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.