DFG-Schwerpunktprogramm 1324

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1 DFG-Schwerunktrogramm 1324 Extraktion quantifizierbarer Information aus komlexen Systemen Satial Besov Regularity for Stochastic Partial Differential Equations on Lischitz Domains P. A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R. L. Schilling Prerint 66

2 Edited by AG Numerik/Otimierung Fachbereich 12 - Mathematik und Informatik Philis-Universität Marburg Hans-Meerwein-Str Marburg

3 DFG-Schwerunktrogramm 1324 Extraktion quantifizierbarer Information aus komlexen Systemen Satial Besov Regularity for Stochastic Partial Differential Equations on Lischitz Domains P. A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, R. L. Schilling Prerint 66

4 The consecutive numbering of the ublications is determined by their chronological order. The aim of this rerint series is to make new research raidly available for scientific discussion. Therefore, the resonsibility for the contents is solely due to the authors. The ublications will be distributed by the authors.

5 Satial Besov Regularity for Stochastic Partial Differential Equations on Lischitz Domains Petru A. Cioica, Stehan Dahlke, Stefan Kinzel, Felix Lindner, Thorsten Raasch, Klaus Ritter, René L. Schilling Abstract We use the scale of Besov saces B α τ,τ(o), α >, 1/τ = α/d + 1/, fixed, to study the satial regularity of the solutions of linear arabolic stochastic artial differential equations on bounded Lischitz domains O R d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear aroximation schemes. The roofs are based on a combination of weighted Sobolev estimates and characterizations of Besov saces by wavelet exansions. Keywords: Stochastic artial differential equation, Besov sace, Lischitz domain, wavelet, weighted Sobolev sace, nonlinear aroximation, adative numerical scheme Mathematics Subject Classification (21): 6H15, Secondary: 46E35, 65C3 1 Introduction In this aer, the satial Besov regularity of the solutions of linear stochastic evolution equations on bounded Lischitz domains is studied. We combine regularity results by Kim [3] on stochastic artial differential equations (SPDEs, for short) on Lischitz domains in terms of weighted Sobolev saces with methods used in Dahlke, DeVore [13], where the Besov regularity of (deterministic) ellitic equations on Lischitz domains is investigated. Our considerations are motivated by the question whether adative and other nonlinear aroximation methods for the solutions of SPDEs on Lischitz domains ay off in the sense that they yield better convergence rates than uniform methods. Thus referring to a This work has been suorted by the Deutsche Forschungsgemeinschaft (DFG, grants DA 36/13-1, RI 599/4-1, SCHI 419/5-1) and a doctoral scholarshi of the Philis-Universität Marburg. 1

6 numerical theme and combining concets and methods from different areas and scientific communities, the article is addressed to readers of both worlds: stochastic analysis and numerical analysis. Therefore, we give a rather detailed account in the first art of the aer, emhasizing concetual and notational clarity. Our setting is as follows. On a finite interval [, T] [, ) let (w κ t ) t [,T], κ N = {1, 2,...}, be indeendent, one-dimensional standard Brownian motions with resect to a filtration (F t ) t [,T] of σ-algebras on a comlete robability sace (Ω, F, P). Throughout the aer we assume that (F t ) t [,T] is normal, i.e. the filtration satisfies the usual hyotheses, see, e.g. [18, Section 3.3.]. Let O R d be a bounded Lischitz domain. We consider the model equation du = d µ,ν=1 a µν u xµx ν dt + g κ dwt κ, u(, ) = u, (1) κ=1 for t [, T] and x O. Here du is Itô s stochastic differential with resect to t, (a µν ) 1 µ,ν d R d d is a strictly ositive definite, symmetric matrix and the coefficients g κ, κ N, are random functions deending on t and x such that the maings Ω [, T] (ω, t) g κ (ω, t, ) are redictable rocesses with values in certain function saces. For details see Section 2.3. Equation (1) is understood in a weak or distributional sense, i.e. u is a solution of (1), if for all smooth and comactly suorted test functions ϕ C (O) the equality u(t, ), ϕ = u, ϕ + d µ,ν=1 t a µν u xµx ν (s, ), ϕ ds + κ=1 t g κ (s, ), ϕ dw κ s holds for all t [, T] P-almost surely. Here and throughout the aer we write u, ϕ for the alication of a distribution u D (O) to a test function ϕ C (O). The existence and uniqueness of solutions of equation (1), resectively equation (3) below, within certain classes H γ,θ (O, T) of stochastic rocesses has been shown in [3]; see also the earlier aers by Krylov, Lototsky and Kim, e.g. [29], [31], [32], [35]. Roughly seaking, the classes H γ,θ (O, T) are L -saces of functions on Ω [, T] with values in weighted Sobolev saces H γ,θ (O) that can be regarded as generalizations of the classical Sobolev saces with zero Dirichlet boundary condition. Again we refer to Section 2.3 for recise definitions. Let us remark that in Examles 17, 18 and 19, illustrating our Besov regularity result in Section 3, the solution of equation (1) in the class H γ 2,θ (O, T) coincides with the unique weak solution with zero Dirichlet boundary condition in the sense of Da Prato, Zabczyk [18], and hence can be reresented by the well known stochastic variation-of-constants formula u(t, ) = e ta u + t e (t s)a G(s) dw s, t [, T]. (2) Here (e ta ) t is the semigrou of contractions on L 2 (O) generated by the artial differential oerator A = d 2 with zero Dirichlet boundary condition considered as an µ,ν=1 aµν x µ x ν 2

7 unbounded oerator on L 2 (O), (G(t)) t [,T] is an oerator-valued rocess and (W t ) t [,T] is a cylindrical Wiener rocess on l 2 (N), see Remarks 13 and 14 in Section 2.3. As already mentioned, our motivation to study the Besov regularity of SPDEs is the theme of nonlinear aroximation of the solution rocesses. For deterministic settings, a detailed overview of nonlinear aroximation and an exosition of the characterization of its efficiency in terms of the Besov smoothness of the target functions can be found in DeVore [21], see also Cohen [8, Chaters 3 and 4]. Let us consider an examle of aroximation by wavelets in L (O), the L -sace of real-valued functions on O, (1, ). To this end, let {ψ λ : λ } be a wavelet basis on O and let f L (O) be a target function which we want to aroximate by functions f N L (O) belonging to certain aroximation saces S N, where N is the number of arameters used to describe the elements of S N. We secify the index set of the wavelet basis by writing = j j 1 j; the wavelets ψ λ, λ j, j j, are those at scale levels j j resectively, and ψ λ, λ j 1, are the scaling functions at the coarsest level j Z. In the case of uniform wavelet aroximation u to a highest scale level j 1 + n, n N, the aroximation saces are { j } 1+n S N = S N(n) = c λ ψ λ : c λ R, λ j, j {j 1,..., j 1 + n}, j=j 1 λ j where N = N(n) = j 1+n j=j 1 j N is the cardinality of the set of all indices u to scale level j 1 + n. Let e N (f) = inf fn S N f f N L(O) be the corresonding aroximation error measured in L (O). It is well known that under certain technical assumtions on the wavelet basis the decay rate of e N (f) is linked to the L -Sobolev smoothness of the target function. More recisely, there exists an uer bound r N deending on the wavelet basis such that, for all s [, r], f W s (O) = e N (f) C N s/d, N = N(n), n N, for some constant C > which does not deend on N. The fractional order Sobolev saces W s (O) are defined in the next section. One can also show the converse C > n N : e N (f) C N s/d, N = N(n) = f W s (O), s < s. If we consider instead best N-term aroximation as a form of nonlinear aroximation, the aroximation saces are { } Σ N = c λ ψ λ : Λ, Λ N, c λ R, λ Λ, λ Λ N N, and in this case the decay rate of the error σ N (f) := inf fn Σ N f f N L(O) is governed by the smoothness of f measured in certain L τ (O)-norms, τ <, which are weaker than the L (O)-norm: For all α [, r], f B α τ,τ (O), 1 τ = α d + 1 = σ N (f) C N α/d, N N, 3

8 s linear arox. line α B α τ,τ (O) nonlinear arox. line: β W β (O) 1 τ = s d Figure 1: Linear vs. nonlinear aroximation illustrated in a DeVore-Triebel diagram. 1 τ 1 τ Bτ,τ α (O) being a Besov sace as defined in Section 2.2. Therefore, if the target function f belongs to Bτ α,τ (O), 1/τ = α/d+1/, for some α [, r], and if in addition β := su{s R : f W s (O)} < α, then the convergence rate of uniform wavelet aroximations is inferior to the convergence rate of the best N-term wavelet aroximation. The latter can be considered as a benchmark for the convergence rate of adative numerical algorithms, see [9], [1], [12]. This situation is illustrated in Figure 1, where each oint (1/τ, s) reresents the smoothness saces of functions with s derivatives in L τ (O). Note that the nonlinear aroximation line {(1/τ, s) [, ) 2 : 1/τ = s/d + 1/} is also the Sobolev embedding line. For bounded domains, all saces left to this line as well as the saces Bτ,τ(O) s on the line are continuously embedded in L (O). Let us return to equation (1) and assume that the solution u = u(ω, t, x), (ω, t, x) Ω [, T] O, vanishes on the boundary O, satisfying a zero Dirichlet boundary condition. It is clear that the smoothness of x u(ω, t, x) deends on the smoothness of the maings x g κ (ω, t, x), κ N. However, even if the satial smoothness of the g κ is high, the Sobolev smoothness of x u(ω, t, x) can be additionally limited by singularities of the satial derivatives of u at the boundary of O, due to the zero Dirichlet boundary condition and the shae of the domain. Such corner singularities are tyical examles for the fact that the satial L -Sobolev regularity of u may be exceeded by the regularity in the scale of Besov saces Bτ,τ α (O), 1/τ = α/d+1/. In this aer, we resent a result on the satial Besov regularity of the solution u to equation (1) which has the following structure: If u L (Ω [, T], P, P λ; W s (O)) 4

9 and if the functions g κ, κ N, are sufficiently regular, then u L τ (Ω [, T], P, P λ; B α τ,τ(o)) for certain α > s and 1/τ = α/d + 1/. Here P is the redictable σ-algebra w.r.t. the filtration (F t ) t [,T] and λ denotes Lebesgue measure on [, T]. This result is imortant for the theoretical foundation of adative numerical methods for the aroximation of u. The roof is based on a wavelet exansion of an extension of O x u(ω, t, x) to R d, which allows us to estimate the Bτ,τ α (O)-norm in terms of the wavelet coefficients. We aly a strategy similar to the one used in Dahlke, DeVore [13], where the Besov regularity of (deterministic) ellitic equations on Lischitz domains is investigated with the hel of an estimate of weighted Sobolev norms of harmonic functions. Our substitute for the latter is an estimate of weighted Sobolev norms of the solution of equation (1) rovided by Kim [3]. There exists an extensive literature on the Besov regularity of SPDEs. In general, however, the assumtions on the domain and the scale of arameters considered do not fit into our setting. To mention an examle, the semigrou aroach to SPDEs of Da Prato, Zabczyk [18], which is laced in a Hilbert sace framework, has been generalized to M-tye 2 Banach saces by Brzeźniak [4], [5], for the urose of gaining better Hölder regularity results. Roughly seaking, the oerator A aearing in equation (2) is considered as the generator of a semigrou on L (O) for some 2, and the stochastic integral in (2) is considered as an stochastic integral in an interolation sace X between L (O) and D(A) L (O), the domain of A, realizing a zero Dirichlet boundary condition. If O is sufficiently smooth, then D(A) = W 2(O) W 1(O) and X Bs,2 (O) for some s [, 2]. In this situation, the Sobolev embedding theorem leads to Hölder regularity results, and these results become better for large. With the hel of a theory of stochastic integration in wider classes of Banach saces, this aroach has been generalized in the works of Van Neerven, Weis, Dettweiler and Veraar, see, e.g. [2], [37], [38], [39], comare also Brzeźniak, Van Neerven [6]. In contrast to these works the roblem considered here is of a different nature. Firstly, we are exlicitly interested in domains with non-smooth boundary. For olygonal non-convex domains, it is well known that W2 2(O) W 2 1 (O) D(A), where D(A) := {u W 2 1 (O) : Au L 2 (O)}, A = = d 2 µ=1, see Grisvard [24], [25], and x 2 µ for more general Lischitz domains see Jerison, Kenig [28]. Secondly, we are interested in the secial scale Bτ,τ(O), α 1/τ = α/d + 1/, τ >, fixed, including in articular saces which are no Banach saces but quasi-banach saces. The arameter τ decreases if α increases and Bτ,τ(O) α fails to be a Banach sace for τ < 1. While our methods work in this setting, any direct aroach requires (at least!) a fully-fledged theory of stochastic integration in quasi-banach saces which is not yet available. Let us emhasize that our result can be extended to more general linear equations of 5

10 the tye du = d µ,ν=1 u(, ) = u, ( a µν u xµx ν + b µ u xµ + cu + f ) dt + ( d σ µκ u xµ + η κ u + g )dw κ t κ, κ=1 µ=1 (3) including, in articular, the case of multilicative noise. Here the coefficients a µν, b µ, c, σ µκ, η κ and the free terms f and g κ are random functions deending on t and x. This extension is ossible because one of our main tools, the weighted Sobolev norm estimate of Corollary 12 in Section 2.3, holds for equations of tye (1) as well as for equations of tye (3). Since this mainly adds notational comlications, we will focus on equation (1) and refer to Aendix B for a short account of how to treat equations of tye (3). The aer is organized as follows: In Section 2 we collect the notations, definitions and reliminary results needed later on. Some general notations are introduced in Section 2.1. Section 2.2 rovides the necessary facts on Besov saces and wavelet decomositions. In Section 2.3 a short introduction to the general L -theory of SPDEs on Lischitz domains due to Kim [3] is given, including definitions of the already mentioned saces H γ,θ (O), Hγ,θ (O, T). Finally, in Section 3 the Besov regularity result (Theorem 15) is stated and roved, and some concrete examles for an alication of the result are given. 2 Preliminaries 2.1 Some notations and conventions In this and the next subsection O R d can be an arbitrary (not necessarily bounded) Lischitz domain. A domain is called Lischitz if each oint on the boundary O has a neighbourhood whose intersection with the boundary after relabeling and reorienting the coordinate axes if necessary is the grah of a Lischitz function By D (O) we denote the sace of Schwartz distributions on O. If not exlicitly stated otherwise, all function saces or saces of distributions are meant to be saces of real-valued functions or distributions. If f D (O) is a generalized function and α = (α 1,...,α d ) N d is a multi-index, we write Dα f = α f x α xα d d for the corresonding derivative w.r.t. x = (x 1,...,x d ) O, where α = α α d. As in equations (1) and (3) we also use the notation f xµx ν = 2 f x µ x ν, f xµ = f x µ. For m N, D m f = {D α f : α = m} is the set ( d+m 1 ) of all m-th order derivatives of f which is identified with an R m -valued distribution. Given [1, ) and m N, W m (O) denotes the classical Sobolev sace consisting of all (equivalence classes of) measurable functions f : O R such that f W m (O) = f L(O) + f W m (O) = ( O f(x) dx) 1/ + α =m ( O Dα f(x) dx) 1/ is finite. For (1, ) and s (m, m + 1), m N, we define the fractional order Sobolev sace W s (O) to be the Besov sace B,(O) s introduced in the next subsection. (This scale of fractional order 6

11 Sobolev saces can also be obtained by real interolation of W n (O), n N. One can show that W2 n(o) = Bn 2,2 (O) for all n N and W n(o) Bn, (O) for all n N, > 2, see, e.g. Triebel [44, Remark 2.3.3/4 and Theorem (b)] together with Disa [23].) Given any countable index set J, the sace of -summable sequences indexed by J is denoted by l = l (J ) and l is the resective norm. Usually we have l = l (N) but, for instance we may also use the notation D m f(x) l = α =m Dα f(x) for f W m (O). Given a distribution f D (O) and a smooth and comactly suorted test function ϕ C (O), we write f, ϕ for the alication of f to ϕ. If H is a Hilbert sace, then, H denotes the inner roduct in H. Given another Hilbert sace U, we denote by L (HS) (H, U) and L (nuc) (H, U) the saces of Hilbert-Schmidt oerators and nuclear oerators from H to U resectively, see, e.g. Pietsch [46, Sections 6 and 15] or Da Prato, Zabczyk [18, Aendix C] for definitions. We also abbreviate L (HS) (H) = L (HS) (H, H) and L (nuc) (H) = L (nuc) (H, H). M 2,c T (H, (F t)) is the sace of continuous, square integrable, H-valued martingales with resect to the filtration (F t ) t [,T]. For Ω [, T] we use the shorthand notation Ω T and P = σ ( {]s, t] F s : s < t T, F s F s } {{} F : F F } ) is the redictable σ-algebra. P λ is the roduct measure of the robability measure P on (Ω, F) and Lebesgue measure λ on ([, T], B([, T])), where B([, T]) denotes the Borel σ-algebra on [, T]. Given any measure sace (A, A, m), any (quasi-)normed sace B with (quasi-)norm B and any summability index >, we denote by L (A, A, m; B) the L - sace of all strongly measurable functions u : A B whose (quasi-)norm u L(A,A,m;B) := ( A u(z) B m(dz)) 1/ is finite. All equalities of random variables or random (generalized) functions aearing in this aer are meant to be P-almost sure equalities. Throughout the aer, C denotes a ositive constant which may change its value from line to line. 2.2 Besov saces and wavelet decomositions In this section we give the definition of Besov saces and describe their characterization in terms of wavelets. Our standard reference in this context is the monograh of Cohen [8]. For a function f : O R and a natural number n N let n n h f(x) := 1 O (x + ih) i= n j= ( ) n ( 1) n j f(x + jh) j be the n-th difference of f with ste h R d. For (, ) the modulus of smoothness is given by ω n (t, f) := su n h f L (O), t >. h <t One aroach to introduce Besov saces is the following. 7

12 Definition 1. Let s,, q (, ) and n N with n > s. Then B s,q(o) is the collection of all functions f L (O) such that ( f B s,q (O) := [ ] ) q 1/q t s ω n dt (t, f) <. t These classes are equied with a (quasi-)norm by taking f B s,q (O) := f L(O) + f B s,q (O). Remark 2. For a more general definition of Besov saces, including the cases where, q = and s < see, e.g. Triebel [45]. We want to describe B s,q (Rd ) by means of wavelet exansions. To this end let ϕ be a scaling function of tensor roduct tye on R d and let ψ i, i = 1,..., 2 d 1, be corresonding multivariate mother wavelets, such that, for a given r N and some N >, the following locality, smoothness and vanishing moment conditions hold. For all i = 1,...,2 d 1, su ϕ, su ψ i [ N, N] d, (4) ϕ, ψ i C r (R d ), (5) x α ψ i (x) dx = for all α N d with α r. (6) We assume that { ϕk, ψ i,j,k : (i, j, k) {1,, 2 d 1} N Z d} is a Riesz basis of L 2 (R d ), where we use the standard abbreviations for dyadic shifts and dilations of the scaling function and the corresonding wavelets ϕ k (x) := ϕ(x k), x R d, for k Z d, and (7) ψ i,j,k (x) := 2 jd/2 ψ i (2 j x k), x R d, for (i, j, k) {1,, 2 d 1} N Z d. (8) Further, we assume that there exists a dual Riesz basis satisfying the same requirements. More recisely, there exist functions ϕ and ψ i, i = 1,...,2 d 1, such that conditions (4), (5) and (6) hold if ϕ and ψ 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 analoguous abbreviations to (7) and (8) for the dyadic shifts and dilations of ϕ and ψ i, and δ k,l denotes the Kronecker symbol. We refer to Cohen [8, Chater 2] for the construction of biorthogonal wavelet bases, see also Daubechies [19] and Cohen, Daubechies, Feauveau [11]. To kee notation simle, we will write ψ i,j,k, := 2 jd(1/ 1/2) ψ i,j,k and ψi,j,k, := 2 jd(1/ 1/2) ψi,j,k, 8

13 for the L -normalized wavelets and the corresondingly modified duals, with := /( 1) if (, ), 1, and :=, 1/ := if = 1. The following theorem shows how Besov saces can be described by decay roerties of the wavelet coefficients, if the arameters fulfil certain conditions. Theorem 3. Let, q (, ) and s > max {, d (1/ 1)}. 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,q(r d ) if, and only if, (convergence in D (R d )) with ( k Z d f, ϕ k f = k Z d f, ϕ k ϕ k + ) 1/ ( 2 d 1 + i=1 2 d 1 i=1 j N 2 jsq j N and (1) is an equivalent (quasi-)norm for B s,q (Rd ). k Z d f, ψ i,j,k, ψ i,j,k, (9) ( f, ψ ) q/ ) 1/q i,j,k, <, (1) k Z d Remark 4. A roof of this theorem for the case 1 can be found in Meyer [36, 1 of Chater 6]. For the general case see for examle Kyriazis [34] or Cohen [8, Theorem 3.7.7]. Of course, if (1) holds then the infinite sum in (9) converges also in B s,q (Rd ). If s > max {, d (1/ 1)} we have the embedding B s,q(r d ) L u (R d ) for some u > 1, see, e.g. Cohen [8, Corollary 3.7.1]. Let us now fix a value (1, ) and consider the scale of Besov saces B s τ,τ (Rd ), 1/τ = s/d + 1/, s >. A simle comutation gives the following result. Corollary 5. Let (1, ), s > and τ R such that 1/τ = s/d + 1/. 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 τ,τ (Rd ) if, and only if, (convergence in D (R d )) with ( f = k Z d f, ϕ k ϕ k + k Z d f, ϕ k τ 2 d 1 i=1 ) 1/τ ( 2 d 1 + i=1 j N and (12) is an equivalent (quasi-)norm for B s τ,τ(r d ). k Z d f, ψ i,j,k, ψ i,j,k, (11) f, ψ ) 1/τ i,j,k, τ <, (12) j N k Z d 9

14 2.3 SPDEs on Lischitz domains and weighted Sobolev saces From now on, let O R d be a bounded Lischitz domain and d 2. We have already mentioned corner singularities as tyical examles where the regularity of a function on O R d in the Besov scale Bτ,τ α (O), 1/τ = α/d+1/, α >, can exceed the regularity in the Sobolev scale W(O), s s >. This reflects the sarsity of the large wavelet coefficients of such a function (given a wavelet basis on the domain O). A general way to deal with smoothness regardless of certain singularities at the boundary is to use weighted Sobolev saces, where the weight function is a ower of the distance to the boundary. The L -theory of SPDEs on Lischitz domains by Kim [3] is based on saces of this tye, namely the weighted Sobolev saces H γ,θ (O), (1, ), θ, γ R, introduced in Lototsky [35]. They are defined in terms of the Bessel-otential saces H γ (R d ) = {u S (R d ) : u H γ (R d ) = (1 ) γ/2 u L(R d ) < }. Here, S (R d ) D (R d ) is the sace of (real valued) temered distributions and (1 ) γ/2 : S (R d ) S (R d ) is the seudo-differential oerator with symbol R d ξ (1 + ξ 2 ) γ/2, i.e. (1 ) γ/2 u = F 1( (1 + ξ 2 ) γ/2 Fu ), where F denotes the Fourier transform on the (comlex valued) temered distributions. For x O we write ρ(x) := dist(x, O) for the distance between x and the boundary of the domain O. Fix c > 1, k > and for n Z consider the subsets O n of O given by O n := {x O : c n k < ρ(x) < c n+k }. Let ζ n, n Z, be non-negative functions satisfying ζ n C (O n), n Z ζ n(x) = 1 and D m ζ n (x) C c mn for all n Z, m N, x O, and a constant C > that does not deend on n, m and x. The functions ζ n can be constructed by mollifying the indicator functions of the sets O n, see, e.g. Hörmander [27, Section 1.4]. If O n is emty we set ζ n. For u D (O) ζ n u is a distribution on O with comact suort which can be extended by zero to R d. This extension is a temered distribution, i.e. ζ n u S (R d ). Definition 6. Let ζ n, n Z, be as above and (1, ), θ, γ R. Then { H γ,θ (O) := u D (O) : u := c nθ ζ H γ,θ (O) n (c n )u(c n ) }. < H γ (R d ) n Z According to Lototsky [35] this definition is indeendent of the secific choice of c, k and ζ n, n N, in the sense that one gets equivalent norms. If γ = m N then the saces can be characterized as H,θ(O) = L,θ (O) := L (O, ρ(x) θ d dx), H m,θ(o) = { u : ρ α D α u L,θ (O) for all α N d with α m }, and one has the norm equivalence C 1 u H,θ m (O) α N d, α m O ρ(x) α D α u(x) ρ(x) θ d dx C u H,θ m (O). (13) 1

15 Analogous notations are used for l 2 = l 2 (N)-valued functions g = (g κ ) κ N. For (1, ), θ, γ R and ζ n, n Z, as above { H γ (Rd ; l 2 ) := g (S (R d )) N : (1 ) γ/2 g κ L (R d ) for all k N and g H γ (R d ;l 2 ) := ( (1 ) γ/2 g κ) L(R κ N l2 }, < d ) { H γ,θ (O; l 2) := g (D (O)) N : g := c nθ ζ H γ,θ (O;l 2) n (c n )g(c n ) H γ (R d ;l 2 }. < ) n Z Remark 7. (a) One can consider the saces H γ,θ (O) as generalizations of the classical Sobolev saces on O with zero boundary conditions. For γ = m N we have the identity H m,d m (O) = W m (O), and the norms in both saces are equivalent, see Theorem 9.7. in Kufner [33]. Here W m (O) is the closure of C (O) in the classical Sobolev sace W m (O). (b) Note that, in contrast to the saces W s(o) = Bs, (O), s (m, m + 1), m N, which can be regarded as real interolation saces of the classical Sobolev saces W m(o), m N (see, e.g. Triebel [45, Section ] and Disa[23]), the saces H γ,θ (O), γ (m, m + 1), m N, are comlex interolants of the resective integer smoothness saces (cf. Lototsky [35, Proosition 2.4]). We can now define saces of stochastic rocesses and random functions in terms of the weighted Sobolev saces introduced above. Definition 8. For γ, θ R and (1, ) we set H γ,θ (O, T) := L ( ΩT, P, P λ; H γ,θ (O)), H γ,θ (O, T; l ( 2) := L ΩT, P, P λ; H γ,θ (O; l 2) ), U γ,θ (O) := L ( Ω, F, P; H γ 2/, θ+2 (O)), and for [2, ), { H γ,θ (O, T) := u H γ,θ (O, T) : u(, ) Uγ,θ (O) and du = f dt + g κ dwt κ κ=1 } for some f H γ 2,θ+ (O, T), g Hγ 1,θ (O, T; l 2), equied with the norm u H γ,θ (O,T) := u H γ,θ (O,T) + f H γ 2,θ+ (O,T) + g H γ 1,θ (O,T;l 2) + u(, ) U γ,θ (O). The equality du = f dt + κ=1 gκ dwt κ above is a shorthand for t t u(t, ), ϕ = u(, ), ϕ + f(s, ), ϕ ds + g κ (s, ), ϕ dws κ (14) for all ϕ C (O), t [, T]. 11 κ=1

16 Remark 9. (a) If [2, ), then the sum of stochastic integrals in (14) converges in the sace M 2,c T (R, (F t)) of continuous, square integrable, R-valued martingales w.r.t (F t ) t [,T]. For the convenience of the readers we include a roof in Aendix A. (b) Using the arguments of Krylov in [31, Remark 3.3], we get the uniqueness (u to indistinguishability) of the air (f, g) H γ 2,θ+ (O, T) Hγ 1,θ (O, T; l 2) which fulfils (14). Consequently, the norm in H γ,θ (O, T) is well defined. Definition 1. We call a redictable D (O)-valued stochastic rocess u = (u(t, )) t [,T] a solution of equation (1) if it is a solution of equation (14) where f is relaced by d µ,ν=1 aµν u xµx ν and u(, ) = u. The next result is taken from Kim [3]. Theorem 11. Let [2, ) and γ R. There exists a constant κ (, 1), deending only on d,, (a µν ) 1 µ,ν d and O, such that for any θ (d κ, d κ ), g H γ 1,θ (O, T; l 2) and u U γ,θ (O) the equation (1) has a unique solution in the class H γ,θ (O, T). For this solution ) u ( g C + u H γ,θ (O,T) H, (15) γ 1,θ (O,T;l 2) U γ,θ (O) where the constant C deends only on d,, γ, θ, (a µν ) 1 µ,ν d, T and O. We will need the following straightforward consequence of this Theorem 11. Recall that if m N and f D (O) is sufficient regular, then D m f l stands for ( α =m Dα f ) 1/, the (ointwise) l -norm of the vector of the m-th order derivatives of f. Corollary 12. In the situation of Theorem 11 with γ = m N, the following inequality holds for every τ [, ]. ( ) τ ρ m δ D m u(ω, t, ) l τ L (O) dt P(dω) C g H m 1,θ (O,T;l 2) + u U m,θ (O), Ω where δ = 1 + d θ. Proof. Theorem 11 imlies, in articular, that ( u H m,θ (O,T) C and we have u H m,θ (O,T) = C Ω Ω g H m 1,θ (O,T) + u U m,θ (O) u(ω, t, ) H,θ m (O,T) dt P(dω) m k= ), ρ k+(θ d)/ D k u(ω, t, ) l L (O) dt P(dω) 12

17 C Ω ρ m δ D m u(ω, t, ) l L (O) dt P(dω) with δ = 1 + d θ ( (2 κ )/, ( + κ )/ ). Now let τ [, ]. Jensen s inequality for concave functions, see, e.g. Schilling [42, Theorem 12.14], yields ) τ/ ρ m δ D m u(ω, t, ) l τ L dt P(dω) C(T) (O) ( g + u H m 1,θ (O,T) U,θ m (O) Ω ( C g H m 1,θ (O,T) + u U m,θ (O) In the last ste we have used the fact that all norms on R 2 are equivalent. Remark 13. Consider the Hilbert sace case = 2 and assume g H γ 2,θ (O, T; l 2). The exression t κ=1 gκ (s, ) dws κ can be considered as an Hγ 2,θ (O)-valued stochastic integral t G(s) dw s with resect to a cylindrical Wiener rocess (W t ) t [,T] on l 2 whose coordinate rocesses are (w κ t ) t [,T], κ N. (See, e.g. Da Prato, Zabczyk [18] or Peszat, Zabczyk [4] for stochastic integration w.r.t. cylindrical rocesses.) Here (G(t)) t [,T] is a stochastic rocess in the sace of Hilbert-Schmidt oerators L (HS) (l 2, H γ 2,θ (O)) defined by G(ω, t) : l 2 H γ 2,θ (O), (xκ ) κ N κ Ng κ (ω, t, )x κ, (ω, t) Ω T, ) τ. and it is an element of the sace L 2 (Ω T ; L (HS) (l 2, H γ 2,θ (O))). Indeed, for fixed (ω, t) Ω T we have G(ω, t) 2 L (HS) (l 2,H γ 2,θ (O)) = κ N g κ (ω, t, ) 2 H γ 2,θ (O) = κ N c nθ ζ n (c n )g κ (ω, t, c n ) 2 H γ 2 (Rd ) n Z = n Z c nθ ζ n (c n )g(ω, t, c n ) 2 H γ 2 (Rd ;l 2 ) by Tonelli s theorem, so that G L2 (Ω T ;L (HS) (l 2 ;H γ 2,θ (O))) = g H γ 2,θ (O,T;l 2). As a consequence, equation (1) can be rewritten in the form du = d µ,ν=1 a µν u xµx ν dt + dm t, u(, ) = u, (16) where (M t ) t [,T] M 2,c T (Hγ 2,θ (O), (F t)) is the H γ 2,θ (O)-valued, square-integrable martingale given by M t := t G(s) dw s, 13 t [, T].

18 Remark 14. In Examles 17, 18 and 19 below the solution u of equation (1) in H γ 2,θ (O, T) as given by Theorem 11 coincides with with the weak solution of equation (16) with zero Dirichlet boundary condition in the sense of Da Prato, Zabczyk [18]. In the examles we consider equation (16) driven by certain Wiener rocesses (M t ) t [,T] in L 2 (O) with u U2,2(O), 2 d = 2 and the solution u is in the class H 2 2,2(O, T) H 2 2, (O, T). (Strictly seaking, in Examle 19 (M t) t [,T] is not a Wiener rocess, but it is one conditioned on the family of random variables Y λ, λ.) Thus, by Remark 7 (a) we know that u is an element of L 2 (Ω T ; (O)). Let us now introduce the oerator (A, D(A)) := ( d µ,ν=1 and consider the equation W 1 2 a µν 2 x µ x ν, { u W 1 2 (O) : d µ,ν=1 } ) a µν u xµx ν L 2 (O) du(t, ) = Au(t, ) dt + dm t, u(, ) = u L 2 (O), t [, T]. (17) A weak solution of equation (17) in the sense of Da Prato, Zabczyk [18] is an L 2 (O)- valued redictable rocess u = (u(t, )) t [,T] with P-almost surely Bochner integrable trajectories t u(ω, t, ) satisfying u(t, ), ζ L2 (O) = u, ζ L2 (O) + t u(s, ), A ζ L2 (O) ds + M t, ζ L2 (O) (18) for all t [, T] and ζ D(A ). It is given by the variation of constants formula u(t, ) = e ta u + t e (t s)a dm s, t [, T], where (e ta ) t is the contraction semigrou on L 2 (O) generated by A. It is clear that the solution u H 2 2,2 (O, T) given by Theorem 11 satisfies u(t, ), ϕ L2 (O) = u, ϕ L2 (O) + t A 1/2 u(s, ), A 1/2 ϕ L2 (O) ds + M t, ϕ L2 (O) (19) for all t [, T] and ϕ C (O). Note that the oerator A is self-adjoint because the coefficients a µν, 1 µ, ν d, are constants. Since every ζ D(A ) = D(A) W 2 1 (O) is the limit in W2 1 (O) of a sequence of test functions (ϕ k ) k N C (O), one can go to the limit k for ϕ = ϕ k in (19) to obtain equation (18). 14

19 3 Besov regularity for SPDEs In this section we state and rove our main result. We give some concrete examles to illustrate its alicability. The result is formulated in terms of the L τ -saces L τ (Ω T ; B s τ,τ (O)) = L τ(ω T, P, P λ; B s τ,τ (O)), τ (, ), s (, ), and the saces introduced in the last section. Theorem 15. Let [2, ) and g H γ 1,θ (O, T; l 2), u U γ,θ (O) for some γ N and θ (d κ, d κ ) with κ = κ (d,, (a µν ), O) (, 1) as in Theorem 11. Let u be the unique solution in the class H γ,θ (O, T) of equation (1) and assume furthermore that Then, we have u L ( ΩT ; B s,(o) ) for some s ( (, γ 1 + d θ u L τ (Ω T ; Bτ,τ α (O)), 1 τ = α d + 1 (, for all α, γ sd ), d 1 )]. (2) and the following estimate holds ( ) u Lτ(Ω T ;Bτ,τ α (O)) C g H γ 1,θ (O,T;l 2) + u U γ,θ (O) + u L(Ω T ;B, s (O)). (21) Here the constant C deends only on d,, γ, α, s, θ, (a µν ) 1 µ,ν d, T and O. Remark 16. Since the constant κ = κ (d,, (a µν ), O) is greater than zero, we can always choose θ = d. In this case, we know from Theorem 11 that for each γ N we have a unique solution u in the class H γ,d (O, T), rovided the free term g and the initial condition u are sufficiently regular. In articular, we get u H γ,d (O, T) = L (Ω T, P, P λ; H γ,d (O)) L (Ω T ; W 1 (O)) L (Ω T ; B 1, (O)). Thus, the additional requirement (2) is fulfilled with s = 1. Since O is an arbitrary bounded Lischitz domain it is in general not clear if u belongs to L (Ω T ; W s (O)) for all s < 2, comare Examle 17 below. However, if γ 2 our result shows that we obtain higher regularity than s = 1 in the nonlinear aroximation scale, namely u L τ (Ω T ; Bτ,τ α (O)), 1 τ = α d + 1 d, for all α < d 1. 15

20 Proof of Theorem 15. We fix α and τ as stated in the theorem and choose a wavelet Rieszbasis { ϕk, ψ i,j,k : (i, j, k) {1,, 2 d 1} N Z d} of L 2 (R d ) which fulfils the assumtions from Section 2.2 with r > γ. Given (j, k) N Z d let Q j,k := 2 j k + 2 j [ N, N] d, such that su ψ i,j,k Q j,k for all i {1,...,2 d 1} and su ϕ k Q,k for all k Z d. Remember that the suorts of the corresonding dual basis fulfil the same requirements. For our urose the set of all indices associated with that wavelets that may have common suort with the domain O will lay an imortant role and we denote them by Λ := { (i, j, k) {1,..., 2 d 1} N Z d Qj,k O }. In articular, we will also use the following notation: Γ := {k Z d : Q,k O }. Due to the assumtion u L (Ω T ; B, s (O)) we have u(ω, t, ) Bs, (O) for P λ- almost every (ω, t) Ω T. As O is a Lischitz domain there exists a linear and bounded extension oerator E : B, s (O) Bs, (Rd ), i.e. there exists a constant C > such that for P λ-almost every (ω, t) Ω T : Eu(ω, t, ) O = u(ω, t, ) and Eu(ω, t, ) B s, (R d ) C u(ω, t, ) B s, (O), see, e.g. Rychkov [41]. In the sequel we will omit the E in our notation and write u instead of Eu. Theorem 3 tells us that for almost all (ω, t) Ω T the following equality holds on the domain O u(ω, t, ) = u(ω, t, ), ϕ k ϕ k + u(ω, t, ), ψ i,j,k, ψ i,j,k,, k Γ (i,j,k) Λ where the sums converge unconditionally in B s,(r d ). Furthermore, cf. Corollary 5, we get for P λ-almost all (ω, t) Ω T ( u(ω, t, ) τ Bτ,τ α (O) C u(ω, t, ), ϕ k τ + k Γ Hence, it is enough to rove that Ω (i,j,k) Λ u(ω, t, ), ψ i,j,k, τ ). (22) u(ω, t, ), ϕ k τ dt P(dω) C u τ L (Ω T ;B (23),(O)) s k Γ 16

21 and Ω (i,j,k) Λ u(ω, t, ), ψ i,j,k, τ dt P(dω) C ( g H γ 1,θ (O,T;l 2) + u U γ,θ (O) + u L(Ω T ; B s, (O)) ) τ. We start with (23). The index set Γ introduced above is finite because of the boundedness of the domain O, so that we can use Jensen s inequality to get for P λ-almost all (ω, t) Ω T ( ( ) ) 1/ τ u(ω, t,.), ϕ k τ C u(ω, t,.), ϕ k C u(ω, t,.) τ B, s (O). k Γ k Γ In the last ste we used Theorem 3 and the boundedness of the extension oerator. Integration with resect to P λ and another alication of Jensen s inequality yield (23). Now let us focus on the inequality (24). To this end, we introduce the following notation ρ j,k := dist(q j,k, O) = inf x Q j,k ρ(x), Λ j := { (i, l, k) Λ : l = j }, Λ j,m := { (i, j, k) Λ j : m2 j ρ j,k < (m + 1)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 (24) into Ω (i,j,k) Λ u(ω, t, ), ψi,j,k, τ dt P(dω) + Ω (i,j,k) Λ\Λ (24) u(ω, t, ), ψi,j,k, τ dt P(dω) =: I + II (25) and estimate each term searately. Let us begin with I. Fix (i, j, k) Λ and (ω, t) Ω T such that ρ(x) γ s D γ u(ω, t, x) l dx <. O By Corollary 12 this holds for P λ-almost all (ω, t) Ω T. By a Whitney-tye inequality, also known as the Deny-Lions lemma, see, e.g. DeVore, Sharley [22, Theorem 3.4], there exists a olynomial P j,k of total degree less than γ such that u(ω, t, ) P j,k L(Q j,k ) C2 jγ u(ω, t, ) W γ (Q j,k ), 17

22 where the last norm is finite since ρ j,k = dist(q j,k, O) >. Since ψ i,j,k, is orthogonal to every olynomial of total degree less than γ, one gets u(ω, t, ), ψi,j,k, = u(ω, t, ) Pj,k, ψ i,j,k, u(ω, t, ) P j,k L(Q j,k ) ψ i,j,k, L (Q j,k ) C 2 jγ u(ω, t, ) W γ (Q j,k ) ( C 2 jγ ρ s γ ρ(x) γ s j,k D γ u(ω, t, x) l dx Q j,k =: C 2 jγ ρ s γ j,k µ j,k(ω, t). Fix j N. Summing over all indices (i, j, k) Λ j and alying Hölder s inequality with exonents > 1 and one finds τ τ u(ω, t, ), ψi,j,k, τ C 2 jγτ ρ (s γ)τ j,k µ j,k (ω, t) τ ) 1/ (i,j,k) Λ j (i,j,k) Λ j ( C µ j,k (ω, t) ) τ ( 2 jγτ τ (s γ)τ ρ τ j,k ) τ. (26) (i,j,k) Λ j (i,j,k) Λ j Since any x O lies outside of all but at most a constant number C > of the cubes Q j,k, k Z d, we get the following bound for the first factor on the right hand side ( (i,j,k) Λ j µ j,k (ω, t) ) τ = ( (i,j,k) Λ j Q j,k ρ(x) γ s D γ u(ω, t, x) l dx ) τ C ρ γ s D γ u(ω, t, ) l τ L (O). (27) In order to estimate the second factor in (26) we use the Lischitz character of the domain O which imlies that Λ j,m C2 j(d 1) for all j, m N. (28) The constant C > does not deend on j or m. Moreover, the boundedness of O yields Λ j,m = for all j, m N with m C2 j. Consequently, ( (i,j,k) Λ j 2 jγτ τ (s γ)τ ρ τ j,k ) τ ( C2 j 2 jγτ τ m=1 (i,j,k) Λ j,m ( C2 j C C m=1 (s γ)τ ρ τ j,k ) τ ) τ 2 j(d 1) γτ j 2 τ (m 2 j ) (s γ)τ τ ( sτ j(d 1 2 τ) + 2 j(d τ) γτ ) τ. (29) 18

23 Now, let us sum over all j N and integrate over Ω T with resect to P λ on both sides of the inequality (26). By using (29) and (27) from above and Corollary 12 we get u(ω, t, ), ψi,j,k, τ dt P(dω) Ω (i,j,k) Λ C j N ( C ( sτ j(d 1 2 τ) + 2 j(d τ) γτ j N 2 j(d 1 sτ τ) τ + ) τ 2 j(d γτ j N Ω τ) τ ρ γ s D γ u(ω, t, ) l τ L (O) dt P(dω) ) ( ) τ g H γ 1,θ (O,T;l 2) + u U γ,θ. (O) One can see that the sums on the right hand side converge if, and only if, α (, γ sd 1) d. Finally, u(ω, t, ), ψi,j,k, ( ) τ τ dt P(dω) C g H γ 1,θ (O,T;l 2) + u U γ,θ. (3) (O) Ω (i,j,k) Λ Now we estimate the term II in (25). First we fix j N and use Hölder s inequality and (28) to get u(ω, t,.), ψi,j,k, ( τ τ j(d 1) C 2 u(ω, t,.), ψi,j,k, ) τ. (i,j,k) Λ j, (i,j,k) Λ j, Summing over all j N and using Hölder s inequality again, yields u(ω, t,.), ψi,j,k, τ = [ u(ω, t,.), ψi,j,k, τ] (i,j,k) Λ\Λ C [2 j(d 1) τ j N ( (i,j,k) Λ j, u(ω, t,.), ψi,j,k, ) τ ] j N ( C 2 j((d 1)( τ) j N (i,j,k) Λ j, s) τ ) τ ( 2 js j N (i,j,k) Λ j, u(ω, t,.), ψi,j,k, ) τ Using Theorem 3 and the boundedness of the extension oerator, one gets for P λ-almost every (ω, t) Ω T that (i,j,k) Λ\Λ u(ω, t,.), ψi,j,k, τ C u(ω, t,.) τ B s, (O) ( j N ) τ 2 j((d 1)( τ) s) τ. The series on the right hand side converges if and only if α (, sd 1) d. But this is art of our assumtions, so that for P λ-almost every (ω, t) Ω T u(ω, t,.), ψi,j,k, τ C u(ω, t,.) τ B, s (O). (i,j,k) Λ\Λ 19.

24 Let us integrate over Ω T with resect to P λ and use Jensen s inequality to get Ω (i,j,k) Λ\Λ u(ω, t, ), ψi,j,k, τ dt P(dω) C u(ω, t, ) τ B, s Ω (O) dt P(dω) [ ] τ C u(ω, t, ) B, s (O) dt P(dω). Because of (3) this roves (24). Now (23) and (22) finish the roof. Next, we give some examles for an alication of Theorem 15. We are mainly interested in the Hilbert sace case = 2. Examle 17. Let us first consider equation (1) in the form (16) where the driving rocess (M t ) t [,T] is a Wiener rocess in W 2 1(O) with covariance oerator Q L (nuc)( W 2 1 (O)). It can be reresented as a stochastic integral rocess ( t G(s) dw s) t [,T] w.r.t. the cylindrical Wiener rocess (W t ) t [,T] on l 2 by defining the integrand rocess (G(t)) t [,T] in the sace of Hilbert-Schmidt oerators L (HS) (l 2, W 2 1 (O)) as the constant deterministic rocess G(ω, t) : l 2 W 1 2 (O), (xκ ) κ N κ N λκ x κ e κ, (ω, t) Ω T, (31) where (e κ ) κ N is an orthonormal basis of W2 1 (O) consisting of eigenvectors of Q with ositive eigenvalues (λ κ ) κ N. This corresonds to defining g = (g κ ) κ N in equation (1) by Ω g κ (ω, t, ) := λ κ e κ, κ N, (ω, t) Ω T. (32) It is easy to see that g is an element of H 1 2,d (O, T; l 2). By definition g 2 H 1 2,d (O,T;l 2) = T 2 c nd ζ n (c n )( λ κ e κ (c n )) κ N 2 H2 1(Rd ;l 2 ) n Z = T 2 λ κ c nd ζ n (c n )e κ (c n ) 2 H2 1(Rd ) κ N n Z (33) = T 2 κ Nλ κ e κ 2 H 1 2,d (O). Using the norm equivalence (13), one has g 2 H 1 2,d (O,T;l 2) CT 2 λ κ ρ α D α e κ 2 L 2 (O) CT 2 λ κ D α e κ 2 L 2 (O) κ N α 1 κ N α 1 = CT 2 κ <. κ Nλ 2

25 2 s B 2 2/3,2/3 (O) 1 τ = α W 1 2 (O) Figure 2: Besov regularity in the scale Bτ,τ α (O), 1/τ = α/2 + 1/2, vs. Sobolev regularity of the solution, illustrated in a DeVore-Triebel diagram τ Thus, in a 2-dimensional setting, Theorem 11 with d = θ = γ = 2 tells us that for every initial condition u U2,2 2 (O) = L 2(Ω, F, P; H2,2 1 (O)) equation (1) has a unique solution u in the class H 2 2,2(O, T) H 2 2,(O, T) = L 2 (Ω T ; H2,(O)). 2 As a trivial consequence, because we have the equality u L 2 (Ω T ; W 1 2 (O)) = L 2(Ω T ; B 1 2,2 (O)) H 2 2, (O) = { u D (O) : ρ α 1 D α u L 2 (O) for all α N 2 with α 2}. (In fact, according to Remark 7 we even know that u L 2 (Ω T ; W 2 1(O)).) Note that in general u does not belong to L 2 (Ω T ; W2(O)) s for all s < 2. Since O is an arbitrary bounded Lischitz domain, certain second derivatives might exlode near the boundary and the norm u(ω, t, ) W 2 2 (O) as well as u(ω, t, ) W s 2 (O), where s (1, 2), might not be finite. If O is a olygonal domain, one can derive an exlicit uer bound for the regularity in the Sobolev scale L 2 (Ω T ; W2 s (O)), s >. Adating techniques used in Grisvard [24], [25] to our stochastic setting, one can show that u / L 2 (Ω T ; W2(O)) s if s > 1 + π/γ, where γ is the measure of the largest interior angle at a corner of O. However, in the considered situation Theorem 15 with s = 1 states that we have u L τ (Ω T ; B α τ,τ (O)) for every α < 2 and 1/τ = α/2 + 1/2. This constellation is illustrated in Figure 2, where each oint (1/τ, s) reresents the smoothness saces of functions with s derivatives in 21

26 L τ (O). Based on the knowledge that u L 2 (Ω T ; W 1 2 (O)) and u L τ (Ω T ; B α τ,τ(o)) for all α < 2, 1/τ = α/2 + 1/2, interolation and embedding theorems yield that u also belongs to each of the saces L τ (Ω T ; B s τ,τ(o)), < τ < 2, s < (1/2 + 1/τ) 2. This is indicated by the shaded area. Examle 18. In view of equality (33) it is clear that we can aly Theorem 11 and Theorem 15 in the same way as in Examle 17, i.e. with d = θ = γ = 2 and s = 1, if the driving rocess (M t ) t [,T] in (16) is a Wiener rocess in W2 1 (O) with covariance oerator Q L (nuc) (W2 1 (O)), and even if it is a Wiener rocess in H2,2(O) 1 with covariance oerator Q L (nuc) (H2,2 1 (O)). In the first case (M t) t [,T] does not satisfy a zero Dirichlet boundary condition as in Examle 17, and in the second case (M t ) t [,T] behaves even more irregularly near the boundary in the sense that the first derivatives are allowed to blow u near O. In these cases we choose (e κ ) κ N in (31) and (32) to be an orthonormal basis of the sace W2 1(O), resectively H1 2,2 (O), consisting of eigenvectors of Q L (nuc)(w2 1 (O)), resectively Q L (nuc) (H2,2(O)), 1 with corresonding eigenvalues (λ κ ) κ N. As in Examle 17 the solution u lies in L 2 (Ω T ; W2 1(O)) = L 2(Ω T ; B2,2 1 (O)) and, by Theorem 15, it also lies in L 2 (Ω T ; Bτ,τ(O)), α 1/τ = α/2 + 1/2, α < 2, see Figure 2. Examle 19. Let the driving rocess (M t ) t [,T] in (16) be a time-deendent version of the stochastic wavelet exansion introduced in Abramovich et al. [1] in the context of Bayesian nonarametric regression and generalized in Bochkina [3], Cioica et al. [7]. 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 (M t ) t [,T]. We first describe the general noise model and then deduce a further examle for the alication of Theorem 15. Let {ψ λ : λ } be a multiscale Riesz basis for L 2 (O) consisting of scaling functions at a fixed scale level j Z and of wavelets at level j and all finer levels. As in the introduction, the notation we use here is different from that used in Section 2.2 because we do not consider a basis on the whole sace R d but on the bounded domain O. Information like scale level, satial location and tye of the wavelets or scaling functions are encoded in the indices λ. We refer to Cohen [8, Sections 2.12, 2.13 and 3.9] and Dahmen, Schneider [15], [16], [17] for detailed descritions of multiscale bases on bounded domains. Adoting the notation of Cohen we write = j j 1 j, where for j j the set j contains the indices of all wavelets ψ λ at scale level j and where j 1 is the index set referring to the scaling functions at scale level j which we denote by ψ λ, λ j 1, for the sake of notational simlicity. We make the following assumtions concerning our basis. Firstly, the cardinalities of the index sets j, j j 1, satisfy C 1 2 jd j C2 jd, j j 1. (34) 22

27 Secondly, we assume that the basis admits norm equivalences similar to those described in Theorem 3. There exists an r N (deending on the smoothness of the scaling functions ψ λ, λ j 1, and on the degree of olynomial exactness of their linear san), such that, given, q >, max{, d(1/ 1)} < s < r, and a real valued distribution f D (O), we have f B s,q (O) if and only if f can be reresented as f = λ c λψ λ, (c λ ) λ R (convergence in D (O)), such that ( j=j 1 ) 2 jq(s+d(1 2 1 ))( ) 1/q q/ c λ <. (35) λ j Furthermore, f B s,q (O) is equivalent to the quasi-norm (35). Concrete constructions of bases satisfying these assumtions can be found in the literature mentioned above. Concerning the family of indeendent standard Brownian motions (wt κ) t [,T], κ N, in (1) resectively (16), we modify our notation and write (wt λ ) t [,T], λ, instead. The descrition of the noise model involves arameters a, b [, 1], c R, with a + b > 1. For every j j 1 we set σ j = (j (j 2)) cd 2 2 a(j (j 1))d 2 and let Y λ, λ j, be Bernoulli distributed random variables on (Ω, F, P) with arameter j = 2 b(j (j 1))d, such that the random variables and rocesses Y λ, (wt λ) t [,T], λ, are stochastically indeendent. Now we are ready to define (M t ) t [,T] by M t := j=j 1 λ j σ j Y λ ψ λ w λ t, t [, T]. (36) Using (35), (34) and a + b > 1, it is easy to check that the infinite sum converges in L 2 (Ω T ; L 2 (O)) as well as in the sace M 2,c T (L 2(O), (F t )) of continuous, square integrable, L 2 (O)-valued martingales w.r.t. the filtration (F t ) t [,T]. Moreover, by the choice of the hyerarameters a, b and c one has an exlicit control of the convergence of the infinite sum in (36) in the (quasi-)banach saces L 2 (Ω T ; B s 1,q (O)), s < r, 1, q >, 2 q. (Comare Cioica et al. [7] which can easily be adated to our setting.) With regard to Theorems 11 and 15 let again d = = γ = θ = 2. Equation (16) with (M t ) t [,T] defined as above corresonds to equation (1) if we set g λ (ω, t, ) := σ j Y λ (ω)ψ λ ( ), λ j, j j 1, (ω, t) Ω T, and sum over all λ instead of κ N. In the following we write l 2 = l 2 ( ). Since a + b > 1 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 and assume that a + b > 2. This is sufficient to ensure that g H 1 2,2 (O, T; l 2): Using (13) one sees that the H 1 2,2 (O, T; l 2)-norm of g = (g λ ) λ satisfies g 2 H 1 2,2 (O,T;l 2) = E c n2 ζ n (c n )g(t, c n ) 2 H2 1(Rd ;l 2 ) dt n Z 23

28 = E = T E C C λ j=j 1 j=j 1 j=j 1 g λ (t, ) 2 H2,2 1 (O) dt σjy 2 λ 2 ψ λ 2 H2,2 1 (O) λ j σj 2 j ρ α D α ψ λ 2 L 2 (O) λ j α 1 σj 2 j ψ λ 2 W 1(O). 2 λ j Since W 1 2 (O) = B1 2,2 (O) with equivalent norms we can use the equivalence (35) with f = ψ λ to get g 2 H 1 2,2 (O,T;l 2) C = C C j=j 1 j=j 1 j=j 1 λ j σ 2 j j2 2j j (j (j 2)) 2c 2 2a(j (j 1)) 2 2b(j (j 1)) 2 2j (j (j 2)) 2c 2 2j(a+b 2). In the last ste we used (34) with d = 2. Thus g H 1 2,2(O, T; l 2 ). As in Examle 17 we may aly Theorems 11 and 15 to conclude that for every initial condition u L 2 (Ω, F, P; H2,2 1 (O)) there exists a unique solution of equation (1) in the class H2 2,2 (O, T), which, in general, is not in L 2 (Ω T, W2(O)) s for all s < 2, but it belongs to every sace L 2 (Ω T ; Bτ,τ α (O)) with α < 2 and τ = 2/(α + 1). Remark 2. In ractice, many adative wavelet-based algorithms are realized with the energy norm of the roblem which is equivalent to a Sobolev norm. Let us denote by {η λ : λ } a wavelet Riesz basis of W s 2(O) for some s >, which can be obtained by rescaling the wavelet basis {ψ λ : λ } of L 2 (O), see, e.g. Cohen [8] or Dahmen [14]. For the best N-term aroximation in this Sobolev norm, it is well known that where σ N,W s 2 (O)(u) := inf u B α τ,τ(o), 1 τ = α s + 1 d 2 = σ N,W s 2 (O)(u) C N (α s)/d, { u u N W s 2 (O) : u N = } c λ η λ : Λ, Λ N, c λ R, λ Λ. λ Λ Therefore, similar to the L 2 (O)-setting, the aroximation order of the best N-term wavelet scheme in W s 2(O) deends on the Besov regularity of the object one wants to aroximate. 24

29 s 1 = α τ B 2 2/3,2/3 (O) W 3/2 2 (O) B 5/3 6/5,6/5 (O) 1 τ = α Figure 3: Besov regularity in the scale Bτ,τ α (O), 1/τ = (α 1)/2 + 1/2, vs. Sobolev regularity of the solution illustrated in a DeVore-Triebel diagram. There exist adative wavelet-based algorithms which are guaranteed to converge and which indeed asymtotically realize the convergence rate of best N-term aroximation with resect to the Sobolev norm. For examle, Cohen, Dahmen, DeVore [9] designed such an adative numerical scheme for solving (deterministic) ellitic PDEs. First results for arabolic roblems were obtained by Schwab, Stevenson [43]. Once again, the use of adative algorithms is justified if the rate of aroximation that can be achieved is higher than in classical uniform schemes. Let u N, N N, denote a uniform aroximation scheme (e.g. a Galerkin aroximation) of u. It is well-known that under certain natural conditions, see, e.g. Dahlke, Dahmen, DeVore [12] or DeVore [21] or Hackbusch [26], u u N W s 2 (O) CN (α s)/d u W α 2 (O). This means that, even in this case, adativity can ay off if the Besov smoothness of the solution is higher than its Sobolev regularity. Let us discuss this relationshi in more detail for the examles from above. We consider aroximation in W2 1 (O). As already mentioned in Examle 17, in general we cannot exect that the satial Sobolev regularity of the solution is higher than 3/2. Therefore, uniform schemes yield an aroximation rate of O(N 1/4 ). On the other hand, our main result shows that u L τ (Ω T ; Bτ,τ α (O)), 1 τ = α for all α < 2. 2 Therefore, by interolation and embedding of Besov saces we can achieve that the solution is contained in all the saces L τ (Ω T ; Bτ,τ(O)) α corresonding to the oints in the traezoid τ

30 with vertices (1/2, ), (1/2, 3/2), (3/2, 2), (3/2, ) and to the oints to the right of this traezoid in the DeVore-Triebel diagram, cf. Figure 3. As a consequence, we get by a short comutation, that u L τ (Ω T ; Bτ,τ α (O)), 1 τ = α for all α < 5 3. Thus, best N-term wavelet aroximation rovides order O(N 1/3 ), so that again the use of adativity is comletely justified. A Convergence of the stochastic integrals In this section we give a roof of the M 2,c T (R, (F t))-convergence of the sum of the stochastic integral rocesses ( t ) gκ (s, ), ϕ dws κ, κ N, aearing in formula (14). Let t [,T] us assume that g H γ,θ (O, T; l 2) for some [2, ) and γ, θ R. We use an analogous strategy to [31], Remark 3.2. Due to the indeendence of the Brownian motions (wt κ) t [,T], κ N, the covariation rocess ([w κ, w l ] t ) t [,T] vanishes if κ l, and we have by Itô s isometry: 2 [ T E g κ (s, ), ϕ dws κ ( ) = E g κ (s, ), ϕ dws κ κ=1, ] ( ) g κ (s, ), ϕ dws κ κ=1 κ=1 T = E g κ (s, ), ϕ 2 ds. κ=1 We are going to show that the last term is less or equal a constant times g 2 H γ,θ (O,T;l 2), which is finite due to our assumtion. Then the convergence of the integral rocesses in M 2,c T (R, (F t)) follows by Doob s maximal inequality for martingales. For u D (O) and n Z we use the notation u n := ζ n (c n )u(c n ) S (R d ). Let us abbreviate L τ (R d ) by L τ for all τ 1 in the sequel. Setting = /( 1), we denote by, L L : L L R the dual form obtained by continuous extension of ϕ, ψ = ϕ(x)ψ(x) dx, ϕ, ψ C (R d ). Now we are ready to estimate as follows: κ=1 = κ=1 g κ (s, ), ϕ 2 ds = κ=1 κ=1 c nd gn κ (s, ), ϕ n n Z c nd (1 ) γ 2 g κ n (s, ), (1 ) γ 2 ϕn n Z [ n Z 2 L L c nd (1 ) γ 2 g κ n (s, ) (1 ) γ 2 ϕn 1/2 L2 (1 ) γ 2 ϕn 1/2 L2 ] 2 ds 26 ds 2 ds

31 κ=1 ( n Z c 2nd (1 ) γ 2 g κ n (s, ) (1 ) γ 2 ϕn 1/2 2 L 2 ) 1/2 ( ) 1/2 (1 ) γ 2 ϕn L1 n Z 2 ds. Here we have used Hölder s inequality twice. Since ϕ has comact suort in O and ζ n equals zero outside O n, the functions ϕ n vanish on R d for all but finitely many n Z. As a consequence, the sum n Z (1 ) γ/2 ϕ n L1 has only finitely many non-zero terms. Therefore, κ=1 C = C κ=1 g κ (s, ), ϕ 2 ds c 2nd (1 ) γ/2 gn(s, κ ) (1 ) γ/2 ϕ n 1/2 2 L 2 ds n Z c 2nd n Z κ=1 (1 ) γ/2 gn κ (s, ) 2, (1 ) γ/2 ϕ n ds, L /2 L /( 2) where the constant C deends on ϕ. In the last ste we used the fact that (1 ) γ/2 g n (s, ) = ( (1 ) γ/2 g κ n(s, ) ) κ N L (R d ; l 2 ) P λ-almost everywhere in Ω T, which results from g being an element of H γ,θ (O; l 2). Alying again Hölder s inequality we obtain c 2nd n Z n Z ( C C κ=1 (1 ) γ/2 gn(s, κ ) 2, (1 ) γ/2 ϕ n L /2 L /( 2) c 2nθ (1 ) γ/2 g n (s, ) l2 1/2 L c 2n(d θ) (1 ) γ/2 ϕ n L/( 2) n Z ( n Z c nθ (1 ) γ/2 g n (s, ) l2 L ) 2 ) 2 c nθ (1 ) γ/2 g n (s, ) l2 L, ( n Z ) 2 c 2n(d θ)/( 2) (1 ) γ/2 ϕ n /( 2) L /( 2) where we have used that 2 and that only finitely many of the ϕ n, n Z, are non-zero. All in all we have shown ( )2 T g κ (s, ), ϕ 2 ds C c nθ (1 ) γ/2 g n (s, ) l2 L ds. κ=1 n Z 27

32 Finally, taking the exectation and alying Jensen s inequality yields E and this finishes the roof. κ=1 g κ (s, ), ϕ 2 ds C g 2 H γ,θ (O,T;l 2) B General linear equations In the introduction we have indicated that our main result can be extended to equation (3). The major reason is, that by a result from Kim [3] an estimate similar to the one roved in Corollary 12 holds not only for the model equation (1) but for equations of the tye (3), rovided the coefficients a µν, b µ, c, σ µκ and η κ, the free terms f and g κ and the initial value u fulfil certain conditions. We can use this fact to extend our regularity result to such equations. In this section we want to get more recise and oint out how to do this. For the convenience of the reader we begin by resenting the result from Kim [3, Theorem 2.8]. Therefore, we need some additional notations. For x, y O we shall write ρ(x, y) := ρ(x) ρ(y). For α R, δ (, 1] and k N we set: [f] (α) k := su ρ k+α (x) D k f(x) and [f] (α) k+δ := su x O f (α) k := k j= [f] (α) j and f (α) k+δ x,y O β =k := f (α) k ρ k+α (x, y) Dβ f(x) D β f(y) x y δ, + [f] (α) k+δ, whenever it makes sense. We shall 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 s fix an arbitrary function µ : [, ) [, ), vanishing only on the set of nonnegative integers, i.e. µ (j) = if and only if j N. We set t + := t + µ (t). Now we are able to resent the assumtions on the coefficients of equation (3) (see Kim [3, Assumtions 2.5 and 2.6]). [K1] For any fixed x O, the coefficients a µν (.,., x), b µ (.,., x), c (.,., x), σ µκ (.,., x), η κ (.,., x) : Ω [, T] R are redictable rocesses with resect to the given normal filtration (F t ) t [,T]. 28

33 [K2] (Stochastic arabolicity) There are constants δ, K >, such that for all (ω, t, x) Ω [, T] O and λ R d : δ λ 2 a µν (ω, t, x)λ µ λ ν K λ 2, where a µν := a µν 1 2 σµ, σ ν l2 for µ, ν {1,...,d}. [K3] For all (ω, t) Ω [, T]: a µν (ω, t,.) () γ + + b µ (ω, t,.) (1) γ + + c(ω, t,.) (2) γ + + σ µ (ω, t,.) () γ + + ν(ω, t,.) (1) γ+1 + K. [K4] The coefficients a µν and σ µ are uniformly continuous in x O, i.e. for any ǫ > there is a δ = δ(ǫ) >, such that a µν (ω, t, x) a µν (ω, t, y) + σ µ (ω, t, x) σ µ (ω, t, y) l2 ǫ, for all (ω, t) Ω [, T], whenever x, y O with x y δ. [K5] The behaviour of the coefficients b µ, c and ν can be controlled near the boundary of O in the following way: lim ρ(x) x O su ω Ω t [,T] Here is the main result of Kim [3]. {ρ(x) b µ (ω, t, x) + ρ 2 (x) c(ω, t, x) + ν(ω, t, x) l2 } =. Theorem 21. Let [2, ) and let assumtions [K1] [K5] be satisfied with K, δ >. Then there exists a constant κ = κ (d,, δ, K, O) (, 1) such that, if θ (d κ, d + κ + 2), for any f H γ 2,θ+ (O, T), g Hγ 1,θ (O, T; l 2) and u U γ,θ (O), equation (3) with initial value u admits a unique solution u H γ,θ (O, T), i.e., there exists an (u to indistinguishability) unique D (O)-valued redictable rocess u H γ,θ (O, T), such that for any ϕ C (O) the equality u(t,.), ϕ = u(,.), ϕ + + t κ=1 a µν (s,.)u xµx ν (s,.) + b µ (s,.)u xµ (s,.) + c(s,.)u(s,.) + f(s,.), ϕ ds t σ µκ (s,.)u xµ (s,.) + η κ (s,.)u(s,.) + g κ (s,.), ϕ dw κ s holds for all t [, T] with robability 1. Moreover, for this solutions we have ) u ( f C + H γ,θ (O,T) H γ 2,θ+ (O,T) g + u H, (37) γ 1,θ (O,T;l 2) U γ,θ (O) where C is a constant deending only on d, γ,, θ, δ, K, T and the domain O. 29

34 An immediate consequence of this theorem is the following estimate. Corollary 22. In the situation of Theorem 21 with γ = m N, the following inequality holds for every τ [, ]: Ω where β = 1 + d θ. ρ m β D m u(ω, t, ) l τ L (O) dt P(dω) ( ) τ C f H m 2 + g,θ+ H m 1,θ (O,T;l 2) + u U m,θ (O), Proof. Just reeat the arguments of the roof of Corollary 12 and use estimate (37) instead of (15) at the beginning. Now we can resent our main result in the generalized setting. Theorem 23. Let [2, ) and let assumtions [K1] [K5] be satisfied with aroriate constants K, δ >. Moreover, let f H γ 2,θ+ (O, T), g Hγ 1,θ (O, T; l 2) and u U γ,θ (O) for some γ N. Denote by u the unique solution of equation (3) in the class H γ,θ (O, T) for a given θ (d κ, d 2++κ (), where κ = κ (d,, δ, K, O) (, 1) as in Theorem 21. Assume furthermore that u L ΩT ; B, s (O)) for some s (, γ (1 + d θ)]. Then, we have u L τ (Ω T ; Bτ,τ α (O)), 1 τ = α d + 1 (, for all α, γ sd ), d 1 and the following estimation holds ( ) u Lτ(Ω T ;Bτ,τ α (O)) C f H γ 2,θ+ (O,T) + g H γ 1,θ (O,T;l 2) + u U γ,θ (O) + u L(Ω T ;B, s (O)). Proof. We can argue like we did in the roof of Theorem 15. We just have to use Corollary 22 where we used Corollary 12. References [1] F. Abramovich, T. Saatinas, B.W. Silverman, Wavelet threshholding via a Bayesian aroach, J. R. Stat. Soc., Ser. B, Stat. Methodol. 6 (1998) [2] R.A. Adams, J.J.F. Fournier, Sobolev Saces, second ed., Elsevier, 23. [3] N. Bochkina, Besov regularity of functions with sarse random wavelet coefficients, Manuscrit (26) URL: htt:// BesovWavelets.df, last accessed: 5/11/21. [4] Z. Brzeźniak, Stochastic Partial Differential Equations in M-Tye 2 Banach Saces, Potential Anal. 4 (1995)

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37 Prerint Series DFG-SPP 1324 htt:// Reorts [1] R. Ramlau, G. Teschke, and M. Zhariy. A Comressive Landweber Iteration for Solving Ill-Posed Inverse Problems. Prerint 1, DFG-SPP 1324, Setember 28. [2] G. Plonka. The Easy Path Wavelet Transform: A New Adative Wavelet Transform for Sarse Reresentation of Two-dimensional Data. Prerint 2, DFG-SPP 1324, Setember 28. [3] E. Novak and H. Woźniakowski. Otimal Order of Convergence and (In-) Tractability of Multivariate Aroximation of Smooth Functions. Prerint 3, DFG-SPP 1324, October 28. [4] M. Esig, L. Grasedyck, and W. Hackbusch. Black Box Low Tensor Rank Aroximation Using Fibre-Crosses. Prerint 4, DFG-SPP 1324, October 28. [5] T. Bonesky, S. Dahlke, P. Maass, and T. Raasch. Adative Wavelet Methods and Sarsity Reconstruction for Inverse Heat Conduction Problems. Prerint 5, DFG- SPP 1324, January 29. [6] E. Novak and H. Woźniakowski. Aroximation of Infinitely Differentiable Multivariate Functions Is Intractable. Prerint 6, DFG-SPP 1324, January 29. [7] J. Ma and G. Plonka. A Review of Curvelets and Recent Alications. Prerint 7, DFG-SPP 1324, February 29. [8] L. Denis, D. A. Lorenz, and D. Trede. Greedy Solution of Ill-Posed Problems: Error Bounds and Exact Inversion. Prerint 8, DFG-SPP 1324, Aril 29. [9] U. Friedrich. A Two Parameter Generalization of Lions Nonoverlaing Domain Decomosition Method for Linear Ellitic PDEs. Prerint 9, DFG-SPP 1324, Aril 29. [1] K. Bredies and D. A. Lorenz. Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding. Prerint 1, DFG-SPP 1324, Aril 29. [11] K. Bredies and D. A. Lorenz. Regularization with Non-convex Searable Constraints. Prerint 11, DFG-SPP 1324, Aril 29.

38 [12] M. Döhler, S. Kunis, and D. Potts. Nonequisaced Hyerbolic Cross Fast Fourier Transform. Prerint 12, DFG-SPP 1324, Aril 29. [13] C. Bender. Dual Pricing of Multi-Exercise Otions under Volume Constraints. Prerint 13, DFG-SPP 1324, Aril 29. [14] T. Müller-Gronbach and K. Ritter. Variable Subsace Samling and Multi-level Algorithms. Prerint 14, DFG-SPP 1324, May 29. [15] G. Plonka, S. Tenorth, and A. Iske. Otimally Sarse Image Reresentation by the Easy Path Wavelet Transform. Prerint 15, DFG-SPP 1324, May 29. [16] S. Dahlke, E. Novak, and W. Sickel. Otimal Aroximation of Ellitic Problems by Linear and Nonlinear Maings IV: Errors in L 2 and Other Norms. Prerint 16, DFG-SPP 1324, June 29. [17] B. Jin, T. Khan, P. Maass, and M. Pidcock. Function Saces and Otimal Currents in Imedance Tomograhy. Prerint 17, DFG-SPP 1324, June 29. [18] G. Plonka and J. Ma. Curvelet-Wavelet Regularized Slit Bregman Iteration for Comressed Sensing. Prerint 18, DFG-SPP 1324, June 29. [19] G. Teschke and C. Borries. Accelerated Projected Steeest Descent Method for Nonlinear Inverse Problems with Sarsity Constraints. Prerint 19, DFG-SPP 1324, July 29. [2] L. Grasedyck. Hierarchical Singular Value Decomosition of Tensors. Prerint 2, DFG-SPP 1324, July 29. [21] D. Rudolf. Error Bounds for Comuting the Exectation by Markov Chain Monte Carlo. Prerint 21, DFG-SPP 1324, July 29. [22] M. Hansen and W. Sickel. Best m-term Aroximation and Lizorkin-Triebel Saces. Prerint 22, DFG-SPP 1324, August 29. [23] F.J. Hickernell, T. Müller-Gronbach, B. Niu, and K. Ritter. Multi-level Monte Carlo Algorithms for Infinite-dimensional Integration on R N. Prerint 23, DFG- SPP 1324, August 29. [24] S. Dereich and F. Heidenreich. A Multilevel Monte Carlo Algorithm for Lévy Driven Stochastic Differential Equations. Prerint 24, DFG-SPP 1324, August 29. [25] S. Dahlke, M. Fornasier, and T. Raasch. Multilevel Preconditioning for Adative Sarse Otimization. Prerint 25, DFG-SPP 1324, August 29.

39 [26] S. Dereich. Multilevel Monte Carlo Algorithms for Lévy-driven SDEs with Gaussian Correction. Prerint 26, DFG-SPP 1324, August 29. [27] G. Plonka, S. Tenorth, and D. Roşca. A New Hybrid Method for Image Aroximation using the Easy Path Wavelet Transform. Prerint 27, DFG-SPP 1324, October 29. [28] O. Koch and C. Lubich. Dynamical Low-rank Aroximation of Tensors. Prerint 28, DFG-SPP 1324, November 29. [29] E. Faou, V. Gradinaru, and C. Lubich. Comuting Semi-classical Quantum Dynamics with Hagedorn Waveackets. Prerint 29, DFG-SPP 1324, November 29. [3] D. Conte and C. Lubich. An Error Analysis of the Multi-configuration Timedeendent Hartree Method of Quantum Dynamics. Prerint 3, DFG-SPP 1324, November 29. [31] C. E. Powell and E. Ullmann. Preconditioning Stochastic Galerkin Saddle Point Problems. Prerint 31, DFG-SPP 1324, November 29. [32] O. G. Ernst and E. Ullmann. Stochastic Galerkin Matrices. Prerint 32, DFG-SPP 1324, November 29. [33] F. Lindner and R. L. Schilling. Weak Order for the Discretization of the Stochastic Heat Equation Driven by Imulsive Noise. Prerint 33, DFG-SPP 1324, November 29. [34] L. Kämmerer and S. Kunis. On the Stability of the Hyerbolic Cross Discrete Fourier Transform. Prerint 34, DFG-SPP 1324, December 29. [35] P. Cerejeiras, M. Ferreira, U. Kähler, and G. Teschke. Inversion of the noisy Radon transform on SO(3) by Gabor frames and sarse recovery rinciles. Prerint 35, DFG-SPP 1324, January 21. [36] T. Jahnke and T. Udrescu. Solving Chemical Master Equations by Adative Wavelet Comression. Prerint 36, DFG-SPP 1324, January 21. [37] P. Kittioom, G. Kutyniok, and W.-Q Lim. Irregular Shearlet Frames: Geometry and Aroximation Proerties. Prerint 37, DFG-SPP 1324, February 21. [38] G. Kutyniok and W.-Q Lim. Comactly Suorted Shearlets are Otimally Sarse. Prerint 38, DFG-SPP 1324, February 21. [39] M. Hansen and W. Sickel. Best m-term Aroximation and Tensor Products of Sobolev and Besov Saces the Case of Non-comact Embeddings. Prerint 39, DFG-SPP 1324, March 21.

40 [4] B. Niu, F.J. Hickernell, T. Müller-Gronbach, and K. Ritter. Deterministic Multilevel Algorithms for Infinite-dimensional Integration on R N. Prerint 4, DFG-SPP 1324, March 21. [41] P. Kittioom, G. Kutyniok, and W.-Q Lim. Construction of Comactly Suorted Shearlet Frames. Prerint 41, DFG-SPP 1324, March 21. [42] C. Bender and J. Steiner. Error Criteria for Numerical Solutions of Backward SDEs. Prerint 42, DFG-SPP 1324, Aril 21. [43] L. Grasedyck. Polynomial Aroximation in Hierarchical Tucker Format by Vector- Tensorization. Prerint 43, DFG-SPP 1324, Aril 21. [44] M. Hansen und W. Sickel. Best m-term Aroximation and Sobolev-Besov Saces of Dominating Mixed Smoothness - the Case of Comact Embeddings. Prerint 44, DFG-SPP 1324, Aril 21. [45] P. Binev, W. Dahmen, and P. Lamby. Fast High-Dimensional Aroximation with Sarse Occuancy Trees. Prerint 45, DFG-SPP 1324, May 21. [46] J. Ballani and L. Grasedyck. A Projection Method to Solve Linear Systems in Tensor Format. Prerint 46, DFG-SPP 1324, May 21. [47] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Convergence Rates for Greedy Algorithms in Reduced Basis Methods. Prerint 47, DFG-SPP 1324, May 21. [48] S. Kestler and K. Urban. Adative Wavelet Methods on Unbounded Domains. Prerint 48, DFG-SPP 1324, June 21. [49] H. Yserentant. The Mixed Regularity of Electronic Wave Functions Multilied by Exlicit Correlation Factors. Prerint 49, DFG-SPP 1324, June 21. [5] H. Yserentant. On the Comlexity of the Electronic Schrödinger Equation. Prerint 5, DFG-SPP 1324, June 21. [51] M. Guillemard and A. Iske. Curvature Analysis of Frequency Modulated Manifolds in Dimensionality Reduction. Prerint 51, DFG-SPP 1324, June 21. [52] E. Herrholz and G. Teschke. Comressive Sensing Princiles and Iterative Sarse Recovery for Inverse and Ill-Posed Problems. Prerint 52, DFG-SPP 1324, July 21. [53] L. Kämmerer, S. Kunis, and D. Potts. Interolation Lattices for Hyerbolic Cross Trigonometric Polynomials. Prerint 53, DFG-SPP 1324, July 21.

41 [54] G. Kutyniok and W.-Q Lim. Shearlets on Bounded Domains. Prerint 54, DFG- SPP 1324, July 21. [55] A. Zeiser. Wavelet Aroximation in Weighted Sobolev Saces of Mixed Order with Alications to the Electronic Schrödinger Equation. Prerint 55, DFG-SPP 1324, July 21. [56] G. Kutyniok, J. Lemvig, and W.-Q Lim. Comactly Suorted Shearlets. Prerint 56, DFG-SPP 1324, July 21. [57] A. Zeiser. On the Otimality of the Inexact Inverse Iteration Couled with Adative Finite Element Methods. Prerint 57, DFG-SPP 1324, July 21. [58] S. Jokar. Sarse Recovery and Kronecker Products. Prerint 58, DFG-SPP 1324, August 21. [59] T. Aboiyar, E. H. Georgoulis, and A. Iske. Adative ADER Methods Using Kernel- Based Polyharmonic Sline WENO Reconstruction. Prerint 59, DFG-SPP 1324, August 21. [6] O. G. Ernst, A. Mugler, H.-J. Starkloff, and E. Ullmann. On the Convergence of Generalized Polynomial Chaos Exansions. Prerint 6, DFG-SPP 1324, August 21. [61] S. Holtz, T. Rohwedder, and R. Schneider. On Manifolds of Tensors of Fixed TT-Rank. Prerint 61, DFG-SPP 1324, Setember 21. [62] J. Ballani, L. Grasedyck, and M. Kluge. Black Box Aroximation of Tensors in Hierarchical Tucker Format. Prerint 62, DFG-SPP 1324, October 21. [63] M. Hansen. On Tensor Products of Quasi-Banach Saces. Prerint 63, DFG-SPP 1324, October 21. [64] S. Dahlke, G. Steidl, and G. Teschke. Shearlet Coorbit Saces: Comactly Suorted Analyzing Shearlets, Traces and Embeddings. Prerint 64, DFG-SPP 1324, October 21. [65] W. Hackbusch. Tensorisation of Vectors and their Efficient Convolution. Prerint 65, DFG-SPP 1324, November 21. [66] P. A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, and R. L. Schilling. Satial Besov Regularity for Stochastic Partial Differential Equations on Lischitz Domains. Prerint 66, DFG-SPP 1324, November 21.