Moreover, the estimate 3.9 holds. iii As mentioned in [86, Remark 3.6], if O is relaced by R d +, the statement of Theorem 3.3 fails to hold for θ d + and θ d. Therefore, in general, we do not exect that the κ and κ can be chosen to be greater than one. Exlicit counterexamles on general bounded Lischitz domains are yet to be constructed. iv Remember that, as mentioned in the introduction, in this thesis we are interested in equations with zero Dirichlet boundary conditions. However, it is not immediately clear in which sense solutions in the class H γ,q O, T fulfil such boundary conditions and therefore can be understood as solutions to Eq... This will be clarified in Chater 4, see in articular Remark 4.3.

73 3.2 An L -theory of linear SPDEs on bounded Lischitz domains 63 v The statement of Theorem 3.3 is roved in [75] not only for bounded Lischitz domains but for any bounded domain G R d which admits the Hardy inequality, i.e., for which ρg x ϕx 2 dx C ϕ x x 2 dx, for all ϕ C G, 3. G G with a constant C which does not deend on ϕ C G; the solution concet is analogous to the one introduced in Definition 3. with G instead of O. It is known that bounded Lischitz domains admit the Hardy inequality, see, e.g., [99] for a roof. The analysis in [75] is done in the framework of the analytic aroach. As ointed out in the introduction, alternatively, equations of the tye 3. can be consider within a semigrou framework. Since many contributions to the regularity analysis of SPDEs use this semigrou aroach, it is imortant to know whether the solution concet used in this thesis matches with the ones used therein. In what follows we resent a secific setting where a solution to Eq. 3. in the sense given above is a weak solution of the corresonding Cauchy roblem in the sense of Da Prato and Zabczyk [32, Section 5..], which is the common solution concet used within the semigrou aroach. We restrict ourselves to the Hilbert sace case i.e., = 2 and articularly to equations in L 2 O. A generalization to Banach saces i.e., > 2 will be discussed in Chater 6. We start by defining what is called a weak solution in the semigrou framework. To this end, we first fix our secific setting. Assumtion 3.5. i The oerator A, D A generates a strongly continuous semigrou { St } t in L 2O. ii W Q t t [,T ] is a Q-Wiener rocess in a real Hilbert sace H,, H adated to the given normal filtration F t t [,T ] with covariance oerator Q L H. iii f : Ω T L 2 O is a redictable stochastic rocess with P-a.s. Bochner integrable trajectories. iv B LH, L 2 O. v u : Ω L 2 O is an F -measurable random variable. Under these conditions we can define what is called a weak solution of the L 2 O-valued SDE } dut + Aut dt = ft dt + B dw Q t, t [, T ], 3.2 u = u, in the semigrou aroach of Da Prato and Zabczyk [32, Section 5..]. Definition 3.6. Let A, DA, W Q t t [,T ], f, B and u fulfil Assumtion 3.5. Then, an L 2 O-valued stochastic rocess u = ut t [,T ] is a weak solution of Eq. 3.2, if it has the following roerties: i u has P-a.s. Bochner integrable trajectories. ii For all ζ DA and t [, T ], we have ut, ζ L2 O = u, ζ L2 O t + t us, A ζ L2 O ds fs, ζ L2 O ds + BW Q t, ζ L2 O P-a.s. 3.3

74 64 Starting oint: Linear SPDEs in weighted Sobolev saces Remark 3.7. i In arts of the literature such as [4] a weak solution in the sense of Definition 3.6 is called analytically weak solution. ii Tyically, if H L 2 O and B is the identity oerator from H into L 2 O, we omit B in 3.2 and 3.3. Now we can rove exemlarily that in a secific setting the solution to Eq. 3. is a weak solution of the corresonding L 2 O-valued SDE of the form 3.2. Proosition 3.8. Assume that the coefficients a ij are constant and symmetric, i.e., they do not deend on ω, t, x Ω T O and a ij = a ji for i, j {,..., d}. Furthermore, let Assumtion 3. be fulfilled with vanishing b i, c, σ ik, and µ k, for i {,..., d} and k N. Fix f H 2,d O, T, g H 2,d O; l 2 and u U2,d 2 O. Then, the solution u H2 2,d O, T of Eq. 3., which exists by Theorem 3.3, is the unique weak solution of Eq. 3.2, where B is the identity oerator from H2,d O into L 2O, and d A, D A := a ij u x i x {u j, W 2 O : i,j= W Q t := d i,j= } a ij u x i x j L 2O, 3.4 g k wt k, t [, T ], 3.5 k= is an H2,d O-valued Q-Wiener rocess with covariance oerator Q L H2,d O given by Qv = g k, v H 2,d O gk, v H2,d O. k= Proof. In the given setting, by Theorem 3.3, the equation du = a ij u x i x j + f } dt + g k dwt k on Ω T O, u = u on Ω O 3.6 has a unique solution u H 2 2,d O, T. In articular, for all ϕ C equality O, with robability one, the ut,, ϕ = u,, ϕ + t d a ij u x i x js, + fs,, ϕ ds + i,j= k= t g k, ϕ dw k s 3.7 holds for all t [, T ]. Fix ζ DA W 2 O. Then, there exists a sequence ϕ n n N C O aroximating ζ in W 2 O. We fix such a sequence and show that for any t [, T ], each side of 3.7 with ϕ n instead of ϕ converges P-a.s. to the corresonding side of 3.3 with with A, B and W Q as defined in 3.4 and 3.5. This obviously would rove the assertion of the theorem. We start with the left hand sides. Since u H 2 2,d 2 O, T, by Theorem 3.8, it has a version with continuous aths, if considered as a rocess with state sace H2,d O. Consequently, with robability one, ut, L 2 O for all t [, T ]. Thus, with robability one, lim ut,, ϕ n = lim ut,, ϕ n L2 O = ut,, ζ L2 O for all t [, T ]. n n We continue with the right hand sides. Since u U 2 2,d O = L 2Ω, F, P; H 2,d O, u L 2 O P-a.s., and lim u, ϕ n = lim u, ϕ n L2 O = u, ζ L2 O n n P-a.s.

75 3.2 An L -theory of linear SPDEs on bounded Lischitz domains 65 Furthermore, since f H 2,d O, T = L 2Ω T, P T, P T ; L 2 O, using Tonelli s theorem and the dominated convergence theorem, we obtain that with robability one, lim n t fs,, ϕ n ds = t fs,, ζ L2 O ds holds for all t [, T ] after ossibly assing to a subsequence. Moreover, since g H2,d O and the Brownian motions wt k t [,T ], k N, are indeendent, an alication of Doob s inequality yields [ ] E su g k wt k 2 C T g 2 H2,d O;l 2. t [,T ] k= H 2,d O In articular, the series k= gk w k converges in L 2 Ω; C[, T ]; H2,d O. Thus, using the roerties of Itô s one-dimensional stochastic integral, yields that with robability one, lim n k= It remains to rove that for all t [, T ], t lim n g k, ϕ n dws k = g k w k, ζ L2 O. k= d a ij u x i x js,, ϕ t n ds = u, A ζ L2 O ds P-a.s., 3.8 i,j= which can be roven by reasoning as follows: The oerator A, D A in 3.4 can be introduced as the variational oerator Ã, D Ã in Section 2.4 starting with the bilinear form a : W 2 O W 2 O R u, v au, v := O i,j= d a ij u x iv x j dx. 3.9 Thus, it is a densely defined, closed, self-adjoint and dissiative oerator generating a contraction semigrou {St} t [,T ] on L 2 O. Since u x H 2,d O, T H 2,d O, T, which easily follows by 2.28 and the fact that u H 2 2,d 2 O, T, the equality a ij u x i x j, ϕ n = a ij u x i, ϕ n x j = a ij u x iϕ n x j dx = a ij u x i, ϕ n x j L2 O holds P T -a.e. for all i, j {,..., d} and n N. Therefore, and, consequently, lim n i,j= lim n i,j= d a ij u x i x j, ϕ n = O d a ij u x i, ζ x j L2 O P T -a.e., i,j= d a ij u x i x j, ϕ n = au, ζ = u, Aζ L2 O = u, A ζ L2 O P T -a.e., where a, is given by 3.9. Now one can use the dominated convergence theorem to rove that 3.8 holds for all t [, T ]. In summary, u is a weak solution of the corresonding infinite-dimensional SDE of the tye 3.2. The uniqueness follows from [32, Theorem 5.4].

76 66 Starting oint: Linear SPDEs in weighted Sobolev saces

77 67 Chater 4 Embeddings of weighted Sobolev saces into Besov saces In this chater we analyse the regularity within the non-linear aroximation scale B α τ,τ O, τ = α d +, α >, of elements from the weighted Sobolev saces H γ O introduced in Subsection Our main goal is to rove that for γ, ν > and 2, the sace H γ,d ν O is embedded into the Besov saces Bτ,τ α O from for certain α < α max = α max γ, ν, d. As before, also in this chater, O denotes a bounded Lischitz domain in R d. Remember that, if we want to clarify whether adative wavelet methods for solving SPDEs bear the otential to be more efficient than their uniform alternatives, we need to analyse the regularity of the corresonding solution in the scale, cf. Section.. In Chater 3 we have seen that there exists a quite satisfactory solvability theory for a wide class of linear SPDEs within the saces H γ O, T = Hγ, O, T with suitable arameters γ R, [2, and θ R cf. Theorem 3.3. For q, [2, and γ, θ R, the elements of H γ,q O, T are L q-integrable stochastic rocesses taking values in H γ O = d θ Hγ,d ν O with ν = +. Thus, a combination of the embedding mentioned above with Theorem 3.3 yields a statement about the satial regularity of linear SPDEs within the scale of Besov saces Theorem 5.2. Even more, this embedding shows that to a certain extent the regualrity analysis for SPDEs in terms of the scale can be traced back to the analysis of such equations in terms of the saces H γ,q O, T see Theorem 5.. Our results also have an imact on the regularity analysis of deterministic artial differential equations. E.g., the results from [76] on the weighted Sobolev regularity of deterministic arabolic and ellitic equations on bounded Cu-domains will automatically lead to regularity results in the scale for these equations. Using the mentioned embedding, one can also derive Besov regularity estimates for degenerate ellitic equations on bounded Lischitz domains as considered, e.g., in [93]. Our results can also be seen as an extension of and a sulement to the Besov regularity results for ellitic equations in [38] and [34 36, 4, 63]. It is worth noting that first results on the regularity in the scale of solutions to deterministic arabolic equations have been obtained in [3], see also the rearative results in [4, 5]. We choose the following outline. First, we will discuss the relationshi between weighted Sobolev saces with and without weights Section 4.. As we have mentioned in Subection 2.3.3,

78 68 Embeddings of weighted Sobolev saces into Besov saces H,d ν m O = W m O for m N, see Lemma 2.5. We generalize this result and rove a general embedding of weighted Sobolev saces into Sobolev saces without weights Proosition 4.. Moreover, we enlighten the fact that, for the range of arameters γ and ν relevant for SPDEs, the elements of H γ,d ν O have zero boundary trace. In articular, this justifies saying that the solutions considered in this thesis fulfil a zero Dirichlet boundary condition. In the intermediate Section 4.2, we recall some fundamental results on the wavelet decomosition of Besov saces. They will be used in Section 4.3, when roving the embedding mentioned above of weighted Sobolev saces into Besov saces from the scale Theorem 4.3. The roof of this theorem is slit into two arts: In Part One, we restrict ourselves to integer γ N. In Part Two the comlex interolation method of A.P. Calderón and its extension to suitable quasi-banach saces by O. Mendez and M. Mitrea [96] is alied in order to rove the embedding for fractional γ R + \ N. In Section 4.4, we resent an alternative roof of Theorem 4.7, which does not require any knowledge about comlex interolation in quasi-banach saces. 4. Weighted Sobolev saces and Sobolev saces without weights We start with a general embedding of weighted Sobolev saces into the closure of C O in the Sobolev saces without weights. Proosition 4.. Let γ, ν, and [2,. Then the following embedding holds: H γ γ ν,d ν O W O. 4. Proof. Since C O is densely embedded in the weighted Sobolev saces, see Lemma 2.45ii, it is enough to rove that H γ γ ν,d ν O W O for the articular arameters. We start the roof by considering the case where γ = ν, i.e., we rove that for γ > and [2, we have H γ,d γ O W γ O. 4.2 For γ = m N this follows from Lemma 2.5. In the case of fractional γ R + \ N we argue as follows. Let γ = m + η with m N and η,. By Lemma 2.45v, Thus, since 4.2 holds for the integer case, By Theorem 2.66ii this yields H m+η,d m+η O = [ H,d m m O, Hm+,d m+ O] η. H m+η,d m+η O [ W m O, W m+ O ] η. H m+η,d m+η O [ F m,2o, F m+,2 O ] η. Since the Triebel-Lizorkin saces constitute a scale of comlex interolation saces, see, e.g., [7, Corollary.], this leads to H m+η m+η,d m+η O F,2 O. Therefore, since F m+η,2 O B, m+η O by Theorem 2.66i, H m+η,d m+η O Bm+η, O = W m+η O,

79 4. Weighted Sobolev saces and Sobolev saces without weights 69 where the last equality follows from Theorem 2.6i. Thus, Embedding 4.2 is roven. The embedding 4. for γ ν follows now by using standard arguments. Indeed, since γ γ ν we have H γ,d ν O Hγ ν,d ν O, see [93, age 3]. Furthermore, the boundedness of the domain O and the fact that d ν d γ ν imly H γ ν,d ν O Hγ ν,d γ ν O, see Lemma 2.45vii. A combination of these two embeddings with 4.2 finally gives the asserted Embedding 4.. The following embedding is a consequence of Corollary 2.67, Theorem 2.6i and Lemma 2.45ii. We use the common notation B s,qo := C O B s,q O for the closure of the test functions C O in the Besov sace Bs,qO for s R and, q,. Corollary 4.2. Let γ, ν, and [2,. Then the following embedding holds: H γ γ ν,d ν O B, O. Remark 4.3. Since O R d is assumed to be a bounded Lischitz domain, we know by [69, Chater VIII, Theorem 2] that for / < s the oerator Tr, initially defined on C O as the restriction to O, extends to a bounded linear oerator from B,O s to B, s / O, see [69] for a definition of Besov saces on O. In this case we denote by B,, s O the subsace of Bs,O with zero boundary trace, i.e., B s,,o := { } u B,O s : Tr u =, < s. If additionally s < + /, then, by [67, Theorem 3.2], these saces coincide with the closure of C O in Bs,O, i.e., B s,o = B s,,o for < s < +. Thus, if / < γ ν < + /, by Corollary 4.2, { } H γ γ ν,d ν O B, O = B γ ν,, O = u B, γ ν O : Tr u =. In Section 3.2 we considered SPDEs in the setting of [75]. The solutions to these equations are stochastic rocesses taking values in H γ d θ,d ν O with ν := +, where the value of θ never leaves the range d < θ < d + ; 4.3 see also Remark 3.4ii and iii. This condition is equivalent to / < ν < + / with ν as introduced before. Hence, if γ > / we deal with solutions fulfilling a zero Dirichlet boundary condition in the sense that they can be considered as stochastic rocesses taking values in B γ ν,, O.

80 7 Embeddings of weighted Sobolev saces into Besov saces 4.2 Wavelet decomosition of Besov saces on R d In this section we resent some fundamental results on the wavelet decomosition of Besov saces. They will serve as a key ingredient in the roof of an embedding of weighted Sobolev saces into Besov saces from the non-linear aroximation scale in the subsequent section. Our standard reference concerning wavelete decomositions of Besov saces is the monograh [27], we also refer to the seminal works [47, 56, 64, 97, 9, 7] for further details. Throughout this chater, let φ be a scaling function of tensor roduct tye on R d and let ψ i, i =,..., 2 d, be corresonding multivariate mother wavelets such that, for a given r N and some M >, the following locality, smoothness and vanishing moment conditions hold. For all i =,..., 2 d, su φ, su ψ i [ M, M] d, 4.4 φ, ψ i C r R d, 4.5 x α ψ i x dx = R d for all α N d with α r. 4.6 For the dyadic shifts and dilations of the scaling function and the corresonding wavelets we use the abbreviations φ k x := φx k, x R d, for k Z d, and 4.7 ψ i,j,k x := 2 jd/2 ψ i 2 j x k, x R d, for i, j, k {,..., 2 d } N Z d, 4.8 and assume that {φ k, ψ i,j,k : i, j, k {,..., 2 d } N Z d} is a Riesz basis of L 2 R d. Further, we assume that there exists a dual Riesz basis satisfying the same requirements. That is, there exist functions φ and ψ i, i =,..., 2 d, such that conditions 4.4, 4.5 and 4.6 hold if φ and ψ i are relaced by φ and ψ i, and such that the biorthogonality relations φ k, ψ i,j,k = ψ i,j,k, φ k =, φ k, φ l = δ k,l, ψ i,j,k, ψ u,v,l = δ i,u δ j,v δ k,l, are fulfilled. Here we use analogous abbreviations to 4.7 and 4.8 for the dyadic shifts and dilations of φ and ψ i, and δ k,l denotes the Kronecker symbol. We refer to [27, Chater 2] for the construction of biorthogonal wavelet bases, see also [45] and [3]. To kee notation simle, we will write ψ i,j,k, := 2 jd 2 ψ i,j,k and ψi,j,k, := 2 jd 2 ψ i,j,k, for the L -normalized wavelets and the corresondingly modified duals, with := / if,,, and :=, / := if =. The following theorem shows how Besov saces on R d can be described by decay roerties of the wavelet coefficients, if the arameters fulfil certain conditions. Theorem 4.4. Let, q, and s > max {, d / }. Choose r N such that r > s and construct a biorthogonal wavelet Riesz basis as described above. Then a locally integrable function f : R d R is in the Besov sace B s,qr d if, and only if, f = f, φ 2 d k φ k + k Z d i= j N k Z d f, ψ i,j,k, ψ i,j,k, 4.9

81 4.3 Weighted Sobolev saces and the non-linear aroximation scale 7 convergence in D R d with k Z d f, φk 2 d + i= j N 2 jsq k Z d and 4. is an equivalent quasi-norm for B s,qr d. f, ψ i,j,k, q q <, 4. Remark 4.5. A roof of this theorem for the case can be found in [97, of Chater 6]. For the general case see for examle [89] or [27, Theorem 3.7.7]. Of course, if 4. holds then the infinite sum in 4.9 converges also in B s,qr d. If s > max {, d / } we have the embedding B s,qr d L s R d for some s >, see, e.g., [7, Theorem.73i]. A simle comutation gives us the following characterization of Besov saces B α τ,τ R d, in the case that the arameters α and τ are linked as in the scale. Corollary 4.6. Let,, α > and τ R such that /τ = α/d + /. Choose r N such that r > α and construct a biorthogonal wavelet Riesz basis as described above. Then a locally integrable function f : R d R is in the Besov sace B α τ,τ R d if, and only if, convergence in D R d with k Z d f = f, φ 2 d k φ k + k Z d i= f, φ k 2 τ τ d + i= j N j N k Z d k Z d f, ψ i,j,k, ψ i,j,k, 4. and 4.2 is an equivalent quasi-norm for B α τ,τ R d. f, ψ i,j,k, τ τ <, Weighted Sobolev saces and the non-linear aroximation scale In this section we rove our main result concerning the relationshi between weighted Sobolev saces and the Besov saces from the non-linear aroximation scale. That is, we rove the following embedding. Theorem 4.7. Let O be a bounded Lischitz domain in R d. Let [2,, and γ, ν,. Then H γ,d ν O Bα τ,τ O, τ = α d + { } d, for all < α < min γ, ν. 4.3 d Before we start roving this result, we make some notes on our strategy. We slit our roof into two arts. In the first art we assume that γ is an integer, i.e., γ N. In this articular case we follow the lines of the roof of [38, Theorem 3.2]. This theorem can be restated as follows: If a harmonic function u lies in the Besov sace B ν,o for some, and ν >, then it is contained in the Besov saces B α τ,τ O, τ = α d +, for all < α < ν d d. 4.4 In order to rove this statement, the authors of [38] use fundamental results on extension oerators [] and on wavelet characterizations of Besov saces as resented in the revious

82 72 Embeddings of weighted Sobolev saces into Besov saces section and estimate the weighted l τ -norm of suitable wavelet coefficients. In this way they rove that any harmonic function u B ν,o fulfils the estimate u B α τ,τ O C u B ν, O, τ = α d +, for all < α < ν d d, 4.5 where the constant C does not deend on u. A close look at the roof reveals that two facts are roven and combined in order to show that 4.5 holds in the rescribed setting. First, without making use of the harmonicity of the considered function u B ν,o, it is roven that u B α τ,τ O C u B ν, O + u H γ,d ν O, τ = α d +, for all < α < γ ν d d, 4.6 rovided the semi-norm u H γ,d ν O = α =γ O ρx α D α ux ρx ν dx, 4.7 is finite for some γ N with γ > ν. Then, it is roven in the same ublication, see [38, Theorem 3.], that for any harmonic function we have u H γ,d ν O C u B ν, O, for all < ν < γ N. 4.8 Finally, a combination of 4.6 and 4.8 yields 4.5 for harmonic functions. However, as already mentioned, 4.6 can be roven without assuming that u is harmonic. Thus, if we assume that u H γ,d ν O Bν,O with γ N, the same strategy yields the estimate u B α τ,τ O C u B ν, O + u H γ,,d ν O τ = α d + {, for all < α < min γ, ν d d Since, as roven in Corollary 4.2, H γ,d ν O Bν,O for [2, and ν < γ, this leads to }. u B α τ,τ O C u H γ,d ν O, τ = α d +, for all < α < γ ν d d, and all u H γ,d ν O, if [2, and < ν γ N. But this is exactly our assertion for γ N and ν γ. Now we resent this roof strategy in detail. The case γ R + \N will be considered thereafter in Part Two. Proof of Theorem 4.7 Part One. In this first art, we rove that the assertion holds for γ N. We fix [2, and start with the case ν > γ. Then, by Corollary 4.2 and Theorem 2.6iii, for any < α < γ, we have H γ,d ν O Bγ,O B α τ,τ O, τ = α d +. Therefore, in this case the assertion follows immediately. From now on, let us assume that < ν γ N. We fix α and τ as stated in the theorem and choose a wavelet Riesz basis {φ k, ψ i,j,k : i, j, k {,..., 2 d } N Z d}

83 4.3 Weighted Sobolev saces and the non-linear aroximation scale 73 of L 2 R d which satisfies the assumtions from Section 4.2 with r > γ and some arbitrary M >. Later on, without loss of generality, 2M N will be assumed. Given j, k N Z d let Q j,k := 2 j k + 2 j [ M, M] d, so that su ψ i,j,k Q j,k for all i {,..., 2 d } and su φ k Q,k for all k Z d. Remember that the suorts of the corresonding dual basis meet the same requirements. For our urose the set of all indices associated with those wavelets and scaling functions that may have common suort with the domain O will lay an imortant role and we denote them by Λ := { i, j, k {,..., 2 d } N Z d : Q j,k O }, and Γ := { } k Z d : Q,k O. After these rearations, we fix u H γ,d ν O. Due to Corollary 4.2 we have u Bν,O. As O is a Lischitz domain there exists a linear and bounded extension oerator E : B ν,o B ν,r d, i.e., there exists a constant C >, such that Eu O = u and Eu B ν, R d C u B ν, O, 4.9 see, e.g., []. The constant in 4.9 as well as all the constants C aearing in the rest of this roof do not deend on u. In the sequel we will omit the E in our notation and write u instead of Eu. Theorem 4.4 tells us that the following equality holds on the domain O: u = u, φ k φ k + k Γ i,j,k Λ u, ψ i,j,k, ψ i,j,k,, where the sums converge unconditionally in B ν,r d. Furthermore, cf. Corollary 4.6, we have u τ B α τ,τ O C k Γ u, φk τ + i,j,k Λ u, ψ i,j,k, τ. Hence, in order to rove Embedding 4.3, it is enough to rove that and i,j,k Λ u, φ k τ C u τ B,O ν 4.2 k Γ u, ψ i,j,k, τ τ C u H γ,d ν O + u B ν, O, 4.2 cf. Corollary 4.2. We start with 4.2. The index set Γ introduced above is finite because of the boundedness of O, so that we can use Jensen s inequality followed by Theorem 4.4 together with the boundedness of the extension oerator to obtain u, φ k τ C k Γ k Γ u, φ k τ C u τ B ν,o.

84 74 Embeddings of weighted Sobolev saces into Besov saces To rove 4.2, we introduce the following notation: ρ j,k := distq j,k, O = inf ρx, x Q j,k { } Λ j := i, l, k Λ : l = j, { Λ j,m := i, j, k Λ j : m2 j ρ j,k < m + 2 j}, Λ j := Λ j \ Λ j,, Λ := j N Λ j, where j, m N and k Z d. We slit the exression on the left hand side of 4.2 into u, ψ i,j,k, τ + u, ψ i,j,k, τ =: I + II 4.22 i,j,k Λ i,j,k Λ\Λ and estimate each term searately. Let us begin with I, i.e., with the coefficients corresonding to wavelets with suort guaranteed to be comletely contained in the domain O. Recall that in this thesis we write A for the interior of a set A R d. Fix i, j, k Λ. In this case, the semi-norm u W γ Q j,k := su D α u LQj,k α =γ is finite, since ρ j,k = distq j,k, O > and since u H γ,d ν O, which imlies u H γ,d ν O = ρx α D α ux ρx ν dx <, α =γ O cf. Remark By a Whitney-tye inequality, also known as the Deny-Lions lemma, see, e.g., [48, Theorem 3.4], there exists a olynomial P j,k of total degree less than γ, and a constant C, which does not deend on j or k, such that u P j,k LQ j,k C 2 jγ u W γ Q j,k. Since ψ i,j,k, is orthogonal to every olynomial of total degree less than γ, we have u, ψ i,j,k, = u P j,k, ψ i,j,k, u P j,k LQ j,k ψ i,j,k, L Q j,k C 2 jγ u W γ Q j,k. The constant C does not deend on j or k, since ψ i,j,k, L Q j,k = ψ i L R d. Inserting the definition of the semi-norm on the right hand side and utting = ρx γ ν ρx ν γ into the integrals, yields u, ψi,j,k, C 2 jγ su α =γ C 2 jγ ρ ν γ j,k =: C 2 jγ ρ ν γ j,k su α =γ µ j,k. D α ux dx Q j,k Q j,k ρx γ ν D α ux dx

85 4.3 Weighted Sobolev saces and the non-linear aroximation scale 75 Fix j N. Summing over all indices i, j, k Λ j exonents /τ > and / τ one finds i,j,k Λ j u, ψ i,j,k, τ C C i,j,k Λ j 2 jγτ ρ ν γτ j,k µ τ j,k µ j,k τ and alying Hölder s inequality with γτ j 2 τ ρj,k ν γτ τ τ, 4.23 i,j,k Λ j i,j,k Λ j where C does not deend on j. In order to estimate the first sum in the roduct on the right hand side of 4.23 we regrou the cubes Q j,k, k Z d, in the following way. Without loss of generality, we assume that 2M N. Let a n, n =,..., 2M d, be an arbitrary arrangement of the d-tules from {,,..., 2M } d, and define R j,n := { Q j,k : k a n + 2MZ d}, n {,..., 2M d } Then, one can check that {R j,n : n {,..., 2M d }} is a finite artition of {Q j,k : k Z d }, i.e., 2M d R j,n = { Q j,k : k Z d} and R j,n R j,n = for n n n= Furthermore, for any fixed n {,..., 2M d }, Thus, setting and using 4.25, we obtain if Q j,k, Q j,k R j,n for some k k, then Q j,k Q j,k = R j,n := { i, j, k Λ j : Q j,k R j,n }, n {,..., 2M d }, i,j,k Λ j µ j,k τ = = i,j,k Λ j 2M d n= su ρx γ ν D α ux τ dx α =γ Q j,k i,j,k Rj,n α =γ su ρx γ ν D α ux dx Q j,k Together with 4.26 and using the norm equivalence 2.28 together with some standard comutations, this yields i,j,k Λ j µ j,k τ C α =γ O ρx γ ν D α ux τ dx τ C u τ H γ,d ν O, 4.27 with a constant C, which does not deend on j. In order to estimate the second sum on the right hand side of 4.23 we use the Lischitz character of the domain O which imlies that Λ j,m C 2 jd for all j, m N

86 76 Embeddings of weighted Sobolev saces into Besov saces Moreover, the boundedness of O yields Λ j,m = for all j, m N with m C2 j, where the constant C does not deend on j or m. Consequently, i,j,k Λ j γτ j 2 τ ρj,k ν γτ τ τ C C C2 j 2 j γτ τ ρj,k m= i,j,k Λ j,m C2 j m= 2 j d ντ τ ν γτ τ τ τ 2 jd γτ j 2 τ m 2 j ν γτ τ τ + 2 j d γτ τ Now, let us sum over all j N. Inequalities 4.29 together with 4.27 and 4.23 imly i,j,k Λ u, ψ i,j,k, τ C j N 2 j d ντ τ + 2 j d γτ τ Obviously, the sums on the right hand side converge if, and only if, α i,j,k Λ u, ψ i,j,k, τ C u τ H γ,d ν O. τ u τ H γ,d ν O., γ ν d d. Finally, Now we estimate the second term II in First we fix j N and use Hölder s inequality and 4.28 to obtain i,j,k Λ j, u, ψ i,j,k, τ C 2 τ jd i,j,k Λ j, u, ψ i,j,k, τ, with a constant C which does not deend on j. Summing over all j N and using Hölder s inequality again yields u, ψi,j,k, τ = u, ψi,j,k, τ i,j,k Λ\Λ C j N 2 C j N 2 j j N i,j,k Λ j, τ jd i,j,k Λ j, d τ ντ τ u, ψ i,j,k, τ τ 2 jν j N i,j,k Λ j, Using Theorem 4.4 and the boundedness of the extension oerator, we obtain i,j,k Λ\Λ u, ψ i,j,k, τ C u τ B,O ν j N 2 j The series on the right hand side converges if, and only if, α i,j,k Λ\Λ u, ψ i,j,k, τ C u τ B ν,o. u, ψ i,j,k, τ. d τ ντ τ τ., ν d d. We thus have

87 4.3 Weighted Sobolev saces and the non-linear aroximation scale 77 So far we have roven the assertion of Theorem 4.7 rovided the smoothness arameter γ is an integer. In what follows, we consider the comlementary case of fractional γ R + \ N. Our strategy for roving Embedding 4.3 in this case relies on a combination of the already roven assertion for γ N and suitable alications of the comlex interolation method of A.P. Calderón. This method is initially defined only for Banach saces and its extension to quasi-banach saces is not a trivial task. However, in our aroach we need such an extension, which reserves the interolation roerty and is alicable to comatible coules of Besov saces B s,q O, B s 2 2,q 2 O on bounded Lischitz domains O R d with s, s 2 R and <, 2, q, q 2 <. Fortunately, such a method has been develoed in [96], see also [7] for more details. We use the notation [ B s,q O, B s 2 2,q 2 O ] η for the extended comlex interolation method from [7, 96] alied to a comatible coule B s,q O, B s 2 2,q 2 O of Besov saces. Then, since the interolation roerty is reserved, B s, O E and B s 2 2, 2 O E 2 imly [ B s, O, B s 2 2, 2 O ] η [ E, E 2 ]η for a comatible coule E, E 2 of Banach saces η,. The following result concerning the comlex interolation of Besov saces is an immediate consequence of [7, Proosition.4], see also [7, Theorem 9.4]. It is a major ingredient in Part Two of the roof of Theorem 4.7. Theorem 4.8. Let O be a bounded Lischitz domain in R d, and [2,. Furthermore, let α < α < and τ, τ, be such that τ = α d + and τ = α d Then, for any η,, where [ B α τ,τ O, B α τ,τ O ] η = Bα τ,τ O, 4.3 α = ηα + ηα and τ = α d +. Proof. By [7, Proosition.4], equality 4.3 holds with Inserting 4.3, the assertion follows. τ = η + η. τ τ This result at hand, we are ready to rove Embedding 4.3 for γ R + \ N. Proof of Theorem 4.7 Part Two. Let γ R + \ N, ν, and [2,. If ν > γ, 4.3 follows with the same arguments as in Part One. Thus, in what follows, we assume that < ν γ. We distinguish five cases. Case. Let γ := m + η, with m N, η,, and ν d d m +. Then, we can argue as follows. Fix an arbitrary ε > with ε η. Set α := m ε, α := m+ ε and let τ, τ, be given by 4.3. Then, by Part One, H m+,d ν O Bα τ,τ O and H,d ν m O Bα τ,τ O. Thus, [ H m,d ν O, H m+,d ν O] η [ B α τ,τ O, B α τ,τ O ] η.

88 78 Embeddings of weighted Sobolev saces into Besov saces Hence, by Theorem 4.8 and Lemma 2.45v, H m+η,d ν O Bα τ,τ O, with α = m + η ε and /τ = α /d + /. This is true for arbitrary ε, η]. We therefore obtain H γ,d ν O = Hm+η,d ν O Bα τ,τ O, τ = α d + { } d, for all < α < m + η = min γ, ν, d by simly alying the first embedding from Theorem 2.6ii. Case 2. Let γ := η, and ν d d. Then, since L O B,O for 2, see Corollary 2.67, we also have H,d ν O = L,d νo B,O for any ν >. Simultaneously, by Part One, for any ε,, H,d ν O B ε τ,τ O with /τ = ε/d + /. Thus, using again Theorem 4.8 and Lemma 2.45v we obtain for any ε,, and therefore H γ,d ν O = Hη,d ν O Bα τ,τ O, H η,d ν O B εη τ,τ O, εη = τ d +, τ = α d + {, for all < α < η = min γ, ν d d Case 3. Let γ := m+η, with m N, η,, and ν d d m. Since in this case, Hγ,d ν O = H m+η,d ν O Hm,d ν O, the embedding 4.3 holds due to the the fact that, by Part One, H,d ν m O Bα τ,τ O, τ = α d + { }, for all < α < ν d d = min d γ, ν. d Case 4. Let γ := m + η, with m N, η,, and m < ν d d m + η. Fix η, with η η, such that ν d d = m + η. Also, let ε, m, and let α := m ε, α := m + ε, and τ, τ, be given by 4.3. If m =, set α :=. By Part One, H m,d m d O Bα τ,τ O and H m+ d,d m+ d O Bα τ,τ O. d Therefore, by Theorem 4.8 and Lemma 2.45v, H m+η,d ν O = Hm+η,d m+η d d Bα O τ,τ O, τ = α d +, with α = m + η ε α = η εη, if m =. Since ε, m is arbitrary, and H γ,d ν O = H m+η,d ν O Hm+η,d ν O, we obtain H γ,d ν O Bα τ,τ O, τ = α d + { }, for all < α < m + η d = ν d = min d γ, ν. d Case 5. Finally, let γ := m + η, with m N, η,, and m + η ν d the lines of Case 5 with η instead of η, we obtain H m+η,d m+η d d Since m + η d d O Bα τ,τ O, d τ = α d +, for all < α < m + η = min ν, and therefore d ν d m + η d d, we have H m+η,d ν O Hm+η,d m+η d d O. These two embeddings rove 4.3 also for this articular case. }. m +. Following { d γ, ν d }.

89 4.4 An alternative roof of Theorem An alternative roof of Theorem 4.7 In this section we resent an alternative roof of Theorem 4.7 for arbitrary γ R, which does not require any knowledge about comlex interolation of quasi-banach saces. However, the arguments are quite involved and not as elegant as in the section before. We use the same notation as in the revious sections of this chater. A close look at Part One of the roof of Theorem 4.7 resented in the revious section reveals that the restriction γ N is required only when estimating the series I = u, ψ i,j,k, τ i,j,k Λ from Let us be more detailed: The restriction γ N is needed for the first time when alying the Deny-Lions lemma [48, Theorem 3.4], which yields the existence of a olynomial P j,k of total degree less than γ, and a constant C, which does not deend on j or k, such that u P j,k LQ j,k C 2 jγ u W γ Q j,k Using this and the orthogonality of ψ i,j,k, to every olynomial of total degree less than γ, we obtain u, ψ i,j,k, C 2 jγ u W γ Q j,k, which is transformed into u, ψ i,j,k, C 2 jγ ρ ν γ j,k su α =γ Q j,k ρx γ ν D α ux dx by utting = ρx ν γ ρx γ ν into the integrals and using ρ j,k = distq j,k, O. Then, alying Hölder inequality, we show that I C i,j,k Λ j su α =γ ρx γ ν D α ux τ dx Q j,k i,j,k Λ j γτ j 2, τ ρj,k ν γτ τ τ. At this oint we use the norm equivalence 2.28, which holds only for γ N, in order to obtain ρx γ ν D α ux dx C u H γ O, Q,d ν j,k i,j,k Λ j su α =γ with a constant C which does not deend on j. This is the last time we use the restriction γ N in Part One of the roof of Theorem 4.7. Let us now assume that we are given the setting of Theorem 4.7 with < ν γ without any additional restriction on γ,. Fix u H γ,d ν O. The exlanations above show that, in this generalized setting, if want to aly the same strategy as in the case of integer γ in order to estimate the sum I from 4.22 by the weighted Sobolev norm of u, we first need an estimate similar to To this end we can use Corollary 4.2, which rovides H γ,d ν O Bν,O in the given setting, together with [48, Theorem 3.5], which is a generalization of the Deny-Lions lemma to arbitrary γ,. From these two facts we obtain the existence of a olynomial P j,k of total degree less than γ, such that u P j,k LQ j,k C 2 jγ u B γ, Q j,k, 4.33

90 8 Embeddings of weighted Sobolev saces into Besov saces where the constant C does not deend on j or k. As in the integer case, we can use the orthogonality of ψ i,j,k, to every olynomial of total degree less than γ, which yields u, ψ i,j,k, C 2 jγ u B γ, Q j,k. If we multily the right hand side with = ρ ν γ jk u, ψ i,j,k, C 2 jγ ρ ν γ jk Thus, an alication of Hölder s inequality leads to i,j,k Λ u, ψi,j,k, τ C i,j,k Λ j ρ γ ν j,k ργ ν j,k, we obtain ργ ν j,k u W γ Q j,k. τ u B,Q γ j,k i,j,k Λ j γτ j 2 τ ρj,k ν γτ τ τ. The following lemma shows that the first sum on the right hand side can be estimated by the weighted Sobolev norm of u times a constant C which does not deend on j or u. Using this estimate and relacing the right laces in Part One of the roof resented in the revious section by the calculations above, Theorem 4.7 can be roven directly for arbitrary γ,. The details are left to the reader. Lemma 4.9. Let O be a bounded Lischitz domain in R d. Let [2,, γ, and ν R with γ ν. Furthermore, assume u H γ,d ν O. Then, for all j N, the inequality i,j,k Λ j ρ γ ν j,k u B γ,q j,k C u H γ,d ν O 4.34 holds, with a constant C, which does not deend on j or u. Proof. Fix j N. Let k be such that 2 + 2M d < 2 k, 4.35 and construct a sequence {ξ n : n Z} C O as in Remark 2.48ii. In order to rove the assertion we are going to show the estimates ρ γ ν j,k u B,Q γ C j,k 2 j nγ ν ξ j n u B,R γ d, 4.36 n N and i,j,k Λ j i,j,k Λ j ξ j n u B γ,r d C 2 j nd γ ξ j n 2 j n u 2 j n H γ R d, 4.37 where the constant C does not deend on j and n. This will rove the assertion since, assuming that 4.36 and 4.37 are true, their combination gives ρ γ ν j,k u B,Q γ C ξ j,k j n 2 j n u 2 j n H γ R d n N 2 j nd ν C n Z 2 nd ν ξ n 2n u 2 n H γ R d, which by Remark 2.48ii and Lemma 2.47 yields i,j,k Λ j ρ γ ν j,k u B γ,q j,k C u H γ,d ν O.

91 4.4 An alternative roof of Theorem Let us first verify inequality To this end, let r be the smallest integer strictly greater than γ. Recall that r h [f] denotes the the r-th difference of a function f : Rd R with ste h R d, comare Subsection Writing out the Besov semi-norm and alying the transformation formula for integrals we see that ξ j n u B,R γ d = t γ su r h [ξ j nu] dt L R h <t d t { = 2 j nd t γ su r h [ξ j nu] 2 j n x } dt dx R d t h <t Since the equality r h [f] cx = r h/c [fc ] x, x Rd, holds for any function f : R d R and c >, we obtain ξ j n u B γ,r d = 2 j nd t γ su h <2 j n t A further alication of the transformation formula for integrals yields { ξ j n u B,R γ d = 2 j nd 2 j nγ t γ su R d [ r h ξ j n 2 j n u 2 j n ] } x dt dx t. r h h <t [ξ j n 2 j n u 2 j n ] = 2 j nd γ ξ j n 2 j n u 2 j n B γ,r d, L R d which imlies 4.37 since the sace H γ R d of Bessel otentials is continuously embedded in the Besov sace B γ,r d, see [6, Theorem 2.3.2d combined with Theorem 2.3.3a]. It remains to rove inequality Recall that the index i referring to the different tyes of wavelets on a cube Q j,k ranges from to 2 d. Since Λ j consists of those indices i, j, k Λ j with 2 j ρ j,k, we have i,j,k Λ j ρ γ ν j,k u B,Q γ j,k = 2 d, ρ γ ν j,k u B,Q γ j,k 4.38 k Λ j dt t where we used the notation Λ j := { } k Z d : i, j, k Λ j. Now we get the required estimate in three stes. Ste. We first show that the cubes suorting the wavelets fit into the stries where the cut-off functions ξ n are identical to one. That is, we claim that the roer choice of k, see 4.35, leads to the fact that, for any k Λ j, there exists a non-negative integer n N such that { } Q j,k S j n := x O : 2 j n 2 k ρx 2 j n 2 k. To rove this, we first note that, since k, k Λ j Q j,k n N S j n. 4.39

92 82 Embeddings of weighted Sobolev saces into Besov saces Fix k Λ j. Because of 4.39, we can define n to be the smallest non-negative integer such that Q j,k S j n, i.e., n := inf { n N : Q j,k S j n }. Then, there are two ossibilities: On the one hand, Q j,k might be contained comletely in S j n, i.e., Q j,k S j n. Then we are done. On the other hand, it might haen that Q j,k is not comletely contained in the strie S j n. In this case, we claim that Q j,k S j n +, i.e., ρx [ 2 j+n + 2 k, 2 j+n + 2 k ] for all x Q j,k. Let us therefore fix x Q j,k. Then, since the length of the diagonal of Q j,k is 2 j 2M d, we have ρx ρ j,k + 2 j 2M d. Also, ρ j,k 2 j+n 2 k since Q j,k S j n. Hence, ρx 2 j+n 2 k + 2 j 2M d. Since 2M d 2 k, we conclude that ρx 2 j+n + 2 k 2 + 2M d 2 n + 2 k 2 j+n + 2 k j+n + 2 k n + It remains to show that ρx 2 j+n + 2 k. We argue as follows: Since Q j,k is not comletely contained in S j n, there exists a oint x Q j,k such that ρx > 2 j+n 2 k. Therefore, since the length of the diagonal of Q j,k is 2 j 2M d and since 4.35 holds, we have ρx > 2 j+n 2 k 2M d 2 j = 2 j+n + 2 k 2 2k 2 2M d 2 k 2 j+n + 2 k n + Thus, since 4.4 and 4.4 hold for arbitrary x Q j,k, we have shown that Q j,k S j n +. Ste 2. Let us fix k Λ j and estimate the Besov semi-norm of the restriction of u to the corresonding cube Q j,k. To this end, we use the results from [68], where the modulus of smoothness ω r t, f, G, t,, of a function f defined on a domain G is comared with the Peetre K-functional { } K r t, f, G := inf f g LG + t g W g W rg r G, t,, f L G. In articular, it is shown therein that for all t,, and f L G for some, ω r t, f, G max { 2 r, d r/2} K r t, f, G holds for r N, see [68, Lemma ]. Using this, we obtain the following estiamte: u B,Q γ j,k = t γ ω r t, u, Q j,k C = C C { t γ inf g W r Q j,k { t γ inf g W r Q j,k dt t t γ K r t r, u, Q j,k dt t } dt u g LQj,k + t r g W r Q j,k t } u g L Q j,k + tr g dt W rq j,k t,

93 4.4 An alternative roof of Theorem where the constant C deends only on r, d and. Recall that r is the smallest integer strictly greater than γ. Ste 3. Now we collect the fruits of our work and aroximate the sum on the right hand side of As in Part One of the roof of Theorem 4.7 from the revious section, we use the artition R j,m, m {,..., 2M d }, of the set {Q j,k : k Z d } defined in Furthermore, we write R j,m := { k Λ j : Q j,k R j,m }, m {,..., 2M d }, 4.42 and S j,n := {k Λ j : Q j,k S j n }, n N. Form Ste we can deduce that Λ j = n N S j,n. Thus, since 4.25 holds, we have Therefore, k Λ j ρ γ ν j,k 2M d Λ j = Λ j Rj,m = m= u B γ,q j,k n N 2M d m= n N 2M d m= k S j,n R j,m S j,n R j,m. ρ γ ν j,k u B γ,q j,k Let us fix n N such that Sj,n as well as m {,..., 2Md }. Then, ρ j,k 2 k 2 j n for k Sj,n, so that using the estimate from Ste 2 we obtain k S j,n R j,m C ρ γ ν j,k u B γ,q j,k k S j,n R j,m C2 j nγ ν { 2 j nγ ν t γ inf g W r Q j,k { t γ inf k Sj,n R g W ro j,m Furthermore, since ξ j n = on Q j,k for any k S j,n, k S j,n R j,m ρ γ ν j,k u B γ,q j,k C2 j nγ ν C2 j nγ ν k S j,n R j,m { t γ inf g W ro t γ inf g W r O k S j,n R j,m } u g L Q j,k + tr g dt W rq j,k t } u g L Q j,k + tr g dt W r Q j,k t. { } ξ j n u g L Q j,k + tr g dt W rq j,k t } ξ j n u g L Q j,k + tr g dt W r Q j,k t.

94 84 Embeddings of weighted Sobolev saces into Besov saces Since Q j,k Q j,l = for k, l R j,m if k l, see 4.26 together with 4.42, we obtain k S j,n R j,m ρ γ ν j,k u B γ,q j,k } C2 j nγ ν t γ inf { ξ j n u g g W ro L + O tr g dt W r O t C2 j nγ ν t γ K r t r, ξ j n u, O dt t. Now we use another result from [68], which shows that the K-functional can be estimated by the modulus of smoothness. That is, [68, Theorem ] yields the existence of a constant C, deending only on r, and O, such that K r t r, ξ j n u, O C ω r t, ξ j n u, O. Putting everything together, we have shown that there exists a constant C which does not deend on j, n or m such that k S j,n R j,m ρ γ ν j,k u B γ,q j,k C2 j nγ ν t γ ω r t, ξ j n u, O = C2 j nγ ν ξ j n u B γ,o C2 j nγ ν ξ j n u B γ,r d. Inserting this estimate into 4.43 yields ρ γ ν j,k u B,Q γ j,k C 2 j nγ ν ξ j n u B,R γ d, n N k Λ j which combined with 4.38 roves dt t

95 85 Chater 5 Satial Besov regularity of SPDEs on bounded Lischitz domains In this chater, we are concerned with the satial regularity of solutions to SPDEs on bounded Lischitz domains O R d in the non-linear aroximation scale B α τ,τ O, τ = α d +, α >, with 2 fixed i.e., toic T in the introduction. We use the same setting and notation as introduced in Chater 3. The embedding of weighted Sobolev saces into Besov saces from the scale roven in the revious chater Theorem 4.7, shows that to a certain extent the analysis of the regularity of SPDEs in terms of the scale can be traced back to the analysis of such equations in terms of the saces H γ,q O, T. In articular, the following embeddings hold. Theorem 5.. Let O be a bounded Lischitz domain in R d. Fix γ,,, q [2,, and θ R. Then H γ,q O, T Hγ,q O, T L qω T ; Bτ,τ α O, for all α and τ with τ = α d + and { < α < min γ, + d θ } d. d Proof. The first embedding follows from the definition of the stochastic arabolic weighted Sobolev saces H γ,q O, T, see Definition 3.3, and holds actually on arbitrary domains with non-emty boundary. Since H γ O = d θ Hγ,d ν O, with ν := +, the second embedding follows immediately from Theorem 4.7. We use this result to rove satial Besov regularity of the solutions to SPDEs in the scale of Besov saces. We divide this chater into two sections. We start with the linear equations introduced in Section 3.2. As outlined therein, in this setting, the L -theory from [75], already rovides existence and uniqueness of solutions in the classes H γ O, T = Hγ, O, T, [2,, γ, θ R. Thus, we can aly Theorem 5. directly and obtain satial regularity results in the right scale, see Theorem 5.2. Afterwards, in Section 5.2, we generalize our results to a class of semi-linear SPDEs: The linear art will be of the same form as in [75], whereas the

96 86 Satial Besov regularity of SPDEs on bounded Lischitz domains non-linearities fulfil certain Lischitz conditions. Since in this case, existence of solutions has not been established yet, we first have to extend the main existence result of the aforementioned L -theory to this class of equations. This will be done in Theorem 5.3. Afterwards, we can aly Theorem 5. and obtain satial regularity in the scale of Besov saces, see Theorem 5.5. The examles and remarks resented in Section 5. have been artially worked out in collaboration with F. Lindner, S. Dahlke, S. Kinzel, T. Raasch, K. Ritter, and R.L. Schilling [25]. 5. Linear equations In this section we use the scale with fixed 2 to analyse the satial regularity of the solutions u H γ O, T of the linear SPDEs of the form 3. studied in [75], see Section 3.2. Since we already have an existence and uniqueness result for this tye of equations in H γ O, T, see Theorem 3.3, we can immediately extract an assertion about the satial regularity of the solution in the scale by alying Theorem 5.. After stating and roving this result, we resent several examles and make some additional remarks. In articular, we enlighten the fact that, on bounded Lischitz domains, the satial smoothness of the solution in the non-linear aroximation scale of Besov saces is generically higher than its satial Sobolev regularity. The relevance of this characteristic from the oint of view of aroximation theory and numerical analysis has been ointed out in Section.. We begin with the main result on the satial regularity in the scale of the solutions to linear SPDEs. It is an imrovement of [25, Theorems 3. and B.3], see also Remark 5.3 below. Theorem 5.2. Let O be a bounded Lischitz domain in R d. Given γ 2,, let a ij, b i, c, σ ik and µ k, i, j {,..., d}, k N, satisfy Assumtion 3. with suitable constants δ and K. Furthermore, assume that u H γ+2 O, T is the unique solution of Eq. 3. with f H γ + O, T, g Hγ+ O, T ; l 2 and u U γ+2 O, where i [2, and θ d + 2 κ, d κ or, alternatively, ii [2, and θ d κ, d + κ, with κ, κ, and > 2 as in Theorem 3.3. Then, u L Ω T ; Bτ,τ α O, τ = α d + {, for all < α < min γ + 2, + d θ } d. 5. d Moreover, for any α and τ fulfilling 5., there exists a constant C, which does not deend on u, f, g and u such that u L Ω T ;B f τ,τ α O C H γ + O,T + g + u. H γ+ O,T ;l 2 U γ+2 O Proof. The assertion is an immediate consequence of Theorem 5. and the existence and uniqueness statements from Theorem 3.3. Remark 5.3. A result similar to Theorem 4.4 has been roven in [25, Theorem 3., see also Theorem B.3]. However, there are three major imrovements in Theorem 5.2 comared to [25, Theorems 3. and B.3]. Firstly, we have no restriction on γ + 2,, whereas in [25] only We remark that the assumtions made in [25, Theorem B.3] are stronger than actually needed. In the notation used therein, the assumtions [K] [K5] only need to be fulfilled for γ 2 instead of γ.

97 5. Linear equations 87 integer γ + 2 N are considered. Secondly, we obtain L -integrability in time of the Bτ,τ α O- valued rocess for arbitrary 2 fulfilling the assumtions i or ii from Theorem 3.3. With the techniques used in [25] just L τ -integrability in time can be established. Thirdly, we do not need the extra assumtion u L [, T ] Ω; B,O s for some s >. Due to Corollary 4.2, it suffices that u H γ+2 O, T. Next, we give some examles of alications of Theorem 5.2 and interret our result from the oint of view of the question whether adativity ays, cf. our motivation for studying toic T from Section.. We are mainly interested in the Hilbert sace case = 2 since, as already ointed out in Section., it rovides a natural setting for numerical discretization techniques like adative wavelet methods, see also the exositions in [5, 25] for more details. We begin with an alication of Theorem 5.2 for articular arameters γ, θ R and = 2. Examle 5.4. Assume that we have given coefficients a ij, b i, c, σ ik, and µ k, with i, j {,..., d} and k N, fulfilling Assumtion 3. with γ =. Furthermore, fix arbitrary f H 2,d+2 O, T, g H 2,d O, T ; l 2 and u U2,d 2 = L 2Ω, F, P; H2,d O. Then, by an alication of Theorem 3.3 with γ =, = 2 and θ = d, Eq. 3. has a unique solution u H 2 2,d O, T. Due to Theorem 5.2, u L 2 Ω T ; B α τ,τ O, In the two-dimensional case, this means that τ = α d + d, for all < α < 2 d. 5.2 u L 2 Ω T ; B α τ,τ O, τ = α 2 +, for all < α < 2. 2 Note that if we assume slightly more regularity on the coefficients, the initial condition u and the free terms f and g, we can included the border case α = d/d in 5.2. To this end, assume that the coefficients a ij, b i, c, σ ik, and µ k, with i, j {,..., d} and k N, fulfil Assumtion 3. for some arbitrary ositive γ >. Furthermore, fix an arbitrary ε > and assume that f H γ 2,d ε+2 O, T, g Hγ+ 2,d ε O, T ; l 2 and u U γ+2 2,d ε = L 2Ω, F, P; H γ+2 2,d ε O. Then, there exists an ε, κ with κ > from Theorem 3.3ii, such that Eq. 3. has a unique solution u H γ+2 2,d ε O, T. Due to Theorem 5.2, u L 2 Ω T ; B α τ,τ O, and therefore, since γ and ε are strictly ositive, τ = α d + {, for all < α < min γ + 2, + ε } d, 2 2 d u L 2 Ω T ; B α τ,τ O, τ = α d + d, for all < α 2 d, which in the two-dimensional case yields u L 2 Ω T ; B α τ,τ O, τ = α d +, for all < α 2. 2 The examle above shows that equations of the tye 3. on general bounded Lischitz domains have satial Besov regularity in the scale u to order α = 2. In order to answer the question whether this is enough for justifying the develoment of satially adative wavelet methods, we have to comare this result with the satial Sobolev regularity of the solution under consideration. We give now a concrete examle of an SPDE of the tye 3. with solution u H 2 2,d O, T whose satial Besov regularity in the scale is strictly higher than its satial Sobolev regularity.

98 88 Satial Besov regularity of SPDEs on bounded Lischitz domains V 7 β 7 V5 V 6 β 6 β 5 β 4 V 4 V 8 β 8 β 3 V 3 β β 2 V V 2 Figure 5.: Polygon in R 2 with β max = β 6 = 5π/4. Examle 5.5. We consider an equation of the tye 3. on a olygonal domain O R 2 and show that, under natural conditions on the data of the equation, if the underlying domain is not convex, the satial Besov regularity of the solution in the scale is strictly higher than its satial Sobolev smoothness. In articular, this shows that, generically, solutions to linear SPDEs on bounded Lischitz domains behave as described in., so that the use of satially adative methods is recommended. For more details on the link between regularity theory and the convergence rates of numerical methods we refer to Section.. Let O R 2 be a simly connected bounded domain in R 2 with a olygonal boundary O such that O lies on one side of O. It can be described by a finite set {V n : n =,..., N} of vertices of the boundary numbered, e.g., according to their order in O in counter-clockwise orientation. For n {,..., N}, we write β n, 2π for the interior angle at the vertex V n and denote by β max the maximal interior angle of O, i.e., β max := max { β n : n =,..., N }. An examle of such a domain with β max = 5π/4 is shown in Figure 5.. Assume that we have an initial condition u U2,2 2 O additionally satisfying u L 2 Ω, F, P; W 2 O L q Ω, F, P; L 2 O for some q > 2. Furthermore, let f L 2 Ω T ; L 2 O H 2,4 O, T and let g H 2,2 O; l 2. Tyically, we make slight abuse of notation and write g also for the constant stochastic rocess g H 2,d O, T ; l 2 with gω, t := g for all ω, t Ω T. Then, due to Theorem 3.3, the stochastic heat equation du = u + f } dt + g k dwt k on Ω T O, 5.3 u = u on Ω O, has a unique solution u H 2 2,2 O, T. We want to comare the satial Besov regularity of the solution to Eq. 5.3 in the scale with its satial Sobolev regularity. Regarding the satial regularity in the non-linear aroximation scale, an alication of Theorem 5.2 yields u L 2 Ω T ; B α τ,τ O, τ = α = α +, for all < α <

99 5. Linear equations 89 Concerning the satial Sobolev regularity of the solution, by our analysis so far, we can only guarantee that u L 2 Ω T ; W 2 O, which is a consequence of Proosition 4.. Together with 5.4, this suggests that the Besov regularity of the solution to Eq. 5.3 in the scale is generically higher than its satial Sobolev regularity in the following sense: There exist olygonal domains O R 2 and free terms f and g fulfilling the assumtions from above, such that s Sob maxu < 2, 5.5 with s Sob maxu as introduced in.2. We can confirm this statement by exloiting the recent results from [92]. Therefore, let us denote by D 2,w : D D 2,w L 2O L 2 O the weak Dirichlet-Lalacian on L 2 O, i.e., D D 2,w := { u W 2 O : u L 2 O }, D 2,wu := u, u D D 2,w. From Proosition 3.8 we already know that our solution u H 2 2,2 O, T is also the unique weak solution in the sense of Da Prato and Zabczyk [32] of the L 2 O-valued ordinary SDE } dut D 2,wut dt = ft dt + dw Q t, t [, T ], 5.6 u = u, driven by the H2,2 O-valued Q-Wiener rocess W Q t t [,T ] := k N gk wt k with covariance oerator Q := k N gk, H t [,T ] 2,2 O gk L H2,2 O. Moreover, due to Theorem 3.8ii, [ ] [ su E ut 2 L 2 O E t [,T ] and by [32, Theorem 5.4], for all t [, T ], su t [,T ] ut 2 L 2 O ] C u 2 H 2 2,2 O,T <, ut = S 2 tu + t S 2 t sfs ds + t S 2 t s dw Q s P-a.s., where { S 2 t } t denotes the contraction semigrou on L 2O generated by D 2,w, D D 2,w. Thus, u is the unique u to modifications mild solution of Eq. 5.6 which is studied in [92], see also [9, Chater 4]. Therein, techniques from [57, 58] have been adated to the stochastic setting, and it has been shown that this solution can be divided into a satially regular and a satially irregular art, regularity being measured by means of Sobolev saces. In articular, if we assume that the range of the covariance oerator Q is dense in H 2,2 O L 2O, it follows from [92, Examle 3.6] that Thus, if O is not convex, we have u / L 2 Ω T ; W s 2 O for any s > + π β max. 5.7 s Sob maxu + π < 2, β max with s Sub maxu as defined in.2. Together with 5.4, this shows that the solution to Eq. 5.3 generically behaves as described in.. Therefore, the develoment of suitable satially adative numerical methods is comletely justified.

100 9 Satial Besov regularity of SPDEs on bounded Lischitz domains r + π/β max = 9/5 3/2 B2/3,2/3 2 O 2 W 2 O q Figure 5.2: Satial Besov regularity in the scale B α τ,τ O, /τ = α + /2, versus satial Sobolev regularity of the solution of Eq. 5.3, illustrated in a DeVore/Triebel diagram. Figure 5.2 shows a DeVore/Triebel diagram illustrating the situation described above see Remark 2.63 for details on the visualisation of Besov saces using this tye of diagrams. The fact that 5.4 holds, is reresented by the solid segment {/τ, α : /τ = α + /2, α < 2} of the L 2 O-non-linear aroximation line and the annulus at 3/2, 2, which stands for the Besov sace B2/3,2/3 2 O. The oint at /2, shows that u L 2Ω T ; W2 O. In this situation, by Theorem 2.6 and standard interolation results, see, e.g. [7, Corollary.], u L 2 Ω T ; Bq,qO r for all /q, r in the interior of the olygon with vertices at the oints /2,, /2,, 3/2, 2, 2, 2, and,. This is indicated by the shaded area. The border at /2, 3/2 illustrates the following consequence of 5.7: For any ε >, there exists a olygonal domain O R 2, such that u / L 2 Ω T ; W 3/2+ε 2 O. The concrete border for the examle in Figure 5. is indicated by the annulus at /2, + π/β max = /2, 9/5, which stands for the Sobolev sace W 9/5 2 O. In the following examle we are concerned with equations of the form 3. driven by a secific tye of noise. Examle 5.6. We consider an equation of the tye 3. driven by a time-deendent version of the stochastic wavelet exansion introduced in [] in the context of Bayesian non-arametric regression and generalized in [4] and [24]. This noise model is formulated in terms of a wavelet basis exansion on the domain O R d with random coefficients of rescribed sarsity and thus tailor-made for alying adative techniques with regard to the numerical aroximation of the corresonding SPDEs. Via the choice of certain arameters secifying the distributions of the wavelet coefficients it also allows for an exlicit control of the satial Besov regularity of the noise. We first describe the general noise model and then deduce a further examle for the alication of Theorem 5.2. Let {ψ λ : λ } be a multiscale Riesz basis of L 2 O consisting of scaling functions at a fixed scale level j Z and of wavelets at level j and all finer levels. We follow [27] and use the same notation as in Section.. Information like scale level, satial location and tye of the wavelets or scaling functions are encoded in the indices λ. In articular, we write

101 5. Linear equations 9 = j j j, where for j j the set j contains the indices of all wavelets ψ λ at scale level j and where j is the index set referring to the scaling functions at scale level j which we denote by ψ λ, λ j, for the sake of notational simlicity; λ := j for all λ j. We make the following assumtions concerning our basis. Firstly, the cardinalities of the index sets j, j j, satisfy C 2 jd j C2 jd, j j, 5.8 with a constant C which does not deend on j. Secondly, we assume that the basis admits norm equivalences similar to those described in Theorem 4.4. That is, there exists an s N deending on the smoothness of the scaling functions ψ λ, λ j, and on the degree of olynomial exactness of their linear san, such that, given, q >, max{, d/ } < s < s, and a real valued distribution v D O, we have v B s,qo if, and only if, v can be reresented as v = λ c λψ λ, c λ λ R convergence in D O, such that j=j s+d 2 jq 2 q q c λ λ j <. 5.9 Furthermore, v B s,q O is equivalent to the quasi-norm 5.9. Concrete constructions of bases satisfying these assumtions can be found e.g. in [42 44] or [9, 2], see also [27, Section 2.2 together with Section 3.9] for a detailed discussion. Concerning the family of indeendent standard Brownian motions wt k t [,T ], k N, in 3., we modify our notation and write wt λ t [,T ], λ, instead. The descrition of the noise model involves arameters a, a 2 [, ], b R, with a + a 2 >. For every j j we set ς j := j j 2 bd 2 2 a j j d 2 and let Y λ, λ j, be {, }-valued Bernoulli distributed random variables on Ω, F, P with arameter j = 2 a 2j j d, such that the random variables and rocesses Y λ, wt λ t [,T ], λ, are stochastically indeendent. The noise in our equation will be described by the L 2 O-valued stochastic rocess M t t [,T ] defined by M t := j=j λ j ς j Y λ ψ λ w λ t, t [, T ]. 5. Using 5.9, 5.8 and a + a 2 >, it is easy to check that the infinite sum converges in L 2 Ω T ; L 2 O as well as in L 2 Ω; C[, T ]; L 2 O. Moreover, by the choice of the arameters a, a 2 and b one has an exlicit control of the convergence of the infinite sum in 5. in the quasi-banach saces L 2 Ω T ; B s,qo, s < s,, q >, 2 q. Comare [24] which can easily be adated to our setting. For simlicity, let us consider the two-dimensional case, i.e., d = 2. Assume that we have a given f H 2,2 O, T, an initial condition u U2,2 2 O, and coefficients aij, b i and c, with i, j {,..., d}, fulfilling Assumtion 3. with σ = and µ =. We consider the equation du = a ij u x i x j + bi u x i + cu + f } dt + ς λ Y λ ψ λ dwt λ on Ω T O, u = u on Ω O, 5. where we sum over all λ instead of k N. That is, we understand this equation similar to equations of the tye 3., where the role of k N in the required definitions is taken by λ. In this setting, let g := g λ λ := ς λ Y λ ψ λ λ. Since a + a 2 > and g H 2,2 O,T ;l 2 = 2/T M L2 Ω T ;L 2 O we have g H 2,2 O, T ; l 2. Let us imose a bit more smoothness on g

102 92 Satial Besov regularity of SPDEs on bounded Lischitz domains and assume that a + a 2 > 2. This is sufficient to ensure that g H 2,2 O, T ; l 2, since so that by 2.28, [ T ] g 2 H 2,2 O,T ;l 2 = E g 2 H2,2 O;l 2 dt g 2 H 2,2 O,T ;l 2 C [ T ] = E g λ t, 2 H2,2 O dt λ [ = T E C j=j j=j j=j λ j ς 2 j j ] ςj 2 Yλ 2 ψ λ 2 H2,2 O, λ j ρ α D α ψ λ 2 L 2 O α λ j ς 2 j j ψ λ 2 W 2 O. Since W 2 O = B 2,2 O, see Theorem 2.6ii, we can use the equivalence 5.9 with v = ψ λ followed by 5.8 with d = 2 to obtain g 2 H 2,2 O,T ;l 2 C = C C j=j j=j j=j λ j ς 2 j j 2 2j Thus g H 2,2 O, T ; l 2 and for any ϕ C O, λ j j j 2 2b 2 2a j j 2 2a 2j j 2 2j j j 2 2b 2 2ja +a 2 2. g λ, ϕ dw λ t = M, ϕ P-a.s. in C[, T ]; R, see also Proosition 3.6 and the definition of stochastic integrals from Subsection for details. As in the examles above, by Theorem 3.3, there exists a unique solution of Eq. 5. in the class H 2 2,2 O, T. As shown in Examles 5.5, in general, the solution rocess is not in L 2 Ω T ; W2 s O for all s < 2, but, by Theorem 5.2, it belongs to every sace L 2 Ω T ; Bτ,τ α O with α < 2 and τ = 2/α +. We make the following note regarding adative versus uniform methods in Sobolev saces. Remark 5.7. As already mentioned in the introduction, in different deterministic settings, there exist adative wavelet-based schemes realising the convergence rate of the best m-term aroximation error in the energy norm. This norm is determined by the equation and is usually equivalent to an L 2 O-Sobolev norm and not to the L 2 O norm itself. Thus, the question arises whether our regularity results underin the use of adativity also in the case that the error is measured in a suitable Sobolev norm. Again this question can be decided after a rigorous regularity analysis of the target function, since the results on the link between regularity theory and the convergence rate of aroximation methods discussed in Section. can be generalised to the case where the error is measured in a Sobolev saces W r 2 O with r > instead of L O.

103 5. Linear equations 93 r q 2 q 2 + π/β max 3/2 W ssob max 2 O B α τ,τ O B 2 2/3,2/3 O W 2 O q Figure 5.3: Satial Besov regularity in the scale B α τ,τ O, /τ = α/2, versus satial Sobolev regularity of the solution to Eq. 5.3, illustrated in a DeVore/Triebel diagram. Let us denote by {η λ : λ } a wavelet basis of W2 r O for some r >. Such a basis can be obtained by rescaling a wavelet basis {ψ λ : λ } of L 2 O as the one used in Examle 5.6 and by using the norm equivalence 5.9, see, e.g., [27] or [4]. For the error of the best m-term wavelet aroximation error in this Sobolev norm, it is well-known that where u B α τ,τ O, τ = α r + d 2 σ m,w r 2 Ou := inf imlies σ m,w r 2 Ou C m α r/d, 5.2 { u u m W r 2 O : u m Σ } m,w r 2 O with { Σ m,w r 2 O := λ Λ c λ η λ : Λ, Λ } = m, c λ R, λ Λ, see, e.g., [25, Corollary 3.2] and the references therein, in articular, [27]. Therefore, similar to the L 2 O-setting, on the one hand, the decay rate of the best m-term wavelet aroximation error in E = W2 r O deends on the Besov regularity of the target function. On the other hand, the convergence rate of uniform numerical methods is determined by the Sobolev regularity of the solution to be aroximated. It is well-known that, under fairly natural conditions, if u m, m N, is a uniform aroximation scheme of u, then, u u m W r 2 O C m s r/d u W s 2 O, m N; see, e.g., [37], [46] or [6] for details. If we consider uniform wavelet aroximation, the following converse assertion also holds: If u / W2 s O, then the convergence rate of the uniform method in W2 r O is limited by s r/d, see, e.g., [25, Proosition 3.2] and the references therein. This means that, if the error is measured in W2 r O, adativity ays if the satial smoothness of the solution in the Besov saces from 5.2 is strictly higher than its satial Sobolev regularity.

104 94 Satial Besov regularity of SPDEs on bounded Lischitz domains Let us consider the setting from Examle 5.5 and discuss the relationshi between the satial Sobolev and Besov regularity in view of aroximation in W2 O, i.e., r =. We use a DeVore/Triebel diagram to visualise our exlanations, see Figure 5.3. Due to 5.7, with s Sob maxu as defined in.2. Thus, { } su s /2 : u L 2 Ω T ; W2 s O s Sob maxu + π, 5.3 β max π 2β max. 5.4 Let us assume that the satial Sobolev regularity of the solution u reaches its maximum, i.e., that u L 2 Ω T ; W +π/βmax 2 O, cf Then, due to 5.4, by Theorem 2.6 and standard interolation results, see, e.g. [7, Corollary.], u L 2 Ω T ; B α τ,τ O, τ = α = α 2 for all < α < α := β max + 3π β max + π. This is illustrated in Figure 5.3 by the solid segment of the line /q 2/q delimited by the origin and the annulus at /τ, α = α /2, α. Thus, the decay rate of the best m-term wavelet aroximation error in W 2 O with resect to the sace coordinates goes u to π/β max + π, which is greater than π/2β max whenever β max > π, i.e., whenever the olygonal domain O is not convex. Therefore, also in this setting, the imlementation of adative wavelet methods is justified. In all the other examles from above we consider general bounded Lischitz domains. In this case, we do not have an exlicit bound for the satial Sobolev regularity of the solution. Thus, we can only assume the limit case β max = 2π. Inserting this into the calculations from above, we can say that, in the worst case, Simultaneously, u L 2 Ω T ; B α τ,τ O, su { s /2 : u L 2 Ω T ; W s 2 O } τ = α = α 2 for all < α < 5 3. Since 5/3 /2 = 2/6 > /4, the develoment of otimal adative algorithms with resect to the sace coordinates, where the error is measured in W2 O, is recommended. We illustrate this limiting case in Figure 5.4 by using again a DeVore/Triebel diagram. We conclude this section with an examle showing that, in contrast to what is known to hold for deterministic equations, adative wavelet methods for SPDEs may ay even if the underlying domain is smooth. Examle 5.8. Let O be a bounded Cu-domain and, therefore, a bounded Lischitz domain in R d. Furthermore, let a ij, b i, c, and µ k, with i, j {,..., d} and k N, be given coefficients satisfying Assumtion 3. with γ =, σ = and suitable constants δ and K. Fix [2, and let f H,d + O, T, g H2,d O, T ; l 2 and u U,d 3 O. Then, by [72, Theorem 2.9] there exists a unique solution u of Eq. 3., which is in the class H 3,d ε O, T for any ε > ; see also Remark 3.4ii. Due to Theorem 5. this yields u L Ω T ; B α τ,τ O, τ = α d +, for all < α < + ε d, ε,. d

105 5.2 Semi-linear equations 95 r q 2 q W 3/2 2 5/3 3/2 2 O B 2 2/3,2/3 O B 5/3 5/6,5/6 O W 2 O q Figure 5.4: Satial Besov regularity in the scale B α τ,τ O, /τ = α/2, versus satial Sobolev regularity of the solution to equations of tye 3., illustrated in a DeVore/Triebel diagram. Thus, in the two-dimensional case, if = 2, we have u L Ω T ; B α τ,τ O, τ = α +, for all < α < 3. 2 What about the satial Sobolev regularity of this solution? It is known from [78, Examle.2] that if we consider R + instead of O, there exists a non-random g, continuously differentiable on [, [, such that the second artial derivatives with resect to the sace coordinates of the solution to the heat equation du = u dt + g k dw k t, u R+ =, u =, on R +, do not lie in L 2 Ω T ; L 2 R +. This is due to the incomatibility of the noise with the zero Dirichlet boundary condition. Exloiting the comatibility results from [55], it is reasonable to exect that we can construct similar examles on smooth bounded domains, with maximal satial Sobolev regularity strictly less than the satial Besov regularity in the non-linear aroximation scale. If this is indeed the case, it shows that in the stochastic setting, adative methods are a serious alternative to uniform methods even if the underlying domain is smooth. It is worth noting that this would be comletely different from what is known to hold in the deterministic setting, where adativity does not ay on smooth domains. 5.2 Semi-linear equations In this section we continue our analysis of the satial regularity of SPDEs in the scale of Besov saces. We generalize the results from the revious section to a class of semi-linear equations on bounded Lischitz domains O R. However, for semi-linear SPDEs, existence of solutions in the classes H γ O, T, γ R, is yet to be roven. Therefore, before we can aly Theorem 5. in order to obtain satial regularity in the non-linear aroximation scale, we have to extend

106 96 Satial Besov regularity of SPDEs on bounded Lischitz domains the main existence and uniqueness result of the L -theory from [75], cf. Theorem 3.3, to semi-linear SPDEs. The equations considered in this section are of the form du = a ij u x i x j + bi u x i + cu + f + Lu dt + σ ik u x i + µ k u + g k + Λu k dwt k on Ω T O, 5.6 u = u on Ω O. While the linear arts of Eq. 5.6 are suosed to satisfy Assumtion 3. as in the revious section, we imose the following conditions on the non-linearities L and Λ. Assumtion 5.9. The functions L : H γ+2 O, T Hγ + O, T and Λ : Hγ+2 O, T Hγ+ O, T ; l 2 fulfil the following conditions: i For all u, v H γ+2 O, T and t [, T ] Lu Lv H γ + O,t + Λu Λv H γ+ O,t;l 2 ε u v + K u v 5.7 H γ+2 O,t H γ+ O,t with ε > and K [, indeendent of u, v and t [, T ]. ii The non-linearities vanish at the origin, i.e., L = and Λ =. We use the following solutions concet, which is a straight-forward generalization of the solution concet resented in Definition 3.. Definition 5.. Given γ R, let a ij, b i, c, σ ik and µ k, i, j {,..., d}, k N, fulfil Assumtion 3.. Furthermore, let Assumtion 5.9i be satisfied for given θ R and [2,. A stochastic rocess u H γ O, T is called a solution of Eq. 5.6 in the class Hγ O, T if, and only if, u H γ O, T with u, = u, Du = a ij u x i x j +bi u x i +cu+f +Lu, and Su = σ ik u x i +µ k u+g k +Λu k k N in the sense of Definition 3.3. Remark 5.. In this thesis, if we call an element u H γ G, T a solution of Eq. 5.6, we mean that u is a solution of Eq. 5.6 in the class H γ G, T. Remark 5.2. As already mentioned, throughout this thesis, for a better readability, we omit the notation of the sums i,j and k when writing down equations and use the so-called summation convention on the reeated indices i, j, k. Thus, the exression du = a ij u x i x j + bi u x i + cu + f + Lu dt + σ ik u x i + µ k u + g k + Λu k dw k t is short-hand for d d d du = a ij u x i x j + b i u x i + cu + f + Lu dt + σ ik u x i + µ k u + g k + Λu k dwt k i,j= i= in the sense of Definition 3.3. We first state our main result on the existence and uniqueness of solutions of Eq. 5.6 in weighted Sobolev saces. i=

107 5.2 Semi-linear equations 97 Theorem 5.3. Let O be a bounded Lischitz domain in R d, and γ R. For i, j {,..., d} and k N, let a ij, b i, c, σ ik, and µ k be given coefficients satisfying Assumtion 3. with suitable constants δ and K. i For [2,, there exists a constant κ,, deending only on d,, δ, K and O, such that for any θ d + 2 κ, d κ, f H γ + O, T, g Hγ+ O, T ; l 2 and u U γ+2 O, the following holds: There exists an ε > such that, if L and Λ fulfil Assumtion 5.9 with ε < ε and K [,, Eq. 5.6 has a unique solution u in the class H γ+2 O, T. For this solution, the a riori estimate u C f H γ+2 O,T H γ + O,T + g + u, 5.8 H γ+ O,T ;l 2 U γ+2 O holds with a constant C which does not deend on u, f, g and u. ii There exists > 2, such that, if [2,, then there exists a constant κ,, deending only on d,, δ, K and O, such that for any θ d κ, d+ κ, f H γ + O, T, g H γ+ O, T ; l 2 and u U γ+2 O, the following holds: There exists an ε > such that, if L and Λ fulfil Assumtion 5.9 with ε < ε and K [,, Eq. 5.6 has a unique solution u in the class H γ+2 O, T. For this solution, the a riori estimate 5.8 holds. Before we resent a roof of this result starting on age, we make some remarks on the assumtions therein. Furthermore, we state and rove the consequences concerning the satial Besov regularity in the scale of solutions to Eq. 5., and rove an auxiliary theorem, which we will need for roving Theorem 5.3. Remark 5.4. i The constants κ and κ coincide with the constants κ and κ, resectively, from Theorem 3.3. Moreover, > 2 is the same as in Theorem 3.3ii. ii The statement of Theorem 5.3i with K = has been already roven in [22], see Theorem 4. together with Remark 4.2 therein. Note that the assumtions made in [22, Theorem 4.] are stronger than actually needed, since the assumtions K K5 therein only need to be fulfilled with γ 2 instead of γ. iii The assumtion that the non-linearities vanish at the origin is made only for convenience. To see this, let all assumtions of Theorem 5.3 be fulfilled excet Assumtion 5.9ii, i.e., we allow L and Λ. Then, Eq. 5.6 can be rewritten as du = a ij u x i x j + bi u x i + cu + f + L + L u dt + σ ik u x i + µ k u + g k + Λ k + Λ u k dwt k on Ω T O, 5.9 u = u on Ω O. with L u := Lu L and Λ u := Λu Λ for u H γ+2 O. Then, Assumtion 5.9 is fulfilled with L and Λ instead of L and Λ, resectively. Thus, since L H γ + O, T and Λ Hγ+ O, T ; l 2, alying Theorem 5.3 yields the existence of a unique solution u H γ+2 O, T of Eq. 5.6, which fulfils the estimate f u C + L H γ+2 O,T H γ + O,T + g + Λ H γ+ O,T ;l 2 + u U γ+2 with a constant C which does not deend on u, f, g, and u. O,

108 98 Satial Besov regularity of SPDEs on bounded Lischitz domains iv Let ε >. Assume that there exists an η, such that the two functions L : H γ+2 O, T Hγ + fulfil the estimate O, T and Λ : Hγ+2 O, T Hγ+ O, T ; l 2 Lu Lv H γ + O,t + Λu Λv H γ+ O,t;l 2 ε u v + K H γ+2 u v 5.2 O,t H γ+2η O,t, + 2η for some ε, ε and K [, indeendent of u, v H γ+2 O, T and t [, T ]. Then, by Lemma 2.45v, the estimate Lu Lv H γ + O,t + Λu Λv H γ+ O,t;l 2 ε u v H γ+2 O,t + K u v H γ + O,t, holds with otentially different ε, ε and K [,, which again do not deend on u, v H γ+2 O, T and t [, T ]. Also, the reverse direction holds, since O Rd is assumed to be bounded and therefore H γ+2η,d+ 2η O Hγ,d+ O for every η,, see also Lemma 2.45vii. Note that 5.2 with η = /2 is exactly 5.7 from Assumtion 5.9i. Next, we resent our main result concerning the satial regularity of semi-linear equations in the non-linear aroximation scale of Besov saces. It is an extension and an imrovement of [22, Theorem 4.4]. Theorem 5.5. Given the setting from Theorem 5.3, let u H γ+2 O, T be the unique solution of Eq Then, u L Ω T ; B α τ,τ, τ = α d + {, for all < α < min γ + 2, + d θ } d. 5.2 d Moreover, for any α and τ fulfilling 5.2, there exists a constant C, which does not deend on u, f, g and u such that u L Ω T ;B f τ,τ α O C H γ + O,T + g + u. H γ+ O,T ;l 2 U γ+2 O Proof. This is an immediate consequence of Theorem 5.3 and Theorem 5.. Now we state and rove an auxiliary result, which we will use later on in order to rove Theorem 5.3. It shows how fixed oint arguments can be used to rove existence and uniqueness of a solution to the semi-linear equation 5.6, rovided this result is already established for the corresonding linear equation. One needs Assumtion 5.9 for the non-linearities and suitable a riori estimates for the linear equations with vanishing initial value. Similar ideas have been already used in the context of SPDEs on the whole sace R d by N.V. Krylov, see [8, Theorem 6.4]. Lemma 5.6. Given the setting from Theorem 3.3, assume that the solution u H γ+2 O, T of Eq. 3. with f H γ + O, T, g Hγ+ O, T ; l 2 and u =, fulfils the estimate u C H γ+2 O,t f H γ + O,t + g 5.22 H γ+ O,t;l 2

109 5.2 Semi-linear equations 99 for all t [, T ] with a constant C, indeendent of t [, T ], u, f and g. Then there exists an ε > deending on T in general, such that, if Assumtion 5.9 is fulfilled with ε < ε and K [,, the following holds: For any f H γ + O, T, g Hγ+ O, T ; l 2 and u U γ+2 O, there exists a unique solution u of Eq. 5.6 in the class H γ+2 O, T. Moreover, there exists a constant C, which does not deend on u, f, g and u, such that u C f H γ+2 O,T H γ + O,T + g + u H γ+ O,T ;l 2 U γ+2 O Proof. For i, j {,..., d} and k N, let a ij, b i, c, σ ik, and µ k, fulfil Assumtion 3. for some γ R. Furthermore, let and θ be as in Theorem 3.3i or ii and fix f H γ + O, T, g H γ+ O, T ; l 2 and u U γ+2 O. Then, the oerator N : H γ+2 O, T Hγ+2 O, T u N u, where N u is the unique solution in the class H γ+2 O, T of the linear equation dv = a ij v x i x j + bi v x i + cv + f + Lu dt + σ ik v x i + µ k v + g k + Λu k dwt k on Ω T O, v = u on Ω O, is well-defined by Theorem 3.3. Fix arbitrary u, v H γ+2 O, T. Then N u N v is the unique solution in the class H γ+2 O, T of the equation dṽ = a ij ṽ x i x j + bi ṽ x i + cṽ + Lu Lv dt + σ ik ṽ x i + µ k ṽ + Λu k Λv k dwt k on Ω T O, ṽ = on Ω O. By 5.22, N u N v H γ+2 O,t C Lu Lv H γ + O,t + Λu Λv H γ+ O,t;l 2 for all t [, T ]. If Assumtion 5.9 is fulfilled with some ε > and K [,, this leads to N u N v H γ+2 O,t C ε u v + C H γ+2 O,t K u v H γ+ O,t Let us first assume that K =. In this case we are done: If we choose ε > small enough, e.g., ε < ε := /C, the oerator N turns out to be a contraction on the Banach sace H γ+2 O, T. Therefore, by the Banach fixed oint theorem, N has a unique fixed oint. Regarding the fact that any solution of Eq. 5.6 in the class H γ+2 O, T is a fixed oint of N and vice versa, we have just roven the fact that, given the setting from Theorem 3.3, if Assumtion 5.9i holds with ε > small enough and K =, Eq. 5.6 has a unique solution u in the class H γ+2 O, T. We can also obtain Estimate 5.23 arguing as follows: Start the fixed oint iteration with u := H γ+2 + O, T and set uj+ := N u j for j. Then u j j N converges to the unique solution u in H γ+2 O, T. Furthermore, since Assumtion 5.9ii is fulfilled and estimate 3.9 holds, we have N u H γ+2 O,T C/ f H γ + O,T + g H γ+ O,T ;l 2 + u U γ+2 O /. 5.25

110 Satial Besov regularity of SPDEs on bounded Lischitz domains Hence, using the a riori estimate from the Banach fixed oint theorem, as it can be found in [66, Theorem 3..2], we obtain u H γ+2 O,T C / C f ε / H γ + O,T + g + u. H γ+ O,T ;l 2 U γ+2 O For K, we still have to work in order to rove that the oerator N has a unique fixed oint under suitable assumtions on ε. To this end, we will rove that, given the setting from Theorem 3.3, if Assumtion 5.9i is fulfilled with ε > small enough, we can find an m N, deending among others on ε, K and T, such that N m becomes a contraction on H γ+2 O, T. Due to the Banach fixed oint theorem, this leads to the existence of a unique fixed oint of N m, which automatically imlies that N has a unique fixed oint, and, therefore, Eq. 5.6 has a unique solution u in the class H γ+2 O, T, cf. Remark 5.7 below. By Theorem 3.8ii, there exists a constant C, deending on T in general, such that for all t [, T ] u v H γ+ O,t C Using this, we obtain from 5.24, that t u v ds. H γ+2 O,s N u N v H γ+2 O,t C ε u v H γ+2 O,t + C K C t u v ds, H γ+2 O,s for all t [, T ]. As a consequence, we can rove by induction, that for any m N the following estimate holds for all t [, T ]: N m u N m v H γ+2 O,t Cm ε m u v H γ+2 m + k= m k O,t C m ε m k K k C k t t s k k! u v ds. H γ+2 O,s For t = T and each k {,..., m} we can aroximate the integrals on the right-hand side by u v H γ+2 O,T T k /k!. Consequently, N m u N m v H γ+2 O,T C m ε m u v H γ+2 + C m ε m m k= O,T m K C T k ε C m ε m + 2 m max k m k k! u v H γ+2 O,T } { K C T ε k k! u v H γ+2 O,T. Now assume that Assumtion 5.9i holds with ε > small enough, e.g., ε /8C, and K,. Then, N m u N m v { H γ+2 O,T 8 m + K } 4 m max C T k u v k N ε k! H γ+2 O,T. Since for any fixed x, the function k x k /k! is decreasing for sufficiently large k, we have { K } C T k max = C 3 <, k N ε k!

111 5.2 Semi-linear equations and, consequently, N m u N m v H γ+2 O,T 4 m + C 3 u v m N H γ+2 O,T, Thus, there exists an m N, such that N m is a contraction on the Banach sace H γ+2 O, T. Consequently, by the Banach fixed oint theorem, N m and therefore N has a unique fixed oint, and, therefore, Eq. 5.6 has a unique solution u in the class H γ+2 O, T. In order to rove Estimate 5.23, we argue as follows: Take the sequence u j j N defined above. Then for any j N, since u =, u j+ H γ+2 O,T j u i+ u i H γ+2 O,T = i= Using 5.26 and u =, we obtain and by 5.25, u j+ H γ+2 O,T u j+ H γ+2 O,T i= j N i+ u N i u H γ+2 i= j 4 i + C 3 N u H γ+2 O,T, + C 3 C f /4 H γ + O,T + g + u. H γ+ O,T ;l 2 U γ+2 O O,T. Since by the Banach fixed oint theorem, there exists a subsequence of u j j N converging to u in H γ+2 O, T, this finishes the roof. Remark 5.7. In the roof of Lemma 5.6 above, we use the following fact: If the m-th ower N m, m N, of a maing N : E E on a Banach sace E, E has a unique fixed oint, then, so does N. This can be seen as follows: Let u be the unique fixed oint of N m for some m N. In articular, N m u = u, and, therefore, due to the associativity of the comosition of functions, N m N u = N u. Consequently, N u is a fixed oint of N m, and, due to the uniqueness assumtion, N u = u. The uniqueness of the fixed oint of N follows immediately from the uniqueness of the fixed oint of N m. After these rearations, we are able to rove Theorem 5.3 in the following way. Proof of Theorem 5.3. We rove that in the different situations of Theorem 3.3, the solution u H γ+2 O, T of Eq. 3. with u = fulfils the a riori estimate 5.22 for all t [, T ] with a constant C which does not deend on t [, T ]. Since we have roven Lemma 5.6, this automatically imlies our assertion with κ = κ and κ = κ from Theorem 3.3. We rove the a riori estimate in four different situations. Case. Assume that a ij and σ ik do not deend on x O and fulfil Assumtion 3. with γ and b i = c = µ k = for all i {,..., d} and k N. Furthermore assume that

112 2 Satial Besov regularity of SPDEs on bounded Lischitz domains θ d + 2 κ, d κ with κ, as in Theorem 3.3i. In this case we can argue as follows: Given f H γ + O, T and g Hγ+ O, T, the solution of the equation du = a ij u x i x j + f dt + σ ik u x i + g k } dwt k on Ω T O, 5.27 u = on Ω O. fulfils the estimate u C f H γ+2 O,T H γ + O,T + g H γ+ O,T ;l with a constant C which does not deend on T. This has been roven in [75], see esecially Corollary 3.6 therein. Therefore, since the restriction u Ω [,t] is the unique solution in the class H γ+2 O, t of the equation dv = a ij v x i x j + f Ω [,t] ds + σ ik v x i + g k } Ω [,t] dw k s on Ω [, t] O, v = on Ω O. for any t [, T ], Estimate 5.22 is fulfilled with C indeendent of t [, T ]. Case 2. Consider the situation from Case and relax the restriction γ allowing γ to be negative. In order to rove that Estimate 5.22 holds also in this situation with a constant indeendent of t [, T ], we will rove that 5.28 holds with a constant C indeendent of T. We follow the lines of Case 2 in the roof of [73, Theorem 4.7]. Let us assume that γ [, ]. The case γ < can be roven analogously by iterating the roof for γ [, ]. For ν [,, let R : H ν + O, T Hν+ O, T ; l 2 H ν+2 O, T f, g Rf, g be the solution oerator for Eq. 5.27, i.e., Rf, g H ν+2 O, T denotes the unique solution of Eq. 5.27, given f H ν + O, T and g Hν+ O, T ; l 2. Notice that by the uniqueness of the solution, this oerator does not deend on ν [,. Furthermore, by the a riori estimate 5.28, it is a bounded oerator with oerator norm indeendent of T, which we will denote by R ν. Fix f, g H γ + O, T Hγ+ O, T ; l 2. Furthermore, let ψ be an infinitely differentiable function on O fulfilling 2.25 and choose c >, such that the oerator L := ψ 2 c is an isomorhism between H γ+2 + O and Hγ + O and between Hγ+3 O; l 2 and H γ+ O; l 2, resectively; this is ossible due to Lemma 2.45vi and Lemma 2.55iii, resectively. Set Then, f, g := L f, g. f = ψd r ψd r f ψψx rd r f c f and g = ψd r ψd r g ψψ x rd r g c g. For r =,..., d, denote u r := RψD r f, ψdr g, u := R ψψ x rd r f c f, ψψx rd r g c g, and set d v := u + ψd r u r. r=

113 5.2 Semi-linear equations 3 Note that, by Theorem 3.3 and Lemma 2.45iii and iv together with Lemma 2.55i and ii, u, u r H γ+3 O, T, r =,..., d, and v Hγ+2 O, T are well-defined. A short calculation shows that for all ϕ C O, with robability one, the equality vt,, ϕ = t a ij sv x i x js, + f s, + fs,, ϕ ds + d t k= σ ik sv x is, + g k s, + g k s,, ϕ dw k s holds for all t [, T ] with f := a ij ψ x i x jdr u r + ψ x id r u r x j + ψ x jd r u r x i and g := σ ik ψ x id r u r k N. Using the decay roerties 2.25 of ψ and its derivatives, Assumtion 3. as well as Lemma 2.45, we can deduce that there exists a constant C, which does not deend on T, such that f H γ+ + O,T + g H γ+2 O,T ;l 2 C d u r H γ+3 O,T r= Thus, by Lemma 2.45iii and Lemma 2.55i, v H γ+2 O, T and solves the equation dv = a ij v x i x j + f + f dt + σ ik v x i + g k + g k } dwt k on Ω T O, v = on Ω O. 5.3 Set ṽ := Rf, g H γ+3 O, T. Then, obviously u := v ṽ Hγ+2 O, T solves Eq Moreover, u H γ+2 O,T v H γ+2 O,T + ṽ H γ+2 O,T. 5.3 We first rove that there exists a constant C, which does not deend on T, such that ṽ H γ+2 O,T C f H γ + O,T + g H γ+ O,T ;l We argue as follows: Since R is a bounded oerator, ṽ H γ+2 O,T ṽ H γ+3 O,T R γ+ f H γ+ + O,T + g H γ+2 O,T ;l. 2 The same argument, together with Lemma 2.45iii and Lemma 2.55i, shows that d u r H γ+3 O,T C 2 R γ+ f H γ+2 + O,T + g H γ+3 r= with a constant C 2 indeendent of T. Consequently, r= O,T ;l 2 d u r H γ+3 O,T C 2 R γ+ L f H γ + O,T + g H γ+ O,T ;l, 2 where L := max { L LH γ+ O;l 2,H γ+3 O;l 2, L LH γ } + O,Hγ+2 + O.

114 4 Satial Besov regularity of SPDEs on bounded Lischitz domains Finally, using estimate 5.29, we obtain 5.32 with a constant C indeendent of T. We move on and estimate v H γ+2 O,T. Since v Hγ+2 O, T solves Eq. 5.3, v H γ+2 O,T v H γ+2 O,T + aij v x i x j H γ + O,T + f H γ + O,T + f H γ + O,T + σ ik u x i H γ+ O,T ;l 2 + g H γ+ O,T ;l 2 + g H γ+ O,T ;l 2. Thus, we can argue as before when estimating ṽ H γ+2 and obtain O,T v H γ+2 O,T C f H γ + O,T + g H γ+ + O,T ;l 2 with a constant C which does not deend on T. This, together with 5.3 and 5.32, roves that 5.28 holds with a constant C indeendent of T. Case 3. Assume that a ij and σ ik do not deend on x O and fulfil Assumtion 3. with γ R and b i = c = µ k = for all i {,..., d} and k N. Furthermore assume that θ d+κ, d+κ with κ, and [2, as in Theorem 3.3ii. In this situation, the assertion for γ can be roven by following the lines of [75, Section 5]. Essentially, this strategy makes use of the fact that the comlex interolation method is an exact interolation method and that for two comatible coules A, B and A, B of Banach saces, [A A, B B ] η = [A, B ] η [A, B ] η, with equivalent norms η,. Additionally, in order to aly this strategy in the case of bounded Lischitz domains, Lemma 2.45v and Lemma 2.55v, concerning comlex interolation of weighted Sobolev saces and of their generalizations H γ O; l 2, are required. Using the argumentation line from Case 2 above, we can obtain the assertion also for γ <. Case 4. Finally, we consider the general case. We assume that θ d+ 2 κ, 2+ 2+κ and [2, or, alternatively, that θ d κ, d + κ and [2, as in the different situations of Theorem 3.3i and ii, resectively. Following the lines of the roof of [75, Theorem 3.7] see also Section 5 in [73] and using what we have roved in the first three cases, we can show that there exists a constant C 3 indeendent of t [, T ] such that u H γ+2 O,t C 3 for all t [, T ]. By Theorem 3.8 this leads to u H γ+2 O,t C 4 t u H γ+ O,t + f H γ + O,t + g H γ+ O,t;l 2 u H γ+2 O,s ds + C 3 f H γ + O,t + g H γ+ O,t;l 2 for all t [, T ] with a constant C 4 indeendent of t [, T ]. Using Gronwall s lemma see, e.g., [7, Corollary 6.2] this roves that for all t [, T ], u C H γ+2 O,t 3 e tc 4 f H γ + O,t + g H γ+ O,t;l 2 Thus, Estimate 5.22 is fulfilled with C := C 3 e tc 4, which does not deend on t [, T ]. We conclude this section with two examles. The first one is ut in a setting similar to the one resented in [2, Section 7]. However, we are able to treat the case of general bounded Lischitz domains, whereas [2, Section 7] is restricted to bounded domains with C 2 boundary. As ointed out in Remark 4.3, we are only concerned with equations fulfilling zero Dirichlet boundary conditions. Using the notation from [2], this means that Γ = O and therefore Γ =.

115 5.2 Semi-linear equations 5 Examle 5.8. For i, j {,..., d} and k N, let a ij, b i, c, σ ik, and µ k be given coefficients fulfilling Assumtion 3. with γ =. Let f L : Ω T H2,d O H 2,d+2 O and gk Λ : Ω T H2,d O H 2,d O, k N, be strongly P T BH2,d O-measurable maings. Assume that f L ω, t, = g k ω, t, = for all ω, t Ω T, k N, and that there exist C L L Ω T ; R and C Λ = CΛ k k N L Ω T ; l 2 such that for all u, v H2,d O, f L ω, t, u f L ω, t, v H 2,d+2 O C Lω, t u v H 2,d O and g k Λω, t, u g k Λω, t, v H 2,d O Ck Λω, t u v H 2,d O, k N. Then, the functions L : H 2 2,d O, T H 2,d+2 O, T u Lu := ω, t f L ω, t, uω, t, and Λ : H 2 2,d O, T H 2,d O, T ; l 2 u Λu := ω, t g k Λ ω, t, uω, t, k N are well-defined and fulfil Assumtion 5.9 with { ε = and K = max C L 2 L Ω T ;R, } C Λ 2 L Ω T ;l 2 <. Therefore, by Theorem 5.3, Eq. 5.6 with L and Λ as defined above has a unique solution u H 2 2,d O, T. Furthermore, due to Theorem 5.5, u L 2 Ω T ; B α τ,τ O, In the two-dimensional case, this yields u L 2 Ω T ; B α τ,τ O, τ = α d + d, for all < α < 2 d. τ = α 2 +, for all < α < 2. 2 Also in this case, we exect that, at least on non-smooth and non-convex domains, s Sob maxu < 2, cf. Examle 5.5. The following examle is insired from [59, Section 6.]. Therein sace time discretization schemes for SPDEs are discussed. Examle 5.9. Let again a ij, b i, c, σ ik, and µ k with i, j {,..., d} and k N be given coefficients fulfilling Assumtion 3. with γ =. Furthermore, let F : Ω T O R d R R be a strongly P T BO BR d BR-measurable function satisfying the following conditions:

116 6 Satial Besov regularity of SPDEs on bounded Lischitz domains [F] There exists a constant C F, which does not deend on ω, t, x Ω T O,, 2 R d, and r, r 2 R such that F ω, t, x,, r F ω, t, x, 2, r 2 C F 2 + ρx r r 2 ; [F2] For all ω, t, x Ω T O: Then, for any u H 2,d O, T, the function F ω, t, x,, =. Ω T ω, t F ω, t,, u x ω, t,, uω, t, H 2,d+2 O is well-defined, strongly P T -measurable and for u, v H 2,d O, T and arbitrary t [, T ], t F ω, s,, u x ω, s,, uω, s, F ω, s,, v x ω, s,, vω, s, 2 H2,d+2 Ω O P λ dω, s t C F ω, s, x, ux ω, s, x, uω, s, x Ω O C Ω t 4CF 2 O F ω, s, x, v x ω, s, x, vω, s, 2 ρx 2 dx P λ dω, s u x ω, s, x v x ω, s, x 2 ρx 2 + uω, s, x vω, s, x 2 dx P λ dω, s C u v 2 H 2,d O,T, where in the last ste we have used the norm equivalence Therefore, there exists a constant K [, such that Assumtion 5.9 is fulfilled with L : H 2 2,d+2 O, T H 2,d+2 O, T u Lu := ω, t F ω, t,, u x ω, t,, uω, t, and ε = Λ =. Thus, by Theorem 5.3, there exists a solution u H 2 2,d O, T of Eq. 5.6 with L as defined above and Λ =. Due to Theorem 5.5, this solution also fulfils u L 2 Ω T ; B α τ,τ O, In the two-dimensional case, this means that τ = α d + d, for all < α < 2 d. u L 2 Ω T ; B α τ,τ O, τ = α 2 +, for all < α < 2. 2 In the light of Examle 5.5, in general, we exect that s Sob maxu < 2; at least on non-smooth and non-convex domains.

117 7 Chater 6 Sace time regularity of the inhomogeneous heat equation with additive noise In this chater we are concerned with the Hölder regularity of the aths of the solution to the inhomogeneous stochastic heat equation with additive noise du = u + f } dt + g k dwt k on Ω T O, 6. u = on Ω O, considered as a stochastic rocess taking values in Besov saces from the scale. Eq. 6. is understood in the sense of Definition 3. with a ij = δ i,j, i, j {,..., d}. It will be sometimes referred to as the stochastic heat equation. As we have already seen in Chater 3, see Theorem 3.3, it is known that for γ R, certain [2, and corresonding θ R, Eq. 6. has a unique solution u within the class H γ+2 O, T, rovided f Hγ + O, T and g Hγ+ O, T ; l 2. Alying Theorem 3.8 we obtain with the restriction E [ u ] C T β β/2 u C β/2 / [,T ];H γ+2 β β O H γ <, O,T 2 < β < β. 6.2 Thus, a simle alication of the embedding 4.3 already yields a first result concerning the Hölder regularity of the aths of u, seen as a stochastic rocess with values in the Besov saces from the scale. That is, E [ u ] C T β β/2 u C β/2 / [,T ];Bτ,τ α O H γ+2 O,T, and therefore [u P ]C β/2 /[,T ];B ατ,τ O < =, 6.3 for all α and τ with τ = α d + and { < α < min γ + 2 β, + d θ } d β. d However, this result turns out to be not at all satisfactory: The range of arameters in Theorem 3.3 is restricted to

118 8 Sace time regularity of the inhomogeneous heat equation with additive noise ˆ [2, and θ d + 2 κ, d κ or, alternatively, ˆ [2, and θ d κ, d + κ, with κ, κ, deending on d, and O, and 2, 4], see also Remark 3.4iii and i. Since we do not have any lower bound for κ, in the first case we can only guarantee that for arbitrary [2, our solution exists in H γ+2 O, T with θ := d + 2 rovided f and g are smooth enough. But, in this case { min γ + 2 β, + d θ } d β <, d so that the calculations above are useless. In the second case, due to the same arguments, we can only guarantee that, for sufficiently smooth f and g, the solution is in H γ+2,d O, T for [2,. Moreover, if we consider general bounded Lischitz domains, we have to assume that 4, comare Remark 3.4i. Thus, by the calculations above, if 2,, this solution fulfils 6.3 for all α and τ with τ = α d + { and < α < min γ + 2 β, β } d <, 6.4 d since 6.2 has to be fulfilled. This is indeed a first result. However, it still has two drawbacks. Firstly, it does not allow to consider the Hilbert sace case = 2. Secondly, in the view of the convergence rate of the best m-term wavelet aroximation error, α > would be desirable, cf. Section.. In order to overcome these difficulties, we aly the following strategy. We start with the analysis of the Hölder regularity of the aths of elements of H γ,q O, T for the case q >. We are motivated by the fact that in [83, Theorem 4.] it has been roved that for q 2, rovided E [ u ] q C T β βq/2 u q C β/2 /q [,T ];H γ+2 β β Rd + H γ,q Rd +,T, 2 q < β < β. Note that a generalization of this result to the case of bounded Lischitz domains O R d instead of R d + would allow us to choose simultaneously, e.g., = 2 and β close to zero, such that α in 6.4 might become greater than. We rove this generalization in Section 6.. After alying Embedding 4.3, this leads to Hölder regularity results for elements of H γ,q O, T, considered as stochastic rocesses with values in the scale of Besov saces. From the oint of non-linear aroximation theory, the ermitted range of and α is much more satisfactory than in the case q =. In Section 6.2 we rove one asect of the suitability of the saces H γ,q O, T for the regularity analysis of SPDEs: We show that, if we have a solution u H γ,q O, T with low regularity γ, but f and g have high L q L -regularity then we can lift u the regularity of the solution. Finally, in Section 6.3 we rove that under suitable assumtions on the L q L - regularity of f and g, the stochastic heat equation has a solution in the sace H γ,q O, T. Then we can aly the results from Section 6. and obtain sace time regularity results for the solution to the stochastic heat equation. Slightly different versions of the results and roofs resented in this chater have been artially worked out in collaboration with K.-H. Kim, K. Lee and F. Lindner [26].

119 6. Sace time regularity of elements from H γ,q O, T 9 6. Sace time regularity of elements from H γ,q O, T In this section we analyse the Hölder regularity of the aths of elements of H γ,q O, T. We are mainly interested in the case where the summability arameters and q in sace and time, resectively, do not coincide. We start by resenting the two main results: the first one is a generalization of Theorem 3.8 to the case q > ; u H γ,q O, T is seen as a stochastic rocess taking values in the weighted Sobolev saces H ν O, ν, θ R. The second one is concerned, θ with the Hölder regularity of u H γ,q O, T, seen as a stochastic rocess taking values in the Besov saces from the scale. Theorem 6.. Let O be a bounded Lischitz domain in R d. Let 2 q <, γ N, θ R, and u H γ,q O, T. Moreover, let 2 q < β < β. Then there exists a constant C, which does not deend on T and u, such that and E[u] q C β/2 /q [,T ];H γ β β O CT β β q 2 CT β β q 2 u q u q H γ,q O,T + Du q + H γ 2,q + O,T Su q H γ,q O,T ;l 2 H γ,q O,T, 6.5 E u q C β/2 /q [,T ];H γ β β O CT β β q 2 CT β β q 2 u q E u q H γ β β O+ u q H γ,q O,T + Du q H γ 2,q + O,T + Su q H γ,q O,T ;l 2 H γ,q O,T. Before we rove this theorem, we resent our second main result, which follows immediately form Theorem 6. by alying Embedding 4.3. Theorem 6.2. Let O be a bounded Lischitz domain in R d. Let 2 q <, γ N, θ R, and u H γ,q O, T. Moreover, let 2 q < β { < min Then, for all α and τ with τ = α d + and, + d θ }. { < α < min γ β, + d θ β d }, d we have E [ u ] q CT u q C β/2 /q [,T ];Bτ,τ α O H γ,q O,T + Du q, 6.7 H γ 2,q + O,T + Su q H γ,q O,T ;l 2 and 6.6 E u q CT C β/2 /q [,T ];Bτ,τ α O u q H γ,q O,T. 6.8 The constants CT in 6.7 and 6.8 are given by CT = C su β [ β,] { T β βq/2 }, with C from 6.5 and 6.6 resectively.

120 Sace time regularity of the inhomogeneous heat equation with additive noise Proof. The assertion is an immediate consequence of Theorem 6. and Theorem 4.7. Now we turn our attention to the roof of Theorem 6.. For the case that the summability arameters in time and sace coincide, i.e., q =, the assertion is covered by Theorem 3.8. A roof of Theorem 3.8 can be found in [75, Theorem 2.9]. It is straightforward and relies on [83, Corollary 4.2], which is the analogue of Theorem 6. on the whole sace R d. However, we are exlicitly interested in the case q > since it allows for a wider range of arameters β and β, and therefore leads to better regularity results. Unfortunately, the roof technique used in [75, Proosition 2.9] does not work any more in this case. Therefore, we use a different aroach: We make use of [83, Proosition 4.], which covers the assertion of Theorem 6. with R d + := {x, x R d : x > } instead of O, and the Lischitz character of O to derive Theorem 6. via a boundary flattening argument. To this end, we need the following two lemmas, which we rove first. We start with a transformation rule for weighted Sobolev saces, where the transformation and its inverse are assumed to be Lischitz. Remember that ρ G x denotes the distance of a oint x G to the boundary G of a domain G R d. Lemma 6.3. Let G, G 2 be two domains in R d with non-emty boundaries, and let φ : G G 2 be a bijective ma, such that φ and φ are Lischitz continuous. Furthermore, assume that there exists a constant C,, such that C ρ G φ y ρ G 2y Cρ G φ y for all y G 2, and that the a.e. existing Jacobians Jφ and Jφ fulfil Det Jφ = and Det Jφ = a.e.. Then, for any γ [, ],,, and θ R, there exists a constant C = Cd, γ,, θ, φ,, which does not deend on u, such that C u H γ G u φ H γ G2 C u H γ G in the sense that, if one of the norms exists, so does the other one and the above inequality holds. Remark 6.4. i A Lischitz continuous function φ : G G 2 with Lischitz continuous inverse, as in the assumtions of Lemma 6.3, is usually called bi-lischitz. ii The Jacobians Jφ and Jφ in Lemma 6.3 exist λ d -a.e. on G and G 2, resectively, due to Rademacher s theorem: Let U R d be an oen set and let m N. A Lischitz continuous function f : U R m is λ d -a.e. totally differentiable in the classical sense. A roof can be found e.g. in [54, Section 3.]. iii The meaning of u φ for u H γ G with γ and φ as in Lemma 6.3 is naturally given as the comosition of the two functions. However, for negative γ [, this definition is not suitable anymore, since in this case u H γ G is not necessarily a function, but only a distribution. We will define u φ in this case during the roof of Lemma 6.3 in such a way that, in articular, the identity holds. u φ, ϕ = u, ϕ φ, ϕ C G 2, 6.9

121 6. Sace time regularity of elements from H γ,q O, T Proof of Lemma 6.3. We consider consecutively the cases γ =,,. For fractional γ,, the statement follows then by using interolation arguments and Lemma 2.45v. Furthermore, we restrict ourselves to the roof of the right inequality in the assertion of the Lemma, i.e., that there exists a constant C = Cd, γ,, θ, φ,, such that for any u H γ G the following inequality holds: u φ H γ G2 C u H γ G. The left inequality can be roven analogously. For γ =, the assertion follows immediately by using the assumtions of the Lemma and the change of variables formula for bi-lischitz transformations, see, e.g., [62, Theorem 3]. Let us go on and look at the case γ =. Because of the density of the test functions C G in H G, see Lemma 2.45ii, it suffices to rove the asserted inequality for u C G. In this case, because of the assumed Lischitzcontinuity of φ, the classical artial derivatives of u φ exist a.e., comare Remark 6.4ii, and u φ y j = d i= x i u φ y j φ i C d x i u φ i= a.e., since the absolute values of the derivatives φ i, j =,..., d, of the coordinates φ i, y j i =,..., d, are bounded by the Lischitz constant of φ. Thus, alying again the change of variables formula for bi-lischitz transformations and the assumed equivalence of ρ G and ρ G 2 φ on G, we can use the norm equivalence 2.28 together with the fact that, since u C G, the classical derivatives u coincide with the generalized derivatives u x i x i, for i {,..., d}, and estimate u φ y d ρ G 2y θ d dy + u φ G 2 j= G 2 y j y ρ G 2y +θ d dy C u φ y d ρ G 2y θ d dy + G 2 G 2 x i u φ y ρ G 2y +θ d dy i= d C ux ρ G x θ d dx + G G x i ux ρ G x +θ d dx C u H G. i= By the norm equivalence 2.28, these calculations yield u φ H G 2 C u H G, if we can guarantee that for any j {,..., d}, the a.e. existing classical artial derivative u φ is a version of the corresonding generalized derivative u φ y j y j. This can be deduced as follows: By the above calculations, u φ and y j u φ are locally integrable functions on G 2. Furthermore, u φ is Lischitz continuous. Thus, u φ has a Lischitz continuous extension to R d, comare, e.g., Theorem in [54, Section 3..], which we also denote by u φ. Moreover, R y j u φ y,..., y j, y j, y j+,..., y d

122 2 Sace time regularity of the inhomogeneous heat equation with additive noise is absolutely continuous on comact subsets of R for every y,..., y j, y j+,..., y d R d see [54, Section 4.9] for a recise definition of absolute continuity. Thus, as in the roof of Theorem 2 in [54, Section 4.9.2], we can do integration by arts and obtain for every ϕ C G2, Therefore, G 2 y j u φ yϕy dy = y j u φ = u φ y j u φ y ϕy dy. G 2 yj a.e., and the assertion for γ = follows. Finally let us consider the case γ =. Assume for a moment that u C G. By the change of variables formula for bi-lischitz transformations, we have u φ, ϕ = u, ϕ φ, ϕ C G 2 ; see also 2.38 in Remark 2.49 for the extended meaning of,. Using Lemma 2.45viii, i.e., the fact that H Gi H,θ Gi, with + = and θ + θ = d i =, 2, 6. see also Remark 2.49, we obtain u, ϕ φ C u H G ϕ φ H,θ G, ϕ C G 2. Thus, an alication of the already roven assertion for γ =, yields u, ϕ φ C u H G ϕ H,θ G2, ϕ C G 2. Hence, by the density of C G2 in H,θ G 2, cf. Lemma 2.45ii, we obtain Alying 6., this shows that u φ, C u H,θ G2 H G. u φ H G2 C u H G, u C G. 6. Let us consider the general case and assume that u H G. We fix a sequence u n n N aroximating u in H G, which exists by another alication of Lemma 2.45ii. By 6., u n φ n N is a Cauchy sequence in the Banach sace H G2 and, therefore, converges in H G2. We set u φ := lim un φ, convergence in H n G2. Then, Equality 6.9 holds, and, by 6., u φ H G2 C u H G, u H G. We use Lemma 6.3 to rove the following rule for bi-lischitz transformations of elements of H,q O, T.

123 6. Sace time regularity of elements from H γ,q O, T 3 Lemma 6.5. Let G, G 2 be bounded domains in R d and let φ : G G 2 satisfy the assumtions of Lemma 6.3. Furthermore, let u H,q G, T with 2 q < and θ R. Then u φ H,q G2, T with deterministic art Du φ = Du φ and stochastic art Su φ = Su φ. In articular, for any ϕ C G2, with robability one, the equality ut, φ, ϕ = u, φ, ϕ holds for all t [, T ]. + t Dus, φ, ϕ ds + k= t S k us, φ, ϕ dw k s 6.2 Proof. We set f := Du and g := Su. Since u H,q G, T, Lemma 6.3 guarantees that u φ H,q G2, T, f φ H,q + G2, T, g φ H,q G2, T ; l 2 and u φ U,q G2. Thus, all the terms in formula 6.2 are well-defined. In articular, since 6.9 holds, showing that for any ϕ C G2, with robability one, the equality ut,, ϕ φ = u,, ϕ φ + t Dus,, ϕ φ ds + k= t S k us,, ϕ φ dw k s 6.3 holds for all t [, T ], roves our assertion with the right meaning of the brackets,, cf. Remark We consider two different cases. Case. Firstly, we assume that > 2. Let us fix ϕ C G2. By Lemma 6.3, ϕ φ H, θ G for any, and θ R, hence also for and := θ := θ , i.e., fulfilling 2 + =, d 2, where θ + θ = d with + =. Moreover, by Lemma 2.45ii we can choose a sequence ϕ n C G aroximating ϕ φ in H, θ G. We know that for all n N, with robability one, the equality t ut,, ϕn = u,, ϕn + fs,, ϕn ds + k= t g k s,, ϕ n dw k s 6.4 holds for all t [, T ]. Thus, if we can show that each side of 6.4 converges in L 2 Ω; C[, T ] to the resective side of 6.3, the assertion follows. We write ṽ n := ϕ n ϕ φ for n N, and start with the right hand side. We estimate [ t t E su u,, ṽn + fs,, ṽn ds + g k ] s,, ṽ n dw k 2 s t [,T ] k= C E[ ] [ u,, ṽn 2 t ] 2 + E su fs,, ṽn ds + t [,T ] [ t E su g k ] s,, ṽ n dw k 2 s t [,T ] =: C I + II + III, k= 6.5

124 4 Sace time regularity of the inhomogeneous heat equation with additive noise and rove that each of the terms on the right hand side converges to zero for n. Before we do this, we show that the following embeddings hold: H, θ G L, θg, 6.6 H, θ G H,θ G, 6.7 H, θ G L,θ G. 6.8 The first one follows immediately from Lemma 2.45vii. In order to rove the second embedding, we argue as follows: Using the fact that = + 2 and <, together with Hölder s inequality, the boundedness of G, and the norm equivalence 2.28 yields ṽ H,θ G C ṽx ρx θ d G + C C ṽx ρx θ d + C G C ṽ H, θ G, α = α = D α ṽx ρx θ d G G D α ṽx ρx θ d with a constant C indeendent of ṽ H, θ G. The third embedding 6.8 follows with similar arguments. Let us return to 6.5. Since L G L,θ G, cf. Lemma 2.45viii, using embedding 6.8 together with the fact that H 2/q 2/q G L G 2, we obtain [ ] I = E u,, ṽn 2 E [ u, 2 L G ] ṽn 2 L,θ G C u, 2 U γ,q G ṽ n 2 H, θ G. 6.9 Furthermore, since H + G H,θ G, cf. Lemma 2.45viii, we can use embedding 6.7 together with Hölder s inequality and estimate the second term as follows: II = E [ C E su t [,T ] [ su t [,T ] t t fs,, ṽn ds 2 ] fs, 2 H + G ds C f 2 H,q + G,T ṽ n 2 H, θ G. ] ṽ n 2 H,θ G 6.2

125 6. Sace time regularity of elements from H γ,q O, T 5 Finally, by Doob s inequality and Itô s isometry, together with Jensen s inequality and Fubini s theorem, [ t III = E su g k ] s,, ṽ n dw k 2 s t [,T ] k= [ T C E ] g k s,, ṽ n 2 ds k= [ T ] = C E g k s, xṽ n x dx 2 ds k= G [ T C E g k s, x 2 ṽ n x ] 2 dx ds. G k= Thus, inserting = ρ 2θ d/ ρ 2θ d/ and using Hölder s inequality twice, followed by an alication of Embedding 6.6, yields [ T III C E G g k s, x 2 2 ρx θ d dx k= C g 2 H,q G,T ;l 2 ṽ n 2 H, θ G. 2 ds ] G ṽ n x ρx θ d dx 6.2 The combination of the estimates 6.9, 6.2 and 6.2 with 6.5 yields the convergence of the right hand side of 6.4 to the right hand side of 6.3 in L 2 Ω; C[, T ]. Let us now consider the corresonding left hand sides. An alication of Theorem 3.8ii and the fact that q lead to [ E su t [,T ] ] ut,, ṽ n 2 [ E su ut, L t [,T ] G C u 2 H, G,T ṽ n 2 L,θ G C u 2 H,q G,T ṽ n 2 L,θ G. ] 2 ṽn 2 L,θ G Thus, by 6.8, [ E su t [,T ] ] 2 ut,, ṽ n C u 2 H,q G,T ṽ n 2 H, θ G. Hence, also the left hand side of 6.4 converges to the left hand side of 6.3 in L 2 Ω; C[, T ] and the assertion is roved for > 2. Case 2. It remains to consider the case = 2. Relacing by 2 and θ by θ = 2d θ and arguing as in the first case using the inequality [ t E su g k s,, ṽ dws k ] 2 C g 2 H,q 2,θ G,T ;l 2 ṽ 2 L 2,θ G t [,T ] k= for the estimate of III, roves the assertion also for = 2. Now we are ready to rove our main result in this section.

126 6 Sace time regularity of the inhomogeneous heat equation with additive noise Proof of Theorem 6.. As before, we simlify notation and write f := Du and g := Su throughout the roof. We will show that 6.5 is true by induction over γ N; estimate 6.6 can be roven analogously. We start with the case γ =. Fix x O and choose r > small enough, e.g., r := r K with r and K > from Definition 2.. Let us assume for a moment that the suorts in the sense of distributions of u, f and g are contained in B r x for each t and ω. With µ from Definition 2., we introduce the function φ : G := O B r x G 2 := φo B r x R d + x = x, x φx := x µ x, x, which fulfils all the assumtions of Lemma 6.3. Note that, since r has been chosen sufficiently small, one has ρ O x = ρ G x for all x O B r x, so that one can easily show that the equivalence v H ν O v, θ H ν, θ G, v D O, su v B r x, holds for all ν, θ R and >. Together with Lemma 6.3 we obtain for any ν [, ], v H ν, θ O v φ H ν, θ G2, v D O, su v B r x. Thus, denoting ũ := u φ, f := f φ and g := g φ, by Lemma 6.5 we know that on G 2 we have dũ = fdt + g k dwt k in the sense of distributions, see Definition 3.3. Furthermore, since ρ G 2y = ρ R d y for all y φo B r x, the equivalence + v φ H ν, θ G2 v φ H ν, θ Rd +, v D O, su v B r x, holds for any ν [, ], where we identify v φ with its extension to R d + by zero. Therefore, by making slight abuse of notation and writing ũ, f and g for the extension by zero on R d + of ũ, f and g resectively, we have ũ H,q Rd +, T, ũ U,q Rd +, f H,q + Rd +, T, g H,q Rd +, T ; l 2, and dũ = fdt + g k dwt k is fulfilled on R d + in the sense of distributions, see Definition 3.3. Thus, we can aly [83, Theorem 4.] and use the equivalences above to obtain Estimate 6.5 in the following way: E[u] q C β/2 /q [,T ];H β β O C E[ũ] q C β/2 /q [,T ];H β +β Rd + C T β βq/2 ũ q H,q Rd +,T + f q H,q + Rd +,T + g q H,q Rd +,T ;l 2 C T β βq/2 u q H,q O,T + f q H,q + O,T + g q H,q O,T ;l 2 Now let us give u the assumtion on the suorts of u, f and g. Let ξ, ξ,..., ξ m, be a finite artition of unity on O, such that ξ C O, and, for i =,..., m, ξ i C B rx i with x i O. Obviously, dξ i u = ξ i fdt + ξ i gt k dwt k for i =,..., m. Since E[u] q Cm, q C β/2 /q [,T ];H β O β m i=. E[ξ i u] q C β/2 /q [,T ];H β β O,

127 6.2 The saces H γ,q O, T and SPDEs 7 we just have to estimate E[ξ i u] q for each i {,..., m}. For i C β/2 /q [,T ];H β O β one obtains the required estimate as before, using the fact that C O-functions are ointwise multiliers in all saces H ν, θ O, ν, θ R, >, see, e.g., [93, Theorem 3.]. The case i = can be treated as follows: Since ξ has comact suort in O, for all ν, θ R and >, we have and consequently vξ H ν, θ O vξ H ν R d, v D O, 6.22 E[ξ u] q C β/2 /q [,T ];H β β O E[ξ u] q C β/2 /q [,T ];H β R d. By [83, Theorem 4.], a further alication of 6.22 and the fact that C ointwise multiliers in all saces H O, we obtain ν, θ O-functions are E[ξ u] q C β/2 /q [,T ];H β R d CT β βq/2 ξ u q + ξ f q + ξ g q H,q R d,t H,q R d,t H,q R d,t ;l 2 CT β βq/2 ξ u q H,q O,T + ξ f q H,q + O,T + ξ g q H,q CT β βq/2 u q H,q O,T + f q H,q + O,T + g q H,q O,T ;l 2. O,T ;l 2 This finishes the roof of estimate 6.5 for the case γ =. Next, let us move to the inductive ste and assume that the assertion is true for some γ = n N. Fix u H n+,q O, T and let ψ denote an infinitely differentiable function on O fulfilling Then v := ψu x H n,q O, T and dv = ψf xdt+ψgxdw k t k comonent-wise. Also, by Lemma 2.45iii and iv, E[u] q C E[u] q C β/2 /q [,T ];H n+ β β O C β/2 /q [,T ];H n β O+ β E[v] q. C β/2 /q [,T ];H n β β O Using the induction hyothesis and alying Lemma 2.45iii and iv once more together with Lemma 2.55i and ii, we see that the induction goes through. 6.2 The saces H γ,q O, T and SPDEs In this section we are concerned with one asect of the suitability of the saces H γ,q O, T for the regularity analysis of SPDEs. We rove that, if we know that the equation du = a ij u x i x j + f dt + σ ik u x i + g k } dwt k on Ω T O, u = on Ω O has a solution u H γ,q O, T, and f and g = gk k N are smooth enough, then we can lift u the regularity of the solution in the scale H ν,q O, T, ν γ, of arabolic weighted Sobolev saces. For simlicity, in this section we make the following restrictions.

128 8 Sace time regularity of the inhomogeneous heat equation with additive noise Assumtion 6.6. i The coefficients a ij and σ ik do not deend on x O and fulfil Assumtion 3. with vanishing b i, c, and µ k, i, j {,..., d} and k N. That is, a ij and σ ik are real valued redictable stochastic rocesses and there exist constants δ, K > such that for any ω, t Ω T and λ R d, δ λ 2 ã ij ω, tλ i λ j a ij ω, tλ i λ j K λ 2, where ã ij ω, t := a ij ω, t 2 σi ω, t, σ j ω, t l2, with σ i ω, t := σ ik ω, t k N l 2. ii a ij is symmetric, i.e., a ij = a ji for i, j {,..., d}. Under these assumtions we can rove the following result. Theorem 6.7. Let O be a bounded Lischitz domain in R d and let a ij and σ ik, i, j {,..., d}, k N, be given coefficients satisfying Assumtion 6.6. Fix γ R, [2, and q := m for some m N. Furthermore, assume that u H γ+,q O, T is a solution to Eq with f H γ,q + O, T and g Hγ+,q O, T ; l 2. Then, u H γ+2,q O, T, and u q C u q + H γ+2,q O,T H γ+,q O,T f q H γ,q + O,T + g q, H γ+,q O,T ;l 2 where the constant C, does not deend on u, f and g. In order to rove this result, we will use the following lemma taken from [82, Lemma 2.3]. Recall that the saces HT γ are the R d -counterarts of the saces H γ, G, T, comare Remark 3.4. Lemma 6.8. Let 2, m N, and, for i =, 2,..., m, Denote Λ i := λ i γ i/2. Then [ T E C λ i,, γ i R, u i H γ i+2 T, u i, =. m ] Λ i u i L dt i= m i= + C [ T E Λ i f i L + Λ i g x i L R d ;l 2 i<j m m ] Λ j u j L dt j= j i [ T E Λ i g x i L R d ;l 2 Λ jg x j L R d ;l 2 where f i := Du i a rs u i x r x s, gik := S k u i σ rk u i x r deends only on m, d,, δ, and K. Now we rove the main result of this section. m k= k i,j ] Λ k u k L dt, and L l 2 := H l 2. The constant C Proof of Theorem 6.7. The case m =, i.e., = q, is covered by [75, Lemma 3.2]. Therefore, let m 2. According to Remark 2.46 it is enough to show that u q H γ,q + O,T C u q + H γ+,q O,T f q H γ,q + O,T + g q. H γ+,q O,T ;l 2

129 6.2 The saces H γ,q O, T and SPDEs 9 Using the definition of weighted Sobolev saces from Subsection 2.3.3, we observe that [ T u q H γ,q + O,T = E e nθ+ ζ n ute n m ] H γ dt n Z [ T C E e nθ+ ζ n ute n H γ n Z + ζ n ute n H γ + ζ nxu x te n H γ m ] dt. Here ζ nx u x is meant to be the scalar roduct in R d. Now we can use Jensen s inequality and Remark 2.48i to obtain [ T u q H γ,q + O,T C E e nθ+ ζ n ute n m H γ n Z + ut q H γ O + u xt q H γ Odt ]. An alication of Lemma 2.45iii and iv leads to [ T u q H γ,q + O,T C E e nθ+ ζ n ute n m ] H γ dt n Z Therefore, it is enough to estimate the first term on the right hand side, i.e., [ T E e nθ+ ζ n ute n m ] H γ dt n Z [ T = E n,...,n m Z e m i= n i Tonelli s theorem together with the relation θ+ m ζ ni ute ni ]. H γ dt i= + C u q H γ+,q O,T. uc H γ = cγ d c 2 γ/2 u L for c,, 6.24 alied to u ni with u n := ζ n u for n Z, show that we only have to handle m [ T m ] e i= n i θ++γ d E e 2n i γ/2 u ni t L dt. n,...,n m Z Note that since u H γ+,q O, T solves Eq with vanishing initial value, i= du n = a rs u n x r x s + f n dt + σ rk u n x r + gnk dw k t, in the sense of distributions on R d, see Definition 3.3, where f n = 2a rs ζ n x su x r a rs ζ n x r x su + ζ nf, g nk = σ rk ζ n x ru + ζ n g k, and u n =. Furthermore, alying [8, Theorem 4.], we have u n H γ+2 T. Thus, we can use Lemma 6.8 to obtain [ T m ] m E e 2n i γ/2 u ni t L dt C Ini + II ni + C III ni n j i= i= i<j m

130 2 Sace time regularity of the inhomogeneous heat equation with additive noise where we denote [ T I ni := E Λ ni f ni t L [ T II ni := E Λ ni g n i m j= j i x t L R d ;l 2 ] Λ nj u nj t L dt, m j= j i ] Λ nj u nj t L dt, [ T III ni n j := E Λ ni g n i x t L R d ;l 2 Λ n j g n j x t L R d ;l 2 m k= k i,j ] Λ nk u nk t L dt, with Λ n := e 2n γ/2. Thus, it is enough to find a roer estimate for e m i= n i θ++γ d m Ini + II ni + n,...,n m Z i= i<j m III ni n j. Alying 6.24 first, followed by Tonelli s theorem, then Hölder s and Young s inequality, leads to n,...,n m Z = e m i= n i n,...,n m Z e m i= n i θ++γ d θ+ n Z m i= I ni m [ T E f ni t, e ni H γ i= m u nj t, e nj ] H γ dt [ T C E e nθ+ f n t, e n H γ e nθ+ u n t, e n m ] H γ dt n Z n Z [ T Cε E e nθ+ f n t, e n q ] H dt γ n Z [ T + ε E e nθ+ u n t, e n q ] H dt γ. j= j i Using the definition of f n and arguing as at the beginning of the roof, we obtain n Z e nθ+ f n t, e n H γ C u x t H γ O + ut H γ O + ft H γ + O C ut + H γ+ O ft H γ + O.

131 6.3 The stochastic heat equation in H γ,q O, T 2 Moreover, e nθ+ u n t, e n H γ n Z e nθ+ ζ n ute n H γ n Z + e nθ+ ζ nx u x te n H γ + n Z n Z C ut H γ O + u xt H γo + u H γ+o C ut + H γ+ O u H γ + O. e nθ+ ζ n ute n H γ Combining the last three estimates, we obtain for any ε > a constant Cε,, such that n,...,n m Z e m i= n i θ++γ d Similar arguments yield also n,...,n m Z e m i= n i which finishes the roof. θ++γ d m i= I ni ε u q H γ,q + O,T + Cε f q H γ,q + O,T + u q. H γ+,q O,T m II ni + i= i<j m III ni n j ε u q H γ,q + O,T + Cε g q H γ+,q + O,T ;l 2 u q, H γ+,q O,T Iterating Theorem 6.7 and using the roerties from Lemma 2.45 of the weighted Sobolev saces leads to the following result. Corollary 6.9. Let O be a bounded Lischitz domain in R d and let a ij and σ ik, i, j {,..., d}, k N, be given coefficients satisfying Assumtion 6.6. Fix γ, [2, and q := m for some m N. Furthermore, assume that u H,q O, T is a solution to Eq with f H γ 2,q + O, T and g Hγ,q O, T ; l 2. Then u H γ,q O, T, and q, u q H γ,q O,T C u q + H,q O,T f q + g H γ 2,q + O,T H γ,q O,T ;l 2 where the constant C, does not deend on u, f and g. Remark 6.. The assertion of Theorem 6.7 and Corollary 6.9 can be roved in the same way for arbitrary domains G R d with non-emty boundary instead of the bounded Lischitz domain O R d, see [26, Theorem 3.8]. Arguing along the lines of [73,75], it can also be extended to the case where the coefficients deend on the sace variable x O. Also, the symmetry of a ij can be droed. Since we are mainly interested in the stochastic heat equation on bounded Lischitz domains, we do not consider these cases in this thesis. 6.3 The stochastic heat equation in H γ,q O, T In this section we develo a first L q L -theory for the stochastic heat equation on bounded Lischitz domains. We rove that under suitable conditions on the free terms, the stochastic heat

132 22 Sace time regularity of the inhomogeneous heat equation with additive noise equation ossesses a unique solution in the class H γ,q,d O, T with q > 2 Subsection This has imortant consequences for the sace time regularity of the solution rocess toic T2 in the introduction. We collect and discuss them in Subsection In articular, we reach our main goal in this chater and rove a result on the Hölder regularity of the aths of the solution to the heat equation, considered as a stochastic rocess taking values in Besov saces from the non-linear aroximation scale Theorem A result on the L q L -regularity We have already seen in Theorem 3.3 that the stochastic heat equation 6. has a solution u in the class H γ,d O, T = Hγ,,d O, T, rovided the free terms f and g fulfil adequate conditions. In this subsection we want to extend this result and rove the existence of solutions to Eq. 6. in the classes H γ,q,d O, T with q > 2 on general bounded Lischitz domains O Rd. Our main goal is to rove the following statement. Theorem 6.. Let O be a bounded Lischitz domain in R d and let γ. There exists an exonent with > 3 when d 3 and > 4 when d = 2, such that for [2, and q <, Eq. 6. has a unique solution u H γ+2,q,d O, T, rovided f H,q,d O, T Hγ,q,d+ O, T and g H,q,d O, T ; l 2 H γ+,q,d O, T ; l Moreover, there exists a constant C,, which does not deend on u, f and g, such that u q C f q + H γ+2,q,d O,T H,q,d O,T f q H γ,q,d+ O,T + g q + H,q,d O,T ;l 2 g q, 6.26 H γ+,q,d O,T ;l 2 and if q = m for some m N, u q C f q + H γ+2,q,d O,T H,q,d O,T f q H γ,q,d+ O,T + g q H γ+,q,d O,T ;l 2 Furthermore, for arbitrary d 2, if O additionally fulfils a uniform outer ball condition, the assertion holds with =. For bounded Cu-domains, this theorem is covered by [74, Theorem 2.7]. Unfortunately, the roof techniques used there do not work if the boundary is assumed to be only Lischitz continuous. Therefore, we use a different strategy. In a first ste we use the stochastic maximal regularity results from [2, 22] to rove that there exists a solution of the stochastic heat equation in H,q,d O, T with q >, i.e., we rove the following statement. Proosition 6.2. Let O be a bounded Lischitz domain in R d. There exists an exonent with > 3 when d 3 and > 4 when d = 2, such that for [2, and q <, Eq. 6. has a unique solution u H,q,d O, T, rovided f H,q,d O, T and g H,q,d O, T ; l 2. Moreover, there exists a constant C,, which does not deend on u, f and g, such that u q H,q,d O,T C f q H,q,d O,T + g q H,q,d O,T ;l Furthermore, for arbitrary d 2, if O additionally fulfils a uniform outer ball condition, the assertion holds with =.

133 6.3 The stochastic heat equation in H γ,q O, T 23 As already mentioned, we want to aly the maximal regularity theory from [2, 22] in order to rove this result. Therefore, we have to rewrite our equation as a Banach sace valued ordinary SDE of the form } dut + Aut dt = ft dt + bt dw H t, t [, T ] u =, where A is a suitable unbounded oerator on some Banach sace and W H is an H-cylindrical Brownian motion on a Hilbert sace H. Thus, before we start with the roof of Proosition 6.2, we introduce a roer oerator and check its relevant roerties. Let O be a bounded Lischitz domain in R d. As in [26, Definition 3.], for arbitrary,, we define the weak Dirichlet-Lalacian D,w on L O as follows: D D,w := { u W } O : u L O, d D,wu := u = δ i,j u x i x j, i,j= u D D,w. Fix [2, with either [C] = 3 + δ when d 3, or [C2] = 4 + δ when d = 2, or [C3] = when d 2 and O additionally fulfils a uniform outer ball condition, where δ > is taken from [26, Proosition 4.]. Then, the unbounded oerator D,w generates a ositive analytic contraction semigrou { S t } t of negative tye on L O, see [26, Theorem 3.8, Corollary 4.2, Lemma 4.4, and Corollary 4.5]. Therefore, the ositive oerator A := D,w admits an H -calculus of angle less than π/2 and ossesses bounded imaginary owers see Theorem 2.69 and Theorem 2.7. Thus, by [6, Theorem.5.3], [ L O, D D,w ] 2 = D D,w 2, 6.29 where the square root of the negative of the weak Dirichlet-Lalacian D,w 2 the inverse of the oerator is defined as D,w 2 := π 2 t 2 S t dt : L O L O 6.3 with domain D D,w 2 := Range D,w 2, see [2, Chater 2.6]. Endowed with the norm u D D,w 2 := D,w 2 u LO, u D D,w 2, D D,w 2 becomes a Banach sace. Exloiting the fundamental results from [26] and [67], we can rove the following identity, which is crucial if we want to aly the results from [2] in our setting.

134 24 Sace time regularity of the inhomogeneous heat equation with additive noise Lemma 6.3. Let O be a bounded Lischitz domain in R d. There is an exonent with > 4 when d = 2 and > 3 when d 3 such that if [2,, then D D,w 2 = W O 6.3 with equivalent norms. Furthermore, for arbitrary d 2, if O additionally fulfils a uniform outer ball condition, 6.3 holds for arbitrary [2, with equivalent norms. Proof. We fix [2, with as in [C], [C2], or [C3] with δ > from [26, Proosition 4.]. As a consequence of [67, Theorem 7.5] we have D 2,w 2 L O = W O 6.32 and D 2,w 2 u LO Du L O, u W O Moreover, by [26, Proosition 4.] the semigrous { S 2 t } t and { S t } t i.e., S 2 tf = S tf, f L O, t, are consistent, and therefore D,w 2 f = D 2,w 2 f, f L O according to 6.3. Thus, by 6.32, W O = Range D,w 2 = D D,w 2, and the norm equivalence follows immediately from Remark 6.4. The comarison of the L -norms of L /2 u and Du for second order ellitic oerators L is known as Kato s square root roblem in L. On the whole sace R d and for = 2, equivalence of the norms for uniformly comlex ellitic oerators in divergence form with bounded measurable coefficients has been established in the seminal work []. Also, on bounded Lischitz domains it has been roven in [2], among other themes, that for symmetric real-valued ellitic oerators the equivalence L /2 LO D LO holds for certain 2. We exect that, if the results concerning the semigrou generated by D,w from [26], which we use in the roof of Lemma 6.3 and in the roof of Proosition 6.2 below, extend to second order ellitic oerators, then Proosition 6.2 and Theorem 6. remain valid for equations of the tye du = Lu + f dt + g k dwt k, u =. In order to kee the exosition at a reasonable level we do not go into details here. After clarifying these roerties of A = D,w, we are ready to rove the existence of a solution u H,q,d O, T to the stochastic heat equation. Proof of Proosition 6.2. As in the roof of Lemma 6.3 we fix [2, with satisfying either [C], [C2], or [C3] with δ > from [26, Proosition 4.]. Furthermore, we fix q and assume that f H,q,d O, T and g H,q,d O, T ; l 2. We will write W l2 = W l2 t t [,T ] for the l 2 -cylindrical Brownian motion defined by l 2 h W l2 th := wt k e k, h l2 L 2 Ω, t [, T ], k=

135 6.3 The stochastic heat equation in H γ,q O, T 25 where {e k : k N} denotes the standard orthonormal basis of l 2, see also Examle Let Φ be the isomorhism between H,d O; l 2 and Γl 2, H,d O from Theorem Then, since H,d O = W O, see Lemma 2.5, Thus, by Lemma 6.3, with Φ g := Φg L q Ω T, P T, P T ; Γl 2, W O X 2 Φ g L q Ω T, P T, P T ; Γl 2, X 2 := [ L O, D D,w ] 2 = D D,w 2, see also Moreover, as already mentioned, D,w admits an H -calculus of angle less than π/2, and X := D D,w X := L O densely, since C O is contained in D D,w. Using all these facts, we can aly [2, Theorem 4.5ii] and obtain the existence of a stochastic rocess u L q Ω T, P T, P T ; D D,w 6.35 solving the infinite dimensional ordinary SDE } dut D,wut dt = ft dt + Φ g t dw l2 t, t [, T ], u =, in the sense of [2, Definition 4.2]. In articular, there exists a modification ũ of u, such that, with robability one, the equality ũt = t ũs ds + t fs ds + t Φ g s dw l2 s in L O 6.36 holds for all t [, T ]. Note that, since 6.34 holds and W O = H,d O is a umd Banach sace with tye 2 see Lemma 2.5, the stochastic integral on the right hand side is well-defined in the sense of [2] as a W O-valued stochastic rocesses, see Theorem Fix ϕ C O. Then, P-a.s., ũt,, ϕ = t ũs,, ϕ ds + t t fs,, ϕ ds + since P-a.s. Equality 6.36 holds for all t [, T ]. Furthermore, since Φ g s dw l2 s, ϕ, t [, T ], k= g k s,, ϕ dw k s = Φ g s dw l2 s, ϕ P-a.s. in C[, T ]; R, cf. Proosition 3.6, the identity ũt,, ϕ = t ũs,, ϕ ds + t fs,, ϕ ds + k= t g k s,, ϕ dw k s, t [, T ], holds with robability one. Therefore, and by 6.35, since D D,w H,d O = W O, u belongs to H,q,d O, T and solves Eq. 6. in the sense of Definition 3.. Since H,q,d O, T O, T, the uniqueness follows from Theorem 3.3. Thus, in order to finish the roof, we show H,2 2,d

136 26 Sace time regularity of the inhomogeneous heat equation with additive noise Estimate To this end we will use the fact that the stochastic rocess V : Ω T L O defined as V t := t S t sfs ds + t S t sφ g s dw l2 s, t [, T ], is a modification of u, see [2, Proosition 4.4]. Since D,w has the deterministic maximal regularity roerty see [26, Proosition 6.] and D D,w D D,w 2 = H,d O, we obtain t ] E[ t q S t sfs ds C f q 6.37 H,q,d O,T. L q[,t ];H,d O Simultaneously, notice that A := D,w and g resectively Φ g fulfil the assumtions of [22, Theorem.]; we have already checked them in our exlanations above. Thus, alying this result, we obtain t ] E[ t q S t sφ g s dw l2 s C g q H,q,d O,T ;l 2 L q[,t ];H,d O The constants in 6.37 and 6.38 do not deend on f and g. Therefore, using the last two estimates we obtain the existence of a constant C, indeendent of f or g, such that V q C f q + H,q,d O,T H,q,d O,T g q. H,q,d O,T ;l 2 Since V is just a modification of the solution u, Estimate 6.28 follows. Now using the lifting argument from Section 6.2 and interolation theory we can rove the main result of this subsection. Proof of Theorem 6.. Let γ. Again, as in the roof of Lemma 6.3, let [2, with satisfying [C], or [C2], or [C3] with δ > as in Theorem [26, Proosition 4.]. We first consider the case q = m for some fixed m N. Assume that f and g fulfil Then, by Proosition 6.2 there exists a unique solution u H,q,d O, T. An alication of Corollary 6.9 yields the estimate u q C u q + H γ+2,q,d O,T H,q,d O,T f q H γ,q,d+ O,T + g q. H γ+,q,d O,T ;l 2 Thus, u H γ+2,q,d O, T, and using Estimate 6.28, leads to u q C f q + H γ+2,q,d O,T H,q,d O,T g q + H,q,d O,T ;l 2 f q H γ,q,d+ O,T + g q H γ+,q,d O,T ;l 2 Hence, we have roven estimate 6.27, since H γ+,d O; l 2 H,d O; l 2. In order to get rid of the restriction q = m with m N and rove the assertion for general q we argue by following the lines of [82, Proof of Theorem 2., age 7]. Let f and g fulfil 6.25 with a fixed q. Denote E γ := H γ,d+ O H,d O H γ+,d O; l 2 H,d O; l 2. By 6.39 and since H,d O; l 2 H,d O; l 2, for any m N, the oerator R m : L m Ω T, P T, P T ; E γ L m Ω T, P T, P T ; H γ+2,d O f, g R m f, g,

137 6.3 The stochastic heat equation in H γ,q O, T 27 where R m f, g is the unique solution in the class H γ+2,m,d O, T of the corresonding stochastic heat equation 6., is well-defined. Moreover, it is a linear and bounded oerator, and, because of the uniqueness of the solution, R = R m is indeendent of m N. Therefore, using interolation results like, e.g., [3, Theorem 5..2], shows that R is a well-defined linear and bounded oerator from L q Ω T, P T, P T ; E γ to L q Ω T, P T, P T ; H γ+2,d O maing any coule f, g L q Ω T, P T, P T ; E γ to the unique solution Rf, g = u H γ+2,q,d O, T of Eq. 6.. When roving Hölder regularity of the solution, considered as a stochastic rocess taking values in Besov saces from the scale, we will mainly use the following consequence of Theorem 6.. Corollary 6.5. Let O be a bounded Lischitz domain in R d. There exists an exonent with > 3 when d 3 and > 4 when d = 2, such that for [2, and q <, Eq. 6. has a unique solution u H 2,q,d O, T, rovided f H,q,d O, T and g H,q,d O, T ; l Moreover, there exists a constant C,, which does not deend on u, f and g, such that and if q = m for some m N, u q C f q + H 2,q,d O,T H,q,d O,T g q, H,q,d O,T ;l 2 u q C f q + H 2,q,d O,T H,q,d O,T g q. H,q,d O,T ;l 2 Furthermore, for d 2, if O additionally fulfils a uniform outer ball condition, the assertion holds with =. Proof. Since H,d O H,d+ O and H,d O; l 2 H,d O; l 2, condition 6.4 imlies 6.25 with γ =, and the assertion follows immediately from Theorem Sace time regularity In this subsection we collect the fruits of our work and resent new results concerning the Hölder regularity of the aths of the solution to the stochastic heat equation 6. on general bounded Lischitz domains. We start with a Hölder-Sobolev regularity result, i.e., with a result concerning the Hölder regularity of the aths of the solution to the stochastic heat equation, considered as a stochastic rocess taking values in weighted Sobolev saces. Theorem 6.6. Let O be a bounded Lischitz domain in R d and fix γ N. Assume that u H γ+2,q,d O, T is the unique solution of Eq. 6. with f H,q,d O, T Hγ,q,d+ O, T and g H,q,d O, T ; l 2 H γ+,q O, T ; l 2, where q < and [2, with,d i > 3 when d 3 and > 4 when d = 2, or, alternatively, ii = for d 2, if O additionally fulfils a uniform outer ball condition, as in Theorem 6.. Furthermore, fix 2 q < β < β.

138 28 Sace time regularity of the inhomogeneous heat equation with additive noise Then there exists a constant C,, which does not deend on u, f and g, such that E u q C β/2 /q [,T ];H γ+2 β,d β O and if q = m for some m N, C f q H,q,d O,T + f q H γ,q,d+ O,T + g q H,q,d O,T ;l 2 + g q H γ+,q,d O,T ;l 2, E u q C β/2 /q [,T ];H γ+2 β,d β O C f q H,q,d O,T + f q H γ,q,d+ O,T + g q H γ+,q,d O,T ;l 2 Proof. The assertion is an immediate consequence of Theorem 6. and Theorem 6.. Now we look at the solution of the stochastic heat equation as a stochastic rocess taking values in the Besov saces from the scale. Given the setting of Theorem 6., an alication of embedding 4.3 shows that the solution u H γ+2,q,d O, T, γ, of the stochastic heat equation fulfils. u L q Ω T, P T, P T ; B α τ,τ O, τ = α d + d, for all < α < d. We are interested in the Hölder regularity of the aths of this B α τ,τ O-valued rocess. Theorem 6.7. Let O be a bounded Lischitz domain in R d and let the setting of Corollary 6.5 be given. That is, let u H 2,q,d O, T be the unique solution of Eq. 6. with f H,q,d O, T and g H,q,d O, T ; l 2, where q < and [2, with i > 3 when d 3 and > 4 when d = 2, or, alternatively, ii = for d 2, if O additionally fulfils a uniform outer ball condition, as in Corollary 6.5. Furthermore, fix 2 q < β <. Then, for all α and τ with τ = α d +, and < α < β d d, 6.4 there exists a constant C, which does not deend on u, f and g such that and if q = m for some m N, E u q C β/2 /q [,T ];B α τ,τ O C f q H,q,d O,T + g q H,q,d O,T ;l 2 E u q C β/2 /q [,T ];B α τ,τ O C f q H,q,d O,T + g q H,q,d O,T ;l 2 Proof. The assertion follows immediately from Theorem 6.2 and Corollary 6.5.,

139 6.3 The stochastic heat equation in H γ,q O, T 29 Remark 6.8. Since β < is assumed in Theorem 6.7, the Hölder regularity of the aths of the solution rocess determined in 6.42 and 6.43 is always strictly less than 2. Moreover, we have a tyical trade-off between time and sace regularity: the higher the Hölder regularity in time, the more restrictive condition 6.4, and therefore, the less the Besov regularity α in sace. If we rise the Hölder regularity in time direction by ε >, we lose 2ε d d from the Besov regularity α in sace. Examle 6.9. Let O be a bounded Lischitz domain in R d. Let [2, with satisfying condition i from Theorem 6.7 above or, alternatively, let [2, if O additionally fulfils a uniform outer ball condition. Furthermore, assume that f L Ω T, P T, P T ; H,d O and g L Ω T, P T, P T ; H,d O; l 2. Then, for any q, f H,q,d O, T and g H,d O, T ; l 2, and, by Corollary 6.5, there exists a unique solution u H 2,q,d O, T to the stochastic heat equation 6.. Chose an arbitrary α > such that < α < d d. Then there exists a β = βα > and a corresonding m = mα N such that simultaneously 2 m < β < and α < β d d. Therefore, an alication of Theorem 6.7 yields E u q C ε [,T ];B α τ,τ O <, τ = α d +, with ε = εα := β 2 m >. Thus, for all α and τ with we have τ = α d +, and < α < d d, u C[, T ]; B α τ,τ O P-a.s.

140 3 Sace time regularity of the inhomogeneous heat equation with additive noise

141 3 Zusammenfassung In der vorliegenden Arbeit wird die Regularität von Lösungen semilinearer arabolischer stochastischer artieller Differentialgleichungen in der Arbeit stets mit SPDEs abgekürzt auf beschränkten Lischitz-Gebieten untersucht. Es werden Itô-Differentialgleichungen zweiter Ordnung mit Dirichlet-Nullrandbedingung betrachtet. Sie haben die allgemeine Form: d d du = a ij u x i x j + b i u x i + cu + f + Lu dt u = i,j= + i= d σ ik u x i + µ k u + g k + Λu k dwt k on Ω [, T ] O, k= i= on Ω, T ] O, u = u on Ω O. Hierbei bezeichnet O ein beschränktes Lischitz-Gebiet in R d d 2, während T, für den Endzeitunkt steht. Mit w k t t [,T ], k N, wird eine Folge unabhängiger reellwertiger Standard-Brown scher Bewegungen bezüglich einer normalen Filtration F t t [,T ] auf einem vollständigen Wahrscheinlichkeitsraum Ω, F, P bezeichnet und du ist im Sinne von Itôs stochastischem Differential nach der Zeit t [, T ] zu verstehen. Die Koeffizienten a ij, b i, c, σ ik und µ k, mit i, j {,..., d} und k N, sind reellwertige Funktionen auf Ω [, T ] O, welche bestimmten Bedingungen genügen. Diese sind in Kaitel 3 genau formuliert, siehe insbesondere Assumtion 3.. Bei den Nichtlinearitäten L und Λ wird davon ausgegangen, dass sie in geeigneten Räumen Lischitz-stetig sind. Wie diese genau aussehen, wird in Kaitel 5 räzisiert, siehe insbesondere Assumtion 5.9. In dieser Arbeit wird eine funktionalanalytische Sichtweise eingenommen. So wird die Lösung u einer SPDE nicht als eine von ω, t, x Ω [, T ] O abhängige reellwertige Funktion betrachtet. Diese wird vielmehr als eine auf Ω [, T ] definierte Funktion mit Werten in dem mit D O bezeichneten Raum reellwertiger Distributionen aufgefasst. Eine der bekanntesten Gleichungen der Form ist die stochastische Wärmeleitungsgleichung mit additivem oder multilikativem Rauschen. Allgemeinere Gleichungen vom Ty mit endlich vielen w k t t [,T ], k {,..., N}, treten beisielsweise in der nichtlinearen stochastischen Filtertheorie auf, vgl. [8, Section 8.]. Werden unendlich viele Brown sche Bewegungen w k t t [,T ], k N, herangezogen, so können auch Systeme, welche einem weißen Rauschen in Raum und Zeit unterliegen, betrachtet werden, vgl. [8, Section 8.]. Diese Gleichungen werden in der Literatur als mathematische Modelle für Reaktionsdiffusionsgleichungen, welche einem nicht vernachlässigbaren Rauschen unterliegen, vorgeschlagen, vgl. [32, Section.7] und die darin angegebene Literatur, insbesondere [9]. Die Frage nach der Existenz und Eindeutigkeit einer Lösung der Gleichung ist weitgehend geklärt, siehe hierzu exemlarisch [75]. Allerdings kann diese eine Lösung in den überwiegenden Fällen nicht exlizit angegeben und ebenso wenig numerisch exakt berechnet werden. Umso

142 32 Zusammenfassung wichtiger sind daher numerische Verfahren, die eine konstruktive Aroximation der Lösung ermöglichen. Grundsätzlich wird hierbei zwischen uniformen und adativen numerischen Verfahren unterschieden. Letztere versrechen eine effizientere Aroximation, deren Imlementierung ist allerdings mit einem erheblich höheren Aufwand verbunden. Daher muss vorab geklärt werden, ob die erzielbare Konvergenzrate tatsächlich höher ist als bei uniformen Alternativen. Diese Frage lässt sich in zahlreichen Fällen nach einer rigorosen Analyse der Regularität des zu aroximierenden Objekts hier: die Lösung der SPDE klären. Dies gilt insbesondere für numerischen Methoden, welche auf Wavelets basieren. In Abschnitt. dieser Arbeit werden die Zusammenhänge zwischen Regularität und Konvergenzordnung numerischer Methoden für den Fall auf Wavelets basierender Verfahren zur Aroximation einer -fach Lebesgue-integrierbaren Funktion u L O ausführlich erläutert. Der Fehler wird in der L O-Norm gemessen. In diesem Setting wird, einerseits, die Konvergenzordnung uniformer Verfahren durch die Regularität der Zielfunktion u in der Skala W s O, s >, von L O-Sobolev-Räumen bestimmt: u W s O imliziert, dass uniforme Verfahren die Konvergenzrate s/d erreichen können. Insbesondere gilt folgende Umkehrung: Falls u / W s O für ein s >, so wird ein uniformes Verfahren nie eine bessere Konvergenzrate als s/d erreichen. Andererseits, wird die Rate der sogenannten best m-term -Aroximation durch die Regularität der Zielfunktion in der seziellen Skala B α τ,τ O, τ = α d +, α >, von Besov-Räumen bestimmt. Bei dieser Methode wird u für jedes m N durch diejenige Linearkombination von m Termen hier: Wavelets aroximiert, die den Fehler minimiert. Daher gilt die Rate der best m-term -Aroximation als Benchmark für die Konvergenzrate konstruktiver Aroximationsverfahren. Die obigen Resultate haben folgende Konsequenzen für die Entscheidung, welche Klasse von Verfahren bei der Lösung von Gleichungen der Form zum Einsatz kommen sollte: Stimmen die räumliche Sobolev-Regularität und die räumliche Besov-Regularität in der Skala der Lösung u = uω, t, überein, so sind uniforme Verfahren aufgrund ihrer relativen Einfachheit vorzuziehen. Ist dies nicht der Fall, srich, ist die räumliche Besov-Regularität in der Skala höher als die räumliche Sobolev-Regularität, dann besteht die berechtigte Hoffnung, dass durch räumlich adativ arbeitende Verfahren eine höhere Konvergenzordnung erreicht werden kann. Vor diesem Hintergrund wird in der vorliegenden Arbeit folgenden beiden Fragestellungen nachgegangen: T Räumliche Regularität. Wie hoch ist die räumliche Regularität der Lösung u = uω, t, der Gleichung in der Skala von Besov-Räumen? Es wird ein möglichst hohes α > gesucht, so dass für alle < α < α und /τ = α/d + /, die Lösung u als -fach Bochner-integrierbarer B α τ,τ O-wertiger stochastischer Prozess aufgefasst werden kann. T2 Raum-zeitliche Regularität. Angenommen die Lösung u lässt sich als B α τ,τ O- wertiger stochastischer Prozess mit α und τ wie in auffassen. Was kann über die Hölder-Regularität der Pfade dieses Prozesses gesagt werden? Die Behandlung des Punktes T dient der Beantwortung der Frage, ob der Einsatz räumlich adativer numerischer Verfahren zur Lösung von SPDEs gerechtfertigt ist. Sollte sich herausstellen, dass u L Ω [, T ]; B α τ,τ O, τ = α d +, mit α > ssob maxu, 2

143 Zusammenfassung 33 wobei { } s Sob maxu := su s : u L Ω [, T ]; W s O, so lässt sich aufgrund der obigen Erläuterungen eine klare Emfehlung für die Entwicklung adativer Verfahren formulieren. Hierbei bezeichnet L Ω [, T ]; E den Raum aller vorhersagbaren, -fach Bochner-integrierbaren stochastischen Prozesse mit Werten in einem Quasi- Banach-Raum E, E. Die Beantwortung der zweiten Frage T2 soll bei der Konvergenzanalyse entsrechender numerischer Raum-Zeit-Schemata eingesetzt werden. Eine solche Analyse wurde erst vor kurzem in [24] begonnen und befindet sich derzeit noch in ihren Anfängen. Nachdem die Ziele formuliert sind und die Motivation erläutert wurde, sollen im Folgenden die erzielten Resultate zusammengefasst werden. Das Gerüst: Eine geeignete L -Theorie für SPDEs Eine direkte Anwendung abstrakter Ansätze für SPDEs, wie zum Beisiel des Halbgruenansatzes für SPDEs von Da Prato und Zabczyk [32] sowie dessen Weiterentwicklung in [2, 22] oder aber des von Pardoux begründeten Variationsansatzes für SPDEs [], liefern keine zufriedenstellenden Antworten auf die unter T und T2 formulierten Fragen. Daher wird in dieser Arbeit ein indirekter Weg eingeschlagen. Die in [75] entwickelte L -Theorie wird als Grundgerüst benutzt und erweitert. Sie garantiert die Existenz und Eindeutigkeit einer Lösung der Gleichung auf allgemeinen beschränkten Lischitz-Gebieten O R d allerdings noch nicht in den assenden Räumen. Daher muss diese Lösung anschließend hinsichtlich der Fragestellungen T und T2 analysiert werden. Einbettungen gewichteter Sobolev-Räume in Besov-Räumen Die in [75] betrachteten Lösungen linearer SPDEs sind Elemente bestimmter Banach-Räume H γ O, T mit [2, sowie γ, θ R, welche aus stochastischen Prozessen mit Werten in gewichteten Sobolev-Räumen H γ O bestehen. Für γ N lässt sich Hγ O als der Raum aller reellwertigen messbaren Funktionen auf O, welche endliche Norm u α γ O ρ O x α D α ux / ρ O x θ d dx besitzen, definieren. Hierbei bezeichnet ρ O x die Distanz zwischen einem Punkt x O und dem Rand O des Gebietes. Für nicht ganzzahlige γ, \ N können diese Räume mittels komlexer Interolation gewonnen werden, während für negative γ < eine Charakterisierung über Dualität möglich ist. Aus der Definition der Banach-Räume H γ O, T lässt sich unmittelbar schließen, dass diese in dem Raum L Ω [, T ]; H γ O der -fach Bochner-integrierbaren, vorhersagbaren H γ O-wertigen stochastischen Prozesse stetig linear eingebettet sind. In Formeln: H γ O, T L Ω [, T ]; H γ O. 3 bedeutet stetig linear eingebettet. Folglich, führt der Nachweis einer Einbettung gewichteter Sobolev-Räume in die Besov-Räume der Skala unmittelbar zu einer Aussage über die räumliche Besov-Regularität von SPDEs im Sinne von T. Die Vermutung, dass eine solche Einbettung tatsächlich nachgewiesen werden kann, ist durch die in [38] bewiesenen Resultate gestützt. Darin wird, unter Ausnutzung gewichteter Sobolev-Normabschätzungen nachgewiesen, Die Gründe dafür werden ausführlich in Abschnitt.2 dieser Arbeit diskutiert.

144 34 Zusammenfassung dass die Lösungen bestimmter deterministischer ellitischer Differentialgleichungen eine hohe Besov-Regularität in der Skala aufweisen. Dies wurde unter anderem dadurch erreicht, dass die Wavelet-Koeffizienten der Lösung mittels gewichteter Sobolev Halb-Normen abgeschätzt werden konnten. Die Äquivalenz von Besov-Normen und entsrechenden gewichteten Folgennormen von Wavelet-Koeffizienten lieferten schließlich die gewünschte Abschätzung der Besov- Norm. Durch den Einsatz ähnlicher Techniken wird in Kaitel 4 dieser Arbeit nachgewiesen, dass für beliebige beschränkte Lischitz-Gebiete O R d und Parameter [2, sowie γ, ν, Folgendes gilt vgl. Theorem 4.7: H γ,d ν O Bα τ,τ O, τ = α d + {, für alle < α < min γ, ν d d }. 4 Die Beweisführung für den seziellen Fall γ N verläuft ähnlich wie in dem Beweis von [38, Theorem 3.2]. Zudem wird auf auf die Tatsache zurückgegriffen, dass unter den gleichen Bedingungen, H γ,d ν γ ν O W O, wobei W s O für den Abschluss in W s O des mit C O bezeichneten Raumes der unendlich oft differenzierbaren Funktionen mit komaktem Träger in O steht. Diese Aussage wird in Proosition 4. bewiesen. Durch den Einsatz der komlexen Interolationsmethode lässt sich die Einbettung 4 auch auf allgemeine γ > übertragen Theorem 4.7. Folgende Konsequenzen des Theorems 4.7 liegen auf der Hand: Bis zu einem gewissen Grad lässt sich die Untersuchung der räumlichen Regularität der Lösungen von SPDEs in der Skala auf die Analyse der räumlichen gewichteten Sobolev-Regularität derselben zurückführen. Mit anderen Worten verbirgt sich hinter jedem Resultat zur gewichteten Sobolev-Regularität der Lösungen von SPDEs eine Aussage über deren räumliche Besov-Regularität in der Skala. T Räumliche Regularität in der Skala von Besov-Räumen Wie bereits erwähnt, sind die in dieser Arbeit betrachteten Lösungen von SPDEs der Form Elemente der Banach-Räume H γ O, T mit [2,, γ, θ R. Aufgrund der Gleichheit θ = d + d θ, folgt aus der Kombination der Einbettungen 3 und 4 dass H γ O, T L Ω [, T ]; Bτ,τ α O, τ = α d +, für alle < α < γ + d θ d d. 5 In Kaitel 5 wird diese Einbettung benutzt, um räumliche Regularität in der Skala für Lösungen linearer und semilinearer SPDEs auf allgemeinen beschränkten Lischitz-Gebieten O R d nachzuweisen. Lineare Gleichungen Die in [75] entwickelte L -Theorie garantiert die Existenz und Eindeutigkeit einer Lösung u H γ O, T für eine große Klasse linearer Gleichungen der From mit L = und Λ =. Die Anwendung der Einbettung 5 zeigt, dass u L Ω [, T ]; Bτ,τ α O, τ = α d +, für alle < α < γ + d θ d d, 6

145 Zusammenfassung 35 siehe hierzu Theorem 5.2. Damit wurde eine Antwort auf die unter T formulierte Frage für den Fall linearer Gleichungen gefunden: Die Lösung u H γ O, T lässt sich als -fach Bochnerintegrierbarer Bτ,τ α O-wertiger stochastischer Prozess mit /τ = α/d + / auffassen, und zwar für alle < α < α, wobei { α := min γ, + d θ } d > d gewählt werden kann. Die genauen Bedingungen an den Gewichtsarameter θ R, unter denen 6 erfüllt ist, finden sich in dem Hautresultat zur räumlichen Regularität der Lösung linearer SPDEs, Theorem 5.2. Beisielsweise gilt die Aussage 6 für = 2, γ = 2 und θ = d = 2, so dass folglich u L 2 Ω [, T ]; B α τ,τ O, τ = α 2 +, für alle < α < 2 2 gilt. In Verbindung mit der in [92] etablierten Schranke für die räumliche Sobolev-Regularität der Lösungen von SPDEs auf nicht-konvexen olygonalen Gebieten zeigen die erzielten Resultate, dass, in der Tat die Lösung von SPDEs das durch 2 beschriebene Verhalten aufweisen kann. Damit haben wir einen klaren Hinweis dafür, dass räumlich adativ arbeitende Verfahren für die Lösung von SPDEs entwickelt werden sollten. Zahlreiche Beisiele, die diese These untermauern sollen, sowie weiterführende Bemerkungen finden sich in Abschnitt 5.. Semilineare Gleichungen Zahlreiche Phänomene aus der Physik oder aus der Chemie verlangen nach einer Modellierung durch nichtlineare Gleichungen. Es ergibt sich daher die Frage, ob sich die weiter oben erzielten Resultate zur Besov-Regularität der Lösungen linearer SPDEs auf nichtlineare Gleichungen übertragen lassen. Als einen ersten Schritt in diese Richtung wird in Abschnitt 5.2 eine Klasse semilinearer SPDEs der Form mit Lischitz-stetigen Nichtlinearitäten L und Λ daraufhin untersucht. Wie zuvor soll die Einbettung 5 für den Nachweis räumlicher Besov-Regularität in der Skala herangezogen werden. Da allerdings für semilineare Gleichungen keine entsrechende L -Theorie existiert, muss zunächst die Existenz einer Lösung u H γ O, T unter geeigneten Bedingungen nachgewiesen werden. Dies geschieht in Theorem 5.3. Die Nichtlinearitäten L und Λ genügen bestimmten Lischitz-Bedingungen siehe Assumtion 5.9, so dass als gestörte lineare Gleichung interretiert werden kann. Die Anwendung geeigneter Fixunkt-Argumente, siehe Lemma 5.6, liefert dann den Beweis für die Existenz einer Lösung u H γ O, T. Diese erfüllt nach 5 zwangsläufig auch 6, so dass für die betrachtete Klasse semilinearer Gleichungen eine Regularitätsaussage in der Skala bewiesen werden kann, siehe hierzu Theorem 5.5. T2 Raum-zeitliche Regularität Nachdem nachgewiesen werden konnte, dass die Lösung u H γ O, T linearer und nichtlinearer SPDEs der Form für < α < α und /τ = α/d + / als Bτ,τ α O-wertiger stochastischer Prozess aufgefasst werden kann, wird die zweite große Fragestellung T2 dieser Arbeit untersucht: Die Hölder-Regularität der Pfade des Lösungsrozesses. Dies geschieht in Kaitel 6. Die Analyse der Hölder-Regularität der Pfade der in den Banach-Räumen H γ O, T enthaltenen stochastischen Prozesse ist bereits Teil der in [75] entwickelten L -Theorie. Ein Element u H γ O, T wird darin als stochastischer Prozess mit Werten in gewichteten Sobolev-Räumen aufgefasst. Insbesondere wird nachgewiesen, dass für 2/ < β < β, u C β/2 / [,T ];H γ β < P-fast sicher. 7 O β

146 36 Zusammenfassung Hierbei wird, wie üblich, für einen beliebigen Quasi-Banach-Raum E, E, der Raum der κ-hölder-stetigen E-wertigen Funktionen auf [, T ] mit C κ [, T ]; E, C κ [,T ];E bezeichnet. Auf den ersten Blick sieht es so aus, als ließe sich daraus unmittelbar eine Aussage über Hölder- Regularität der Pfade der Lösungen u H γ O, T, aufgefasst als stochastische Prozesse mit Werten in der Skala von Besov-Räumen, herleiten. Eine Anwendung der Einbettung 4 würde genügen. Allerdings sind die sich daraus ergebenden Resultate nicht zufriedenstellend. Dies liegt vorwiegend an der Kombination der folgenden beiden Umstände: Der Hölder-Exonent κ = β/2 / hängt von dem Parameter, der gleichzeitig die Integrabilität in Raumrichtung misst, ab. Gleichzeitig müssen bestimmte Annahmen über den Gewichtsarameter θ R getroffen werden, um überhaut die Existenz einer Lösung u H γ O zu erhalten. Um trotz dieser Hürden geeignete Resultate zu erzielen, bedienen wir uns folgender Strategie. Zunächst wird die Hölder-Regularität der Pfade der Klasse H γ,q O, T stochastischer Prozesse untersucht. Die Elemente dieser Banach-Räume sind q-fach Bochner-integrierbare H γ O- wertige stochastische Prozesse, welche bestimmten Bedingungen genügen. Im Grunde genommen sind es Erweiterungen der Klassen H γ O, T, wobei jetzt der Parameter q, der die Integrabilität nach der Zeit und nach ω Ω misst, sich ausdrücklich von dem Parameter, der die Integrabilität in Raumrichtung angibt, unterscheiden darf. Es lässt sich nachweisen, dass für u H γ,q O, T mit 2 q <, γ N und 2/q < β < β, gilt: u C β/2 /q [,T ];H γ β β O < P-fast sicher, siehe Theorem 6.. Insbesondere, hängt jetzt der Hölder-Exonent nicht mehr vom räumlichen Integrabilitätsarameter ab. Daher ergibt die Anwendung der Einbettung 7 brauchbare Resultate zur Hölder-Besov-Regularität stochastischer Prozesse aus H γ,q O, T auch für den Fall, dass der Gewichtsarameter θ R den oben erwähnten Einschränkungen genügen muss. Diese Resultate lassen sich nur dann für die Beantwortung der unter T2 formulierten Frage heranziehen, wenn nachgewiesen werden kann, dass die Lösungen zu den SPDEs der Form in der Klasse H γ,q O, T mit q enthalten sind. Das bedeutet, dass die in [75] entwickelte L -Theorie soweit wie möglich zu einer L q L -Theorie ausgebaut werden muss. In dieser Arbeit wird eine erste L q L -Theorie für die stochastische Wärmeleitungsgleichung mit additivem Rauschen auf allgemeinen beschränkten Lischitz-Gebieten entwickelt, siehe hierzu insbesondere Theorem 6.. Die Beweise basieren auf einer Kombination von Resultaten aus dem Halbgruenansatz mit Techniken aus dem von N.V. Krylov begründeten analytischen Ansatz für SPDEs. Aus dem Halbgruenansatz kann die Existenz einer Lösung, die allerdings geringe räumliche gewichtete Sobolev-Regularität aufweist, gezeigt werden, siehe Theorem 6.2. Indem Techniken aus dem analytischen Ansatz benutzen werden, kann anschließend nachgewiesen werden, dass diese Regularität anwächst, sobald die Koeffizienten der Gleichung eine höhere Regularität haben, siehe Theorem 6.7. Um diese beiden Ansätze zusammenbringen zu können, muss zunächst nachgewiesen werden, dass die jeweiligen Lösungsbegriffe übereinstimmen. Dazu werden im Laufe der Arbeit einzelne Hilfsresultate bewiesen, siehe etwa Theorem 2.54 sowie Proosition 3.6. Die in Kaitel 6 durchgeführte Analyse führt schließlich zu einer zufriedenstellenden Aussage über die Hölder Regularität der Pfade der Lösung der Wärmeleitungsgleichung u H γ,q,d O, T, aufgefasst als stochastischer Prozess mit Werten in den Besov-Räumen aus der Skala, siehe Theorem 6.7. Insbesondere lässt sich unter geeigneten Bedingungen an die Komonenten der Gleichung nachweisen, dass für alle Parameter, die der Bedingung genügen, gilt: 2 q < β <, τ = α d +, und < α < β d d u C β/2 /q [,T ];B α τ,τ O < P-fast sicher.

147 37 Notation We collect here frequently used notations from this thesis. The number in the right column refers to the age where the symbol is introduced or where it aears first. General mathematics N set of ositive integers {, 2,...} N set of non-negative integers {,, 2,...} Z set of integers R set of real numbers R + set of ositive real numbers, C set of comlex numbers d N, d 2 dimension R d d-dimensional Euclidian sace {x,..., x d : x,..., x d R} R d + half sace in R d, {x,..., x d R d : x > } B r x oen ball with radius r > centred at x, {y R d : y x < r} A interior of a set A R d A B set of all maings from a set B to a set A. absolute value, Euclidian norm on R d, or cardinality of a finite set; in Examle 5.6 also used for the scale level see. 9 Σ σ the sector {z C \ {} : arg z < σ} C 52 δ j,k Kronecker symbol norm equivalence 22 isomorhic 22 = isometrically isomorhic 22 Id identity oerator A indicator or characteristic function u G restriction of u to G continuously linearly embedded 22 T > time horizon 2 A B, A closure of A B in B, B [E, E 2 ] η comlex interolation sace of exonent η, 22, 77 E E 2 intersection sace for a comatible coule E, E 2 22 E E 2 cartesian roduct 22 u d i= u x i xi, whenever it makes sense existential quantifier A B A is a subset of B A B A is a roer or strict subset of B, i.e., A B and A B A B same meaning as A B, emhasizing that A B z comlex conjugate of a comlex number z C, a b min{a, b}

148 38 Notation Oerators LE, E 2 vector sace of all linear and bounded oerators from E to E 2 5 LE LE, E 5 L f H, E vector sace of finite rank oerators from H to E 25 L H, U vector sace of nuclear oerators from H to U 6 L 2 H, U vector sace of Hilbert-Schmidt oerators from H to U 6, 27 E dual sace of E, i.e., E := LE, R 5 x, x E E dual airing of x E and x E 6 x, x dual airing of x E and x E 6 Γ H, E vector sace of γ-summing oerators from H to E 25 Γ H,E, norm on Γ H, E, equivalent to Γ H,E 25 ΓH, E vector sace of γ-radonifying oerators 26 ΓH,E, norm on ΓH, E, equivalent to ΓH,E 26 h x rank one oerator h, H x LH, E 25 rans range of an oerator S : E E 2, {Sx : x E } DA domain of an unbounded oerator A 5 ρa resolvent set of an oerator A 5 σa sectrum of an oerator A 5 D,w weak Dirichlet-Lalacian 23 Domains G arbitrary domain, i.e., an oen and connected subset of R d 6 G boundary of a domain G R d 6 ρx, ρ G x distance of a oint x G to the boundary G 6 ψ infinitely differentiable function on G, equivalent to ρ G 37 O bounded Lischitz domain in R d 6 Measure theory and robability M, A, µ σ-finite measure sace 7 L M, A, µ; E sace of µ-equivalence classes of -integrable strongly A- 7 measurable functions from M to E,, L M, A, µ; E sace of µ-equivalence classes of strongly A-measurable functions 7 with a.e. finite norm L M; E shorthand for L M, A, µ; E,, ] 7 L M shorthand for L M, A, µ; R,, ] 7 L L R d, BR d, λ d ; R 8 BE Borel σ-field on a quasi-normed sace E, i.e., the σ-field generated 7 by the standard toology on E PI ower set of I 8 δ i Dirac measure 8 l I L I, PI, i I δ i; R 8, l2 I scalar roduct on l 2 I 8 l2 I norm, l2 I on l 2 I 8 l 2 l 2 N 8 λ d Lebesgue measure on R d, BR d and restrictions on A, BA for 8 A BR d λ λ

149 Notation 39 gλ d measure with density g with resect to λ d 8 dx shorthand for λ d dx 8 f, g G fg dx for fg L G, BG, λ d ; R 8 Ω, F, P comlete robability sace 8 P comlete robability measure on Ω, F 8 E[ ], E exectation 8 F t t [,T ] normal filtration on Ω, F, P 2 L F Ω;... closure of the finite rank F t t [,T ] -adated ste rocesses in 3 L Ω; ΓL 2 [, T ]; H, E {wt k t [,T ] } k N sequence of stochastically indeendent real-valued standard Brownian 2 motions with resect to a normal filtration F t t [,T ] W H H-cylindrical Brownian motion 27 T Φt dw Ht stochastic integral of Φ with resect to W H 28 3 Ω T Ω [, T ] 2 P T redictable σ-field on Ω T 2 P T roduct measure P λ on Ω T, F B[, T ] and on Ω T, P T 2 a.e., µ-a.e. almost everywhere a.s., P-a.s. almost surely Distributions and derivatives C G sace of infinitely differentiable real-valued functions with comact 2 suort in the domain G D G sace of real-valued distributions 2 SR d Schwartz sace of raidly decreasing real-valued functions on R d 2 S R d sace of real-valued temered distributions 2 S R d ; C sace of comlex-valued temered distributions 2 F, F Fourier transform on S R d ; C and its inverse 2 u, ϕ alication of u D G u S R d to ϕ C G ϕ SRd ; 2 see also 2.24 and 2.38 for generalizations D α u, α N d classical derivative 2 D α u, α N d generalized/weak/distributional derivative 2 D m u, m N generalized/weak/distributional derivative of order m and the vector 2 of all generalized/weak/distributional derivatives of order m u x, u xx, u x i, u x i x j generalized/weak/distributional derivatives of first and second order 2 su u suort of a distribution u D G H Σ σ set of all bounded analytic functions on the sector Σ σ 52 H Σ σ subset H Σ σ consisting of all functions fulfilling Function saces non-linear aroximation scale 2 CG sace of real-valued continuous functions on a domain G 2 C r G, r N sace of real-valued r-times continuously differentiable functions 2 CG sace of real-valued continuous functions on G 2 C r G, r N sace of real-valued r-times continuously differentiable functions 2 with derivatives which can be extended to G C κ [, T ]; E sace of Hölder continuous functions taking values in the quasi- Banach sace E κ, 9

150 4 Notation W s G L -Sobolev sace of order s 3 W s G closure of C G in W s G 32 H, s HR s d sace of Bessel otentials 33 HR s d ; l 2 sace of Bessel otentials for l 2 -valued functions 35 Hl s 2 shorthand for HR s d ; l 2 35 H γ G weighted Sobolev sace of order γ with summability arameter 37 and weight arameter θ H γ G; l 2 weighted Sobolev sace of l 2 -valued functions 42 L G H G 37 B,qG s Besov sace 46 B,qG s closure of C G in Bs,qG 69 F,qG s Triebel-Lizorkin sace 5 ω n t, u, G n-th order L -modulus of smoothness 47 K r t, u, G Peetre K-functional 82 n h u, n h [u] n-th difference of a function u with ste h Rd 47 Saces of stochastic rocesses and random variables L Ω T ; E L Ω T, P T, P T ; E,, ] 2 H γ,q G, T L qω T, P T, P T ; H γ G 57 H γ,q G, T ; l 2 L q Ω T, P T, P T ; H γ G; l 2 57 U γ,q G L qω, F, P; H γ 2/q 2/q G 57 H γ,q G, T see Definition H γ,q T see Remark H γ,q T see Remark Du, Su deterministic and stochastic art of an element u H γ,q G, T, 57 see Definition 3.3 H γ G, T Hγ, G, T H γ G, T ; l 2 H γ, G, T ; l 2 U γ G U γ, G H γ G, T Hγ, G, T Wavelets φ scaling function of tensor roduct tye on R d 7 ψ i multivariate mother wavelets corresonding to φ i =,..., 2 d 7 ψ i,j,k, φ k dyadic shifts and dilations of the scaling function and the corresonding 7 wavelets j, k N Z d ψ i, ψ i,j,k, φ k elements of the corresonding dual Riesz basis 7 ψ i,j,k, L -normed wavelets 7 ψ i,j,k, L -normed dual wavelet, = / 7 {ψ λ : λ } wavelet Riesz basis of L 2 O 3 {ψ λ : λ j } wavelet basis at scale level j + j 2 {φ λ : λ j } scaling functions at level j j 2 {ψ λ : λ j } scaling functions at level j 2 S j j j multiresolution analysis 2 Σ m sace of m-term aroximation in L O 4 Σ m,w r O sace of m-term aroximation in W r O 93

151 Notation 4 Semi-Quasi-Norms [ ] W m G 32 W m G, m N 74 W s G, s / N 32 H k G 38 B s,q G 47 [u] C κ [,T ];E, u C[,T ];E, u C κ [,T ];E 9 [ ] α m, α m, t + 56

152 42 Notation

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161 Erklärung Ich versichere, dass ich meine Dissertation Besov Regularity of Stochastic Partial Differential Equations on Bounded Lischitz Domains selbstständig, ohne unerlaubte Hilfe angefertigt und mich dabei keiner anderen als der von mir ausdrücklich bezeichneten Quellen und Hilfen bedient habe. Die Dissertation wurde in der jetzigen oder einer ähnlichen Form noch bei keiner anderen Hochschule eingereicht und hat noch keinen sonstigen Prüfungszwecken gedient. Marburg, 9. Dezember 23