Onthe L ( L ) -regularityandBesovsmoothnessofstochasticparabolicequationsonboundedLipschitzdomains

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.18(2013), no. 82, 1–41.

ISSN:1083-6489 DOI:10.1214/EJP.v18-2478

On the L

(L

) -regularity and Besov smoothness of stochastic parabolic equations

on bounded Lipschitz domains

Petru A. Cioica

Kyeong-Hun Kim

Kijung Lee

Felix Lindner

Abstract

We investigate the regularity of linear stochastic parabolic equations with zero Diri- chlet boundary condition on bounded Lipschitz domainsO ⊂R^dwith both theoreti- cal and numerical purpose. We use N.V. Krylov’s framework of stochastic parabolic weighted Sobolev spacesH^γ,q_p,θ(O, T). The summability parameterspandqin space and time may differ. Existence and uniqueness of solutions in these spaces is estab- lished and the Hölder regularity in time is analysed. Moreover, we prove a general embedding of weightedLp(O)-Sobolev spaces into the scale of Besov spacesB^α_τ,τ(O), 1/τ=α/d+ 1/p,α >0. This leads to a Hölder-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.

Keywords:Stochastic partial differential equation ; Lipschitz domain ;Lq(Lp)-theory; weight- ed Sobolev space ; Besov space ; quasi-Banach space ; embedding theorem ; Hölder regularity in time ; nonlinear approximation ; wavelet ; adaptive numerical method ; square root of Laplacian operator .

AMS MSC 2010:Primary 60H15, Secondary 46E35 ; 35R60.

Submitted to EJP on December 4, 2012, final version accepted on September 11, 2013.

1 Introduction

LetO ⊂R^d be a bounded Lipschitz domain,T ∈(0,∞)and let(w^k_t)_t∈[0,T],k∈N^{, be} independent one-dimensional standard Wiener processes defined on a probability space (Ω,F,P). We are interested in the regularity of the solutions to parabolic stochastic partial differential equations (SPDEs, for short) with zero Dirichlet boundary condition

∗Support: Deutsche Forschungsgemeinschaft (DFG, grants DA 360/13-1, DA 360/13-2, DA 360/11-1, DA 360/11-2, SCHI 419/5-1, SCHI 419/5-2), and doctoral scholarship of the Philipps-Universität Marburg, and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0027680).

†Philipps-Universität Marburg, Germany. E-mail:cioica@mathematik.uni-marburg.de

‡Korea University, South Korea. E-mail:kyeonghun@korea.ac.kr

§Ajou University, South Korea. E-mail:kijung@ajou.ac.kr

¶TU Dresden, Germany. E-mail:felix.lindner@tu-dresden.de

of the form

du= (a^iju_xix^j +f)dt+ (σ^iku_xi+g^k)dw_t^k onΩ×[0, T]× O, u= 0 onΩ×(0, T]×∂O,

u(0,·) =u0 onΩ× O,







(1.1)

where the indicesiand jrun from1 todand the indexkruns throughN={1,2, . . .}. Here and in the sequel we use the summation convention on the repeated indicesi, j, k. The coefficientsa^ij andσ^ik depend on(ω, t)∈Ω×[0, T]. The force termsf andg^k de- pend on(ω, t, x)∈Ω×[0, T]× O. By the nature of the problem, in particular by the bad contribution of the infinitesimal differences of the Wiener processes, the second spatial derivatives of the solution may blow up at the boundary ∂O even if the boundary is smooth, see, e.g., [31]. Hence, a natural way to deal with problems of type (1.1) is to consider uas a stochastic process with values in weighted Sobolev spaces on O that allow the derivatives of functions from these spaces to blow up near the boundary. This approach has been initiated and developed by N.V. Krylov and collaborators, first as anL2-theory for smooth domainsO(see [31]), then as anLp-theory (p≥2) for the half space ([36, 37]), for smooth domains ([26, 30]), and for general bounded domains allow- ing Hardy’s inequality such as bounded Lipschitz domains ([29]). Existence and unique- ness of solutions have been established within specific stochastic parabolic weighted Sobolev spaces, denoted by H^γ_p,θ(O, T) in [29]. These spaces consist of elements u˜ of the formd˜u= ˜f dt+ ˜g^kdw^k_t, whereu˜,f˜and˜g^k, considered as stochastic processes with values in certain weighted Lp(O)-Sobolev spaces, are Lp-integrable w.r.t. P⊗dt. We refer to Section 3 for the exact definition.

In this article we treat regularity issues concerning the solutionuof problem (1.1) which arise, besides others, in the context of adaptive numerical approximation meth- ods.

The starting point of our considerations was the question whether we can improve the Besov regularity results in [6] in time direction. In [6] the spatial regularity ofuis measured in the scale of Besov spaces

B_τ,τ^α (O), 1 τ =α

d +1

p, α >0, (∗)

wherep≥2is fixed. Note that forα >(p−1)d/pthe sumability parameterτ becomes less than one, so that in this case B_τ,τ^α (O) is not a Banach space but a quasi-Banach space. It is a known result from approximation theory that the smoothness of a target functionf ∈Lp(O)within the scale (∗) determines the rate of convergence that can be achieved by adaptive and other nonlinear approximation methods if the approximation error is measured inL_p(O); see [7, Chapter 4], [15] or the introduction of [6]. Based on theL_p-theory in [29], it is shown in [6] that the solutionuto problem (1.1) satisfies

u∈Lτ(Ω×[0, T],P,P⊗dt;B^α_τ,τ(O)), 1 τ =α

d +1

p, (1.2)

for certainα > 0depending on the smoothness ofu0, f and g^k,k∈N. In general, the spatial regularity ofuin the Sobolev scale W_p^s(O), s≥0, which determines the order of convergence for uniform approximation methods inL_p(O), is strictly less than the spatial regularity ofuin the scale (∗). It can be due to, e.g., the irregular behaviour of the noise at the boundary or the irregularities of the boundary itself; see [40] for the latter case. This justifies the use of nonlinear approximation methods such as adaptive wavelet methods for the numerical treatment of SPDEs, cf. [5, 4]. The proof of (1.2) relies on characterizations of Besov spaces by wavelet expansions and on weighted

Sobolev norm estimates foru, resulting from the solvability of the problem (1.1) within the spacesH^γ_p,θ(O, T).

An obvious approach to improve (1.2) with respect to regularity in time is to try to combine the existing Hölder estimates in time for the elements of the spacesH^γ_p,θ(O, T) (see [29, Theorem 2.9]) with the wavelet arguments in [6]. However, it turns out that a satisfactory result requires a more subtle strategy in three different aspects.

Firstly, we need an extension of theLp-theory in [29] to anLq(Lp)-theory for SPDEs dealing with stochastic parabolic weighted Sobolev spacesH^γ,q_p,θ(O, T)with possibly dif- ferent summability parametersq and p in time and space respectively. These spaces consist of elementsu˜ of the formd˜u= ˜f dt+ ˜g^kdw_t^k, whereu˜,f˜and ˜g^k, considered as stochastic processes with values in suitable weighted L_p(O)-Sobolev spaces, are L_q- integrable w.r.t.P⊗dt. Such an extension is needed to obtain better Hölder estimates in time in a second step. Satisfactory existence and uniqueness results concerning so- lutions in the spacesH^γ,q_p,θ(O, T)have been established in [28] for domainsO withC¹- boundary. Unfortunately, the techniques used there do not work on general Lipschitz domains. Also, theL_q(L_p)-results that have been obtained in [50] within the semigroup approach to SPDEs do not directly suit our purpose: On the one hand, for general Lip- schitz domainsOthe domains of the fractional powers of the leading linear differential operator cannot be characterized in terms of Sobolev or Besov spaces as in the case of a smooth domains O; see, e.g., the introduction of [6] for details. On the other hand, even in the case of a smooth domainOwe need regularity in terms ofweightedSobolev spaces to obtain the optimal regularity in the scale (∗).

Secondly, once we have established the solvability of SPDEs within the stochastic parabolic weighted Sobolev spacesH^γ,q_p,θ(O, T), we have to exploit theLq(Lp)–regularity of the solution and derive improved results on the Hölder regularity in time for large q. ForO =R^d+ this has been done by Krylov [35]. It takes quite delicate arguments to apply these results to the case of bounded Lipschitz domains via a boundary flattening argument.

Thirdly, in order to obtain a reasonable Hölder-Besov regularity result, it is neces- sary to generalize the wavelet arguments applied in [6] to a wider range of smoothness parameters. This requires more sophisticated estimates.

In this article we tackle and solve the tasks described above. We organize the article as follows. In Section 2 we recall the definition and basic properties of the (determin- istic) weighted Sobolev spacesH_p,θ^γ (G)introduced in [41] (see also [46, Chapter 6]) on general domainsG⊂R^d with non-empty boundary. In Section 3 we give the definition of the spacesH^γ,q_p,θ(G, T)and specify the concept of a solution for equations of type (1.1) in these spaces. Moreover, we show that if we have a solutionu∈H^γ,q_p,θ(G, T)with low regularityγ ≥ 0, but f and the g^k’s have highLq(Lp)-regularity, then we can lift up the regularity of the solution (Theorem 3.8). In this sense the spacesH^γ,q_p,θ(G, T)are the right ones for our regularity analysis of SPDEs.

Section 4 is devoted to the solvability of Eq. (1.1) in H^γ,q_p,θ(O, T), O ⊆ R^{d being a} bounded Lipschitz domain. The focus lies on the caseq > p≥2and we restrict our con- siderations to equations with additive noise, i.e.σ^ik≡0. In Subsection 4.1 we consider equations on domains with small Lipschitz constants and derive a result for general integrability parametersq ≥ p≥ 2 (Theorem 4.2). We use anLq(Lp)-regularity result for deterministic parabolic equations from [18] and an estimate for stochastic integrals in UMD spaces from [49] to obtain a certain lowL_q(L_p)-regularity of the solution. Then the regularity is lifted up with the help of Theorem 3.8. In Subsection 4.2, we consider the stochastic heat equation on general bounded Lipschitz domains. Here we use the results from [50] on maximalLq-regularity of stochastic evolution equations (see also [51] and [49]) to derive existence and uniqueness of a solution with low regularity. A

main ingredient will be the fact that the domain of the square root of the weak Dirich- let Laplacian on L_p(O) coincides with the closure of the test functionsC₀^∞(O)in the Lp(O)-Sobolev space of order one (Lemma 4.5). This stays true only for a certain range ofp∈[2, p0)withp0>3. Thus, so does our result (Theorem 4.4). In a second step, we again lift up the regularity by using Theorem 3.8. In both settings we derive suitable a-priori estimates.

In Section 5 we present our result on the Hölder regularity in time of the elements ofH^γ,q_p,θ(O, T)(Theorem 5.1). It is an extension of the Hölder estimates in time for the elements ofH^γ,q_p,θ(T) = H^γ,q_p,θ(R^d+, T)in [34] to the case of bounded Lipschitz domains.

The implications for the Hölder regularity of the solutions of SPDEs are described in Theorem 5.3.

In Section 6 we pave the way for the analysis of the spatial regularity of the solutions of SPDEs in the scale (∗). We discuss the relationship between the weighted Sobolev spaces H_p,θ^γ (O) and Besov spaces. Our main result in this section, Theorem 6.9, is a general embedding of the spaces H_p,d−νp^γ (O), γ, ν > 0, into the Besov scale (∗). Its proof is an extension of the wavelet arguments in the proof of [6, Theorem 3.1], where only integer valued smoothness parametersγare considered. It can also be seen as an extension of and a supplement to the Besov regularity results for deterministic elliptic equations in [12] and [9, 10, 11, 13]. To the best of our knowledge, no such general embedding has been proven before. In the course of the discussion we also enlighten the fact that, for the relevant range of parametersγandν, the spacesH_p,d−νp^γ (O)act like Besov spacesB_p,p^γ∧ν(O)with zero trace on the boundary (Remark 6.7). Let us note that related embedding results have been obtained, simultaneously with and independently from our work, in [21].

In Section 7 the results of the previous sections are combined in order to deter- mine the Hölder-Besov regularity of the elements of the stochastic parabolic spaces H^γ,q_p,θ(O, T)and of the solutions of SPDEs within these spaces. The related result in [6]

is significantly improved in several aspects; see Remark 7.3 for a detailed comparison.

We obtain an estimate of the form

Ekuk^q_Cκ([0,T];B^α_τ,τ(O))≤Nkuk^q_Hγ,q

p,θ(O,T), 1 τ = α

d +1 p,

for certainαdepending on the smoothness and weight parametersγandθand for cer- tainκdepending onqandα(Theorem 7.4). Using the a-priori estimates from Section 4, the right hand side of the above inequality can be estimated by suitable norms off and gifuis the solution to the corresponding SPDE (Theorem 7.5).

Let us also mention the related work [1] on the Besov regularity for the deterministic heat equation. The authors study the regularity of temperatures in terms of anisotropic Besov spaces of typeBτ,τ^α/2,α((0, T)× O),1/τ =α/d+ 1/p. However, the range of admis- sible values for the parameterτ is a priori restricted to(1,∞), so thatαis always less thand(1−1/p). In our article the parameter τ in (∗) may be any positive number, in- cluding in particular the case whereτis less than1and whereB_τ,τ^α (O)is not a Banach space but a quasi-Banach space.

Notation and conventions.Throughout this paper,Oalways denotes a bounded Lips- chitz domain inR^d,d≥1, as specified in Definition 2.5 below. General subsets ofR^dare denoted byG. We write∂Gfor their boundary (if it is not empty) andG^◦for the interior.

N:={1,2, . . .} denotes the set of strictly positive integers whereasN0 :=N∪ {0}. Let (Ω,F,P)be a complete probability space and{Ft, t ≥0} be an increasing filtration of σ-fields Ft ⊂ F, each of which contains all(F,P)-null sets. ByP we denote the pre- dictableσ-field generated by{Ft, t≥0}and we assume that{(w¹_t)_t∈[0,T],(w_t²)_t∈[0,T], . . .}

are independent one-dimensional Wiener processes w.r.t.{Ft, t≥0}. Forκ∈(0,1)and a quasi-Banach space(X,k·k_X)we denote byC^κ([0, T];X)the Hölder space of continuous X-valued functions on[0, T]with finite normk·k_Cκ([0,T];X)defined by

[u]_Cκ([0,T];X):= sup

s,t∈[0,T]

ku(t)−u(s)kX

|t−s|^κ , kuk_C([0,T];X):= sup

t∈[0,T]

ku(t)kX,

kuk_Cκ([0,T];X)=kuk_C([0,T];X)+ [u]_Cκ([0,T];X).

For1< p <∞,Lp(A,Σ, µ;X)denotes the space ofµ-strongly measurable andp-Bochner integrable functions with values inX on aσ-finite measure space (A,Σ, µ), endowed with the usual Lp-Norm. We write Lp(G) instead of Lp(G,B(G), λ^d;R) if G ∈ B(R^d), where B(G) and B(R^d) are the Borel-σ-fields on G and R^d. Recall the Hilbert space

`<sub>2</sub> := `₂(N) = {a = (a¹,a², . . .) : |a|`2 = (P

k|a^k|²)^1/2 < ∞} with the inner product ha,bi_`2 =P

ia^kb^k, fora,b∈`₂. The notationC₀^∞(G)is used for the space of infinitely differentiable test functions with compact support in a domainG⊆R^d. For any distribu- tionf onGand anyϕ∈C₀^∞(G),(f, ϕ)denotes the application off toϕ. Furthermore, for any multi-indexα = (α1, . . . , αd) ∈ N^d0, we write D^αf = ^∂|α|f

∂x^α₁¹...∂x^αd_d for the corre- sponding (generalized) derivative w.r.t.x= (x1, . . . , xd)∈G, where|α|=α1+. . .+αd. By making slight abuse of notation, form ∈ N^{0, we write} D^mf for any (generalized) m-th order derivative of f and for the vector of all m-th order derivatives of f. E.g.

if we write D^mf ∈ X, whereX is a function space on G, we meanD^αf ∈ X for all α∈N^d0with|α|=m. We also use the notationf_xix^j =_∂x^∂i²∂x^fj, f_xi= _∂x^∂fi. The notationfx

(respectivelyfxx) is used synonymously forDf :=D¹f (respectively forD²f), whereas kfxkX :=P

ikuxⁱkX (respectivelykfxxkX :=P

i,jkfxⁱx^jkX). Moreover, ∆f :=P

if_xixⁱ, whenever it makes sense. Givenp ∈ [1,∞) and m ∈ N^, W_p^m(G)denotes the classical Sobolev space consisting of allf ∈Lp(G)such that|f|_Wl

p(G):= sup_α∈Nd

0,|α|=lkD^αfk_Lp(G) is finite for alll ∈ {1,2, . . . , m}. It is normed viakfk^p_Wm

p(G) := kfk^p_L

p(G)+|f|^p_Wm p (G). We also set |f|_W0

p(G) := kfk_Lp(G). The closure of C₀^∞(O) in W_p¹(O) is denoted by W˚_p¹(O) and is normed bykfkW˚_p¹(O) := (P

ikf_xik^p_L

p(O))^1/p. If we have two quasi-normed spaces (Xi,k·kX_i),i= 1,2,X1 ,→X2means thatX1is continuously linearly embedded inX2. For a compatible couple(X₁, X₂)of Banach spaces,[X₁, X₂]_ηdenotes the interpolation space of exponentη∈(0,1)arising from the complex interpolation method. In general, N will denote a positive finite constant, which may differ from line to line. The notation N = N(a1, a2, . . .)is used to emphasize the dependence of the constant N on the set of parameters {a1, a2, . . .}. In general, this set will not contain all the parameters N depends on.A∼Bmeans thatAandBare equivalent.

2 Weighted Sobolev spaces

We start by recalling the definition and some basic properties of the (deterministic and stationary) weighted Sobolev spacesH_p,θ^γ (G)introduced in [41]. These spaces will serve as state spaces for the solution processesu= (u(t))_t∈[0,T]to SPDEs of type (1.1) and they will play a fundamental role in all the forthcoming sections.

Forp∈(1,∞)and γ∈R^{, let}H_p^γ :=H_p^γ(R^d) := (1−∆)^−γ/2Lp(R^d)be the spaces of Bessel potentials, endowed with the norm

kuk_H^γ_p :=k(1−∆)^γ/2uk_Lp(Rd):=kF⁻¹[(1 +|ξ|²)^γ/2F(u)(ξ)]k_Lp(Rd),

where F denotes the Fourier transform. It is well known that if γ is a nonnegative

integer, then

H_p^γ =

u∈L_p : D^αu∈L_pfor allα∈N^d0with|α| ≤γ .

Let G ⊂ R^d be an arbitrary domain with non-empty boundary ∂G. We denote by ρ(x) :=ρG(x) :=dist(x, ∂G)the distance of a pointx∈Gto the boundary∂G. Further- more, we fix a bounded infinitely differentiable functionψdefined onGsuch that for all x∈G,

ρ(x)≤N ψ(x), ρ(x)^m−1|D^mψ(x)| ≤N(m)<∞for allm∈N0, (2.1) where N and N(m) do not depend on x ∈ G. For a detailed construction of such a function see, e.g., [46, Chapter 3, Section 3.2.3]. Letζ ∈ C₀^∞(R⁺) be a non-negative function satisfying

X

n∈Z

ζ(e^n+t)> c >0for allt∈R. (2.2) Note that any non-negative smooth functionζ∈C₀^∞(R+)withζ >0on[e⁻¹, e]satisfies (2.2). Forx∈Gandn∈Z^{, define}

ζn(x) :=ζ(eⁿψ(x)).

Then, there existsk0 >0 such that, for alln∈ Z^{, supp}ζn ⊂Gn :={x∈G: e^−n−k0 <

ρ(x)< e^−n+k0}, i.e.,ζn ∈C₀^∞(Gn). Moreover,|D^mζn(x)| ≤N(ζ, m)e^mnfor allx∈Gand m∈N^{0, and}P

n∈Zζn(x)≥δ >0for allx∈G. Forp∈(1,∞)andγ, θ∈R, we denote by H_p,θ^γ (G)the space of all distributionsuonGsuch that

kuk^p_Hγ

p,θ(G):=X

n∈Z

e^nθkζ_−n(eⁿ·)u(eⁿ·)k^p_Hγ p <∞.

It is well-known that

Lp,θ(G) :=H_p,θ⁰ (G) =Lp(G, ρ^θ−ddx), and that, ifγis a positive integer,

H_p,θ^γ (G) =

u∈Lp,θ(G) : ρ^|α|D^αu∈Lp,θ(G)for allα∈N^d0with|α| ≤γ , kuk^p_Hγ

p,θ(G)∼ X

|α|≤γ

Z

G

ρ^|α|D^αu

pρ^θ−ddx; (2.3)

see, e.g., [41, Proposition 2.2]. This is the reason why the space H_p,θ^γ (G) is called weighted Sobolev space of orderγ, with summability parameterpand weight parame- terθ.

For p ∈ (1,∞) and γ ∈ R^{we write} H_p^γ(`2)for the collection of all sequences g = (g¹, g², . . .)of distributions onR^dwithg^k ∈H_p^γ for eachk∈N^and

kgk_Hp^γ<sub>(`2)</sub>:=kgk<sub>Hp</sub><sup>γ</sup><sub>(R</sub>d;`₂):=k|(1−∆)^γ/2g|`₂kL_p :=

X^∞

k=1

|(1−∆)^γ/2g^k|^21/2 _L

p

<∞.

Analogously, forθ∈R, a sequenceg= (g¹, g², . . .)of distributions onGis inH_p,θ^γ (G;`2) if, and only if,g^k∈H_p,θ^γ (G)for eachk∈N^and

kgk^p_Hγ

p(G;`₂):=X

n∈Z

e^nθkζ−n(eⁿ·)g(eⁿ·)k^p_Hγ

p(`₂)<∞.

Now we present some useful properties of the space H_p,θ^γ (G) taken from [41], see also [32, 33].

Lemma 2.1. Let G ⊂ R^d be a domain with non-empty boundary ∂G, γ, θ ∈ R^{, and} p∈(1,∞).

(i)The spaceC₀^∞(G)is dense inH_p,θ^γ (G).

(ii)Assume that γ−d/p = m+ν for somem ∈ N^0, ν ∈ (0,1]and that i, j ∈ N^d0 are multi-indices such that|i| ≤mand|j|=m. Then for anyu∈H_p,θ^γ (G), we have

ψ^|i|+θ/pDⁱu∈C(G), ψ^m+ν+θ/pD^ju∈C^ν(G),

|ψ^|i|+θ/pDⁱu|_C(G)+ [ψ^m+ν+θ/pD^ju]_Cν(G)≤Nkuk_H^γ

p,θ(G). (iii)u∈H_p,θ^γ (G)if, and only if,u, ψux∈H_p,θ^γ−1(G)and

kuk_H^γ

p,θ(G)≤Nkψu_xk_Hγ−1

p,θ (G)+Nkuk_Hγ−1

p,θ (G)≤Nkuk_H^γ

p,θ(G). Also,u∈H_p,θ^γ (G)if, and only if,u,(ψu)_x∈H_p,θ^γ−1(G)and

kuk_H^γ

p,θ(G)≤Nk(ψu)xk_H^γ−1

p,θ (G)+Nkuk_H^γ−1

p,θ (G)≤Nkuk_H^γ

p,θ(G). (iv)For anyν, γ∈R^,ψ^νH_p,θ^γ (G) =H_p,θ−pν^γ (G)and

kuk_H^γ

p,θ−pν(G)≤Nkψ^−νuk_H^γ

p,θ(G)≤Nkuk_H^γ

p,θ−pν(G).

(v)Ifγ∈(γ0, γ1)then, for anyε >0, there exists a constantN =N(γ0, γ1, θ, p, ε), such that

kuk_H^γ

p,θ(G)≤εkuk_H^γ1

p,θ(G)+N(γ₀, γ₁, θ, p, ε)kuk_H^γ0 p,θ(G).

Also, ifθ∈(θ0, θ1)then, for anyε >0, there exists a constantN =N(θ0, θ1, γ, p, ε), such that

kuk_H^γ

p,θ(G)≤εkuk_H^γ

p,θ0(G)+N(θ0, θ1, γ, p, ε)kuk_H^γ

p,θ1(G).

(vi)There exists a constantc0 >0depending onp,θ,γ and the functionψsuch that, for allc≥c₀, the operatorψ²∆−cis a homeomorphism fromH_p,θ^γ+1(G)toH_p,θ^γ−1(G). Remark 2.2. Assertions (vi) and (iv) in Lemma 2.1 imply the following: Ifu∈H_p,θ−p^γ (G) and∆u∈H_p,θ+p^γ (G), thenu∈H_p,θ−p^γ+2 (G)and there exists a constantN, which does not depend onu, such that

kuk_Hγ+2

p,θ−p(G)≤Nk∆uk_H^γ

p,θ+p(G)+Nkuk_H^γ

p,θ−p(G).

A proof of the following equivalent characterization of the weighted Sobolev spaces H_p,θ^γ (G)can be found in [41, Proposition 2.2].

Lemma 2.3. Let{ξ_n:n∈Z} ⊆C₀^∞(G)be such that for alln∈Z^andm∈N0,

|D^mξn| ≤N(m)c^nm and suppξn ⊆ {x∈G:c^−n−k0 < ρ(x)< c^−n+k0} (2.4) for somec > 1 and k₀ >0, where the constantN(m)does not depend on n ∈ Z ^and x∈G. Then, for anyu∈H_p,θ^γ (G),

X

n∈Z

c^nθkξ−n(cⁿ·)u(cⁿ·)k^p_Hγ

p ≤Nkuk^p_Hγ p,θ(G). If in addition

X

n∈Z

ξn(x)≥δ >0for allx∈G (2.5) then the converse inequality also holds.

Remark 2.4. (i)It is easy to check that both ξ⁽¹⁾_n :=e⁻ⁿ(ζn)_xi : n∈Z ^and

ξ_n⁽²⁾:=e⁻²ⁿ(ζn)_xix^j : n∈Z satisfy(2.4)withc:=e. Therefore,

X

n∈Z

e^nθ

keⁿ(ζ_−n)_xi(eⁿ·)u(eⁿ·)k^p_Hγ

p+ke²ⁿ(ζ_−n)_xix^j(eⁿ·)u(eⁿ·)k^p_Hγ p

≤Nkuk^p_Hγ p,θ(G). (ii)Givenk1≥1, fix a functionζ˜∈C₀^∞(R⁺)with

ζ(t) = 1˜ for all t∈h1

N 2^−k1, N(0) 2^k1i ,

whereN andN(0)are as in(2.1). Then, the sequence{ξn:n∈Z} ⊆C₀^∞(G)defined by ξn:= ˜ζ(2ⁿψ(·)), n∈Z,

fulfils the conditions (2.4)and (2.5)from Lemma 2.3 with c= 2and a suitablek₀ >0. Furthermore,

ξn(x) = 1 for all x∈ρ⁻¹ 2⁻ⁿ

2^−k1,2^k1 .

In this paper, O will always denote a bounded Lipschitz domain in R^{d. More} precisely:

Definition 2.5. We call a bounded domain O ⊂R^d a Lipschitz domain if, and only if, for anyx0= (x¹₀, x⁰₀)∈∂O, there exists a Lipschitz continuous functionµ0 :R^d−1→R such that, upon relabeling and reorienting the coordinate axes if necessary, we have

(i) O ∩Br₀(x0) ={x= (x¹, x⁰)∈Br₀(x0) :x¹> µ0(x⁰)}, and (ii) |µ0(x⁰)−µ0(y⁰)| ≤K0|x⁰−y⁰|, for anyx⁰, y⁰∈R^d−1, wherer0, K0are independent ofx0.

Remark 2.6. Recall that for a bounded Lipschitz domainO ⊂R^d, W˚_p¹(O) =H_p,d−p¹ (O)

with equivalent norms. This follows from [38, Theorem 9.7] and Poincaré’s inequality.

3 Stochastic parabolic weighted Sobolev spaces and SPDEs

In this section, we first introduce the stochastic parabolic spacesH^γ,q_p,θ(G, T)for ar- bitrary domainsG⊂R^d with non-empty boundary in analogy to the spaces H^γ,q_p,θ(T) = H^γ,q_p,θ(R^d+, T) from [34, 35]. Then we show that they are suitable to serve as solution spaces for equations of type (1.1) in the following sense: If we have a solution u ∈ H^γ,q_p,θ(G, T)with low regularity γ ≥ 0, but f and the g^k’s have high Lq(Lp)-regularity, then we can lift up the regularity of the solution (Theorem 3.8).

Definition 3.1. LetGbe a domain inR^d with non-empty boundary. Forp, q ∈ (1,∞), γ, θ∈R^andT ∈(0,∞)we define

H^γ,qp,θ(G, T) :=Lq(Ω×[0, T],P,P⊗dt;H_p,θ^γ (G)), H^γ,qp,θ(G, T;`<sub>2</sub>) :=L<sub>q</sub>(Ω×[0, T],P,P⊗dt;H<sub>p,θ</sub><sup>γ</sup> (G;`₂)),

U_p,θ^γ,q(G) :=Lq(Ω,F0,P;ψ^1−2/qH_p,θ^γ−2/q(G)).

If p = q we also write H^γp,θ(G, T), H^γp,θ(G, T;`2) and U<sub>p,θ</sub><sup>γ</sup> (G) instead of H<sup>γ,p</sup>p,θ(G, T), H<sup>γ,p</sup>p,θ(G, T;`2)andU_p,θ^γ,p(G)respectively.

From now on let

p∈[2,∞), q∈[2,∞), γ∈R, θ∈R.

Definition 3.2. Let G be a domain in R^d with non-empty boundary. We write u ∈ H^γ,q_p,θ(G, T) if, and only if, u ∈ H^γ,qp,θ−p(G, T), u(0,·) ∈ U_p,θ^γ,q(G), and there exist some f ∈H^γ−2,qp,θ+p(G, T)andg∈H^γ−1,qp,θ (G, T;`2)such that

du=f dt+g^kdw_t^k

in the sense of distributions. That is, for any ϕ ∈ C₀^∞(G), with probability one, the equality

(u(t,·), ϕ) = (u(0,·), ϕ) + Z t

0

(f(s,·), ϕ)ds+

∞

X

k=1

Z t 0

(g^k(s,·), ϕ)dw_s^k

holds for allt ∈ [0, T], where the series is assumed to converge uniformly on[0, T]in probability. In this situation we writeDu := f and Su := g. The norm in H^γ,q_p,θ(G, T)is defined as

kuk_H^γ,q

p,θ(G,T):=kuk_H^γ,q

p,θ−p(G,T)+kDuk_Hγ−2,q

p,θ+p(G,T)+kSuk_Hγ−1,q

p,θ (G,T;`₂)+ku(0,·)k_U^γ,q

p,θ(G). Ifp=qwe also writeH^γ_p,θ(G, T)instead ofH^γ,p_p,θ(G, T).

Remark 3.3. ReplacingGbyR^dand omitting the weight parameterθand the weight function ψ in the definitions above, one obtains the spaces H^γ,qp (T) = H^γ,qp (R^d, T), H^γ,qp (T;`<sub>2</sub>) = H<sup>γ,q</sup>p (R<sup>d</sup>, T;`₂),U_p^γ,q = U_p^γ,q(R^d), andHp^γ,q(T)as introduced in [35, Def- inition 3.5]. The latter are denoted byH^γ,q_p (T) in [34]; ifq = pthey coincide with the spacesH^γ_p(T)introduced in [32, Definition 3.1].

We consider initial value problems of the form

du= (a^ijuxⁱx^j+f)dt+ (σ^ikuxⁱ+g^k)dw_t^k, u(0,·) =u0, (3.1) on an arbitrary domainG⊂R^dwith non-empty boundary. We use the following solution concept.

Definition 3.4. We say that a stochastic processu∈H^γ,q_p,θ(G, T)is a solution of Eq.(3.1) if, and only if,

u(0,·) =u0, Du=a^iju_xix^j +f, and Su= σ^iku_xi+g^k

k∈N, in the sense of Definition 3.2.

Remark 3.5. Here and in the sequel we use the summation convention on the repeated indices i, j, k. The question, in which sense, for a bounded Lipschitz domainO ⊂ R^d, the elements ofH^γ,q_p,θ(O, T)fulfil a zero Dirichlet boundary condition as in Eq.(1.1), will be answered in Remark 6.7.

We make the following assumptions on the coefficients in Eq. (3.1). Throughout this paper, whenever we will talk about this equation, we will assume that they are fulfilled.

Assumption 3.6. (i)The coefficientsa^ij =a^ij(ω, t)andσ^ik=σ^ik(ω, t)are predictable.

They do not depend onx∈G. Furthermore,a^ij =a^jifori, j∈ {1, . . . , d}.

(ii)There exist constantsδ0, K >0such that for any(ω, t)∈Ω×[0, T]andλ∈R^d, δ₀|λ|²≤¯a^ij(ω, t)λⁱλ^j≤a^ij(ω, t)λⁱλ^j ≤K|λ|²,

where¯a^ij(ω, t) :=a^ij(ω, t)−¹₂(σ^i·(ω, t), σ^j·(ω, t))`₂, withσ^i·(ω, t) = σ^ik(ω, t)

k∈N∈`2.

We will use the following result taken from [34, Lemma 2.3].

Lemma 3.7. Letp≥2,m∈N^{, and, for}i= 1,2, . . . , m,

λi∈(0,∞), γi∈R, u⁽ⁱ⁾∈ H^γ_pⁱ⁺²(T), u⁽ⁱ⁾(0,·) = 0.

DenoteΛi:= (λi−∆)^γi/2. Then

EhZ T 0

m

Y

i=1

kΛi∆u⁽ⁱ⁾k^p_L

pdti

≤N

m

X

i=1

EhZ T 0

kΛ_if⁽ⁱ⁾k^p_L

p+kΛ_ig_x⁽ⁱ⁾k^p_L

p(`₂)

Y^m

j=1 j6=i

kΛ_j∆u^(j)k^p_L

pdti

+N X

1≤i<j≤m

EhZ T 0

kΛig⁽ⁱ⁾_x k^p_L

p(`₂)kΛjg^(j)_x k^p_L

p(`₂) m

Y

k6=i,jk=1

kΛk∆u^(k)k^p_L

pdti ,

wheref⁽ⁱ⁾:=Du⁽ⁱ⁾−a^rsu⁽ⁱ⁾_xrx^s,g^(i)k :=S^ku⁽ⁱ⁾−σ^rku⁽ⁱ⁾_xr andL_p(`<sub>2</sub>) :=H<sub>p</sub><sup>0</sup>(`₂). The constant N depends only onm,d,p,δ0, andK.

Now we are able to prove that if we have a solutionu∈H^γ+1,q_p,θ (G, T)to Eq. (3.1) and if the regularity of the forcing termsf andg is high then we can lift the regularity of the solution. Note that in the next theorem there is no restriction, neither on the shape of the domainG⊂R^dnor on the parametersθ, γ∈R^.

Theorem 3.8. LetG⊂R^dbe an arbitrary domain with non-empty boundary. Letγ∈R^, p≥ 2 andq = pmfor somem ∈ N^{. Let}f ∈ H^γ,qp,θ+p(G, T), g ∈ H^γ+1,qp,θ (G, T;`2)and let u∈H^γ+1,q_p,θ (G, T)be a solution to Eq.(3.1)withu0= 0. Thenu∈H^γ+2,q_p,θ (G, T), and

kuk^q

H^γ+2,qp,θ−p(G,T)≤N kuk^q

H^γ+1,qp,θ−p(G,T)+kfk^q

H^γ,q_p,θ+p(G,T)+kgk^q

H^γ+1,qp,θ (G,T;`2)

, where the constantN ∈(0,∞)does not depend onu,f andg.

Proof. The casem= 1, i.e.,p=qis covered by [29, Lemma 3.2]. Therefore, letm≥2. According to Remark 2.2 it is enough to show that

k∆uk^q_Hγ,q

p,θ+p(G,T)≤N kuk^q

H^γ+1,qp,θ−p(G,T)+kfk^q_Hγ,q

p,θ+p(G,T)+kgk^q

H^γ+1,q_p,θ (G,T;`₂)

. Using the definition of weighted Sobolev spaces from Section 2, we observe that

k∆uk^q

H^γ,qp,θ+p(G,T)=EhZ T 0

X

n∈Z

e^n(θ+p)k(ζ_−n∆u(t))(eⁿ·)k^p_Hγ p

m

dti

≤NEhZ T 0

X

n∈Z

e^n(θ+p)

k∆(ζ_−nu(t))(eⁿ·)k^p_Hγ p

+k(∆ζ_−nu(t))(eⁿ·)k^p_Hγ

p +k(ζ_−nxux(t))(eⁿ·)k^p_Hγ p

m

dti . (Hereζ_−nxu_xis meant to be a scalar product inR^d.) Now we can use Jensen’s inequality and Remark 2.4(i) to obtain

k∆uk^q_Hγ,q

p,θ+p(G,T)≤NEhZ T 0

X

n∈Z

e^n(θ+p)k∆(ζ−nu(t))(eⁿ·)k^p_Hγ p

^m

+ku(t)k^q_Hγ

p,θ−p(G)+kux(t)k^q_Hγ p,θ(G)dti

.

An application of Lemma 2.1(iii) and (iv) leads to

k∆uk^q_Hγ,q

p,θ+p(G,T)≤NEhZ T 0

X

n∈Z

e^n(θ+p)k∆(ζ_−nu(t))(eⁿ·)k^p_Hγ p

^m dti

+Nkuk^q

H^γ+1,qp,θ−p(G,T). Therefore, it is enough to estimate the first term on the right hand side,

EhZ T 0

X

n∈Z

e^n(θ+p)k∆(ζ−nu(t))(eⁿ·)k^p_Hγ p

^m dti

=EhZ T 0

X

n₁,...,n_m∈Z

e ^Pmi=1ni (θ+p)

m

Y

i=1

k∆(ζ−niu(t))(eⁿⁱ·)k^p_Hγ pdti

.

Tonelli’s theorem together with the relation ku(c·)k^p_Hγ

p =c^pγ−dk(c⁻²−∆)^γ/2uk^p_L

p forc∈(0,∞), (3.2) applied to∆u⁽ⁿⁱ⁾withu⁽ⁿ⁾:=ζ−nuforn∈Z, show that we only have to handle

X

n1,...,nm∈Z

e ^Pmi=1ni

(θ+p+pγ−d)

EhZ T 0

m

Y

i=1

k(e⁻²ⁿⁱ−∆)^γ/2∆u⁽ⁿⁱ⁾(t)k^p_L

pdti .

Note that sinceu∈ H^γ+1,q_p,θ (G, T)solves Eq. (3.1) with vanishing initial value,u⁽ⁿ⁾ is a solution of the equation

dv= (a^rsvx^rx^s+f⁽ⁿ⁾)dt+ (σ^rkvx^r+g^(n)k)dw_t^k, v(0,·) = 0,

onR^{d, where}f⁽ⁿ⁾=−2a^rs(ζ_−n)x^sux^r−a^rs(ζ_−n)x^rx^su+ζ_−nf andg^(n)k=−σ^rk(ζ_−n)x^ru+

ζ_−ng^k. Furthermore, applying [32, Theorem 4.10], we have u⁽ⁿ⁾ ∈ H^γ+2_p (T). Thus, we can use Lemma 3.7 to obtain

EhZ T 0

m

Y

i=1

k(e⁻²ⁿⁱ−∆)^γ/2∆u⁽ⁿⁱ⁾(t)k^p_L

pdti

≤N

m

X

i=1

Ini+IIni

+N X

1≤i<j≤m

IIIninj

where we denote In_i :=EhZ T

0

kΛn_if⁽ⁿⁱ⁾(t)k^p_L

p

m

Y

j=1 j6=i

kΛn_j∆u^(nj)(t)k^p_L

pdti ,

II_ni :=EhZ T 0

kΛnig⁽ⁿ_xⁱ⁾(t)k^p_L

p(`2) m

Y

j=1 j6=i

kΛnj∆u^(nj)(t)k^p_Lpdti ,

IIIn_in_j :=EhZ T 0

kΛn_ig⁽ⁿ_xⁱ⁾(t)k^p_L

p(`₂)kΛn_jg_x^(nj)(t)k^p_L

p(`₂) m

Y

k6=i,jk=1

kΛn_k∆u^(nk)(t)k^p_L

pdti ,

withΛ_n := (e⁻²ⁿ−∆)^γ/2. Thus, it is enough to find a proper estimate for

X

n1,...,nm∈Z

e ^Pmi=1ni

(θ+p+pγ−d)X^m

i=1

In_i+IIn_i

+ X

1≤i<j≤m

IIIn_in_j

.

Applying (3.2) first, followed by Tonelli’s theorem, then Hölder’s and Young’s inequality, leads to

X

n1,...,nm∈Z

e ^Pmi=1ni

(θ+p+pγ−d) m

X

i=1

In_i

= X

n1,...,nm∈Z

e ^Pmi=1ni (θ+p)

m

X

i=1

EhZ T 0

kf⁽ⁿⁱ⁾(t, eⁿⁱ·)k^p_Hγ p

m

Y

j=1 j6=i

k∆u^(nj)(t, e^nj·)k^p_Hγ pdti

≤NEhZ T 0

X

n∈Z

e^n(θ+p)kf⁽ⁿ⁾(t, eⁿ·)k^p_Hγ p

X

n∈Z

e^n(θ+p)k∆u⁽ⁿ⁾(t, eⁿ·)k^p_Hγ p

m−1

dti

≤N(ε)EhZ T 0

X

n∈Z

e^n(θ+p)kf⁽ⁿ⁾(t, eⁿ·)k^p_Hγ p

^q_p dti

+εEhZ T 0

X

n∈Z

e^n(θ+p)k∆u⁽ⁿ⁾(t, eⁿ·)k^p_Hγ p

^q_p dti

.

Using the definition off⁽ⁿ⁾and arguing as at the beginning of the proof, we obtain X

n∈Z

e^n(θ+p)kf⁽ⁿ⁾(t, eⁿ·)k^p_Hγ

p ≤N

kux(t)k^p_Hγ

p,θ(G)+ku(t)k^p_Hγ

p,θ−p(G)+kf(t)k^p_Hγ p,θ+p(G)

≤N ku(t)k^p

H_p,θ−p^γ+1 (G)+kf(t)k^p_Hγ p,θ+p(G)

. Moreover,

X

n∈Z

e^n(θ+p)k∆u⁽ⁿ⁾(t, eⁿ·)k^p_Hγ

p ≤X

n∈Z

e^n(θ+p)k(∆ζ−nu(t))(eⁿ·)k^p_Hγ p

+X

n∈Z

e^n(θ+p)k(ζ−nxux(t))(eⁿ·)k^p_Hγ p

+X

n∈Z

e^n(θ+p)k(ζ−n∆u(t))(eⁿ·)k^p_Hγ p

≤N

ku(t)k^p_Hγ

p,θ−p(G)+kux(t)k^p_Hγ

p,θ(G)+k∆uk^p_Hγ p,θ+p(G)

≤N ku(t)k^p

H^γ+1_p,θ−p(G)+k∆uk^p_Hγ p,θ+p(G)

.

Combining the last three estimates, we obtain for anyε >0a constantN(ε)∈(0,∞), such that

X

n₁,...,n_m∈Z

e ^Pmi=1ni

(θ+p+pγ−d) m

X

i=1

In_i≤εk∆uk^q

H^γ,qp,θ+p(G,T)

+N(ε) kfk^q_Hγ,q

p,θ+p(G,T)+kuk^q

H^γ+1,qp,θ−p(G,T)

. Using similar arguments we obtain

X

n₁,...,n_m∈Z

e ^Pmi=1ni

(θ+p+pγ−d)X^m

i=1

IIn_i+ X

1≤i<j≤m

IIIn_in_j

≤εk∆uk^q_Hγ,q

p,θ+p(G,T)+N(ε) kgk^q

H^γ+1,q_p,θ (G,T;`₂)+kuk^q

H^γ+1,qp,θ−p(G,T)

, which finishes the proof.