**E**l e c t ro nic
**J**

o f

**P**r

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

q### (L

p### ) **-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^{d}with 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^{ij}u_{x}ix^{j} +f)dt+ (σ^{ik}u_{x}i+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^{k}dw^{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^{k}dw_{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 withC^{1}-
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 functionsC_{0}^{∞}(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}_{C}κ([0,T];B^{α}_{τ,τ}(O))≤Nkuk^{q}_{H}γ,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^{d}^{are}
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{(w^{1}_{t})_{t∈[0,T]},(w_{t}^{2})_{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_{C}κ([0,T];X)defined by

[u]_{C}κ([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_{C}κ([0,T];X)=kuk_{C([0,T}_{];X)}+ [u]_{C}κ([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

`_{2} := `_{2}(N) = {a = (a^{1},a^{2}, . . .) : |a|`2 = (P

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

ia^{k}b^{k}, fora,b∈`_{2}. The notationC_{0}^{∞}(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∈C_{0}^{∞}(G),(f, ϕ)denotes the application off toϕ. Furthermore,
for any multi-indexα = (α1, . . . , αd) ∈ N^{d}0, we write D^{α}f = ^{∂}^{|α|}^{f}

∂x^{α}_{1}^{1}...∂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^{m}f 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^{m}f ∈ X, whereX is a function space on G, we meanD^{α}f ∈ X for all
α∈N^{d}0with|α|=m. We also use the notationf_{x}ix^{j} =_{∂x}^{∂}i^{2}∂x^{f}^{j}, f_{x}i= _{∂x}^{∂f}i. The notationfx

(respectivelyfxx) is used synonymously forDf :=D^{1}f (respectively forD^{2}f), whereas
kfxkX :=P

ikux^{i}kX (respectivelykfxxkX :=P

i,jkfx^{i}x^{j}kX). Moreover, ∆f :=P

if_{x}ix^{i},
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|_{W}l

p(G):= sup_{α∈}_{N}d

0,|α|=lkD^{α}fk_{L}_{p}_{(G)}
is finite for alll ∈ {1,2, . . . , m}. It is normed viakfk^{p}_{W}m

p(G) := kfk^{p}_{L}

p(G)+|f|^{p}_{W}m
p (G). We
also set |f|_{W}0

p(G) := kfk_{L}_{p}_{(G)}. The closure of C_{0}^{∞}(O) in W_{p}^{1}(O) is denoted by W˚_{p}^{1}(O)
and is normed bykfkW˚_{p}^{1}(O) := (P

ikf_{x}ik^{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(X_{1}, X_{2})of Banach spaces,[X_{1}, X_{2}]_{η}denotes 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/2}Lp(R^{d})be the spaces of
Bessel potentials, endowed with the norm

kuk_{H}^{γ}_{p} :=k(1−∆)^{γ/2}uk_{L}_{p}_{(}_{R}d):=kF^{−1}[(1 +|ξ|^{2})^{γ/2}F(u)(ξ)]k_{L}_{p}_{(}_{R}d),

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^{α}u∈L_{p}for allα∈N^{d}0with|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^{m}ψ(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 ∈ C_{0}^{∞}(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∈C_{0}^{∞}(R+)withζ >0on[e^{−1}, e]satisfies
(2.2). Forx∈Gandn∈Z^{, define}

ζn(x) :=ζ(e^{n}ψ(x)).

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

ρ(x)< e^{−n+k}^{0}}, i.e.,ζn ∈C_{0}^{∞}(Gn). Moreover,|D^{m}ζn(x)| ≤N(ζ, m)e^{mn}for 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}_{H}γ

p,θ(G):=X

n∈Z

e^{nθ}kζ_{−n}(e^{n}·)u(e^{n}·)k^{p}_{H}γ
p <∞.

It is well-known that

Lp,θ(G) :=H_{p,θ}^{0} (G) =Lp(G, ρ^{θ−d}dx),
and that, ifγis a positive integer,

H_{p,θ}^{γ} (G) =

u∈Lp,θ(G) : ρ^{|α|}D^{α}u∈Lp,θ(G)for allα∈N^{d}0with|α| ≤γ ,
kuk^{p}_{H}γ

p,θ(G)∼ X

|α|≤γ

Z

G

ρ^{|α|}D^{α}u

pρ^{θ−d}dx; (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 =
(g^{1}, g^{2}, . . .)of distributions onR^{d}^{with}g^{k} ∈H_{p}^{γ} for eachk∈N^{and}

kgk_{H}_{p}^{γ}_{(`}_{2}_{)}:=kgk_{H}_{p}^{γ}_{(}_{R}d;`_{2}):=k|(1−∆)^{γ/2}g|`_{2}kL_{p} :=

X^{∞}

k=1

|(1−∆)^{γ/2}g^{k}|^{2}^{1/2}
_{L}

p

<∞.

Analogously, forθ∈R, a sequenceg= (g^{1}, g^{2}, . . .)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}_{H}γ

p(G;`_{2}):=X

n∈Z

e^{nθ}kζ−n(e^{n}·)g(e^{n}·)k^{p}_{H}γ

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 spaceC_{0}^{∞}(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^{d}0 are
multi-indices such that|i| ≤mand|j|=m. Then for anyu∈H_{p,θ}^{γ} (G), we have

ψ^{|i|+θ/p}D^{i}u∈C(G), ψ^{m+ν+θ/p}D^{j}u∈C^{ν}(G),

|ψ^{|i|+θ/p}D^{i}u|_{C(G)}+ [ψ^{m+ν+θ/p}D^{j}u]_{C}ν(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_{x}k_{H}γ−1

p,θ (G)+Nkuk_{H}γ−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^{ν}H_{p,θ}^{γ} (G) =H_{p,θ−pν}^{γ} (G)and

kuk_{H}^{γ}

p,θ−pν(G)≤Nkψ^{−ν}uk_{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}^{γ}1

p,θ(G)+N(γ_{0}, γ_{1}, θ, p, ε)kuk_{H}^{γ}0
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≥c_{0}, the operatorψ^{2}∆−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_{H}γ+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} ⊆C_{0}^{∞}(G)be such that for alln∈Z^{and}m∈N0,

|D^{m}ξn| ≤N(m)c^{nm} and suppξn ⊆ {x∈G:c^{−n−k}^{0} < ρ(x)< c^{−n+k}^{0}} (2.4)
for somec > 1 and k_{0} >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^{nθ}kξ−n(c^{n}·)u(c^{n}·)k^{p}_{H}γ

p ≤Nkuk^{p}_{H}γ
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)_{x}i : n∈Z ^{and}

ξ_{n}^{(2)}:=e^{−2n}(ζn)_{x}ix^{j} : n∈Z
satisfy(2.4)withc:=e. Therefore,

X

n∈Z

e^{nθ}

ke^{n}(ζ_{−n})_{x}i(e^{n}·)u(e^{n}·)k^{p}_{H}γ

p+ke^{2n}(ζ_{−n})_{x}ix^{j}(e^{n}·)u(e^{n}·)k^{p}_{H}γ
p

≤Nkuk^{p}_{H}γ
p,θ(G).
**(ii)**Givenk1≥1, fix a functionζ˜∈C_{0}^{∞}(R^{+})with

ζ(t) = 1˜ for all t∈h1

N 2^{−k}^{1}, N(0) 2^{k}^{1}i
,

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

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

ξn(x) = 1 for all x∈ρ^{−1} 2^{−n}

2^{−k}^{1},2^{k}^{1}
.

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= (x^{1}_{0}, x^{0}_{0})∈∂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 ∩Br_{0}(x0) ={x= (x^{1}, x^{0})∈Br_{0}(x0) :x^{1}> µ0(x^{0})}, and
**(ii)** |µ0(x^{0})−µ0(y^{0})| ≤K0|x^{0}−y^{0}|, for anyx^{0}, y^{0}∈R^{d−1}^{,}
wherer0, K0are independent ofx0.

**Remark 2.6.** Recall that for a bounded Lipschitz domainO ⊂R^{d}^{,}
W˚_{p}^{1}(O) =H_{p,d−p}^{1} (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^{and}T ∈(0,∞)we define

H^{γ,q}p,θ(G, T) :=Lq(Ω×[0, T],P,P⊗dt;H_{p,θ}^{γ} (G)),
H^{γ,q}p,θ(G, T;`_{2}) :=L_{q}(Ω×[0, T],P,P⊗dt;H_{p,θ}^{γ} (G;`_{2})),

U_{p,θ}^{γ,q}(G) :=Lq(Ω,F0,P;ψ^{1−2/q}H_{p,θ}^{γ−2/q}(G)).

If p = q we also write H^{γ}p,θ(G, T), H^{γ}p,θ(G, T;`2) and U_{p,θ}^{γ} (G) instead of H^{γ,p}p,θ(G, T),
H^{γ,p}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,q}p,θ−p(G, T), u(0,·) ∈ U_{p,θ}^{γ,q}(G), and there exist some
f ∈H^{γ−2,q}p,θ+p(G, T)andg∈H^{γ−1,q}p,θ (G, T;`2)such that

du=f dt+g^{k}dw_{t}^{k}

in the sense of distributions. That is, for any ϕ ∈ C_{0}^{∞}(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_{H}γ−2,q

p,θ+p(G,T)+kSuk_{H}γ−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^{d}and omitting the weight parameterθand the weight
function ψ in the definitions above, one obtains the spaces H^{γ,q}p (T) = H^{γ,q}p (R^{d}, T),
H^{γ,q}p (T;`_{2}) = H^{γ,q}p (R^{d}, 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^{ij}ux^{i}x^{j}+f)dt+ (σ^{ik}ux^{i}+g^{k})dw_{t}^{k}, u(0,·) =u0, (3.1)
on an arbitrary domainG⊂R^{d}with 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^{ij}u_{x}ix^{j} +f, and Su= σ^{ik}u_{x}i+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^{ji}fori, j∈ {1, . . . , d}.

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

where¯a^{ij}(ω, t) :=a^{ij}(ω, t)−^{1}_{2}(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}_{p}^{i}^{+2}(T), u^{(i)}(0,·) = 0.

DenoteΛi:= (λi−∆)^{γ}^{i}^{/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_{i}f^{(i)}k^{p}_{L}

p+kΛ_{i}g_{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^{rs}u^{(i)}_{x}rx^{s},g^{(i)k} :=S^{k}u^{(i)}−σ^{rk}u^{(i)}_{x}r andL_{p}(`_{2}) :=H_{p}^{0}(`_{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^{d}nor on the parametersθ, γ∈R^{.}

**Theorem 3.8.** LetG⊂R^{d}be an arbitrary domain with non-empty boundary. Letγ∈R^{,}
p≥ 2 andq = pmfor somem ∈ N^{. Let}f ∈ H^{γ,q}p,θ+p(G, T), g ∈ H^{γ+1,q}p,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,q}p,θ−p(G,T)≤N
kuk^{q}

H^{γ+1,q}p,θ−p(G,T)+kfk^{q}

H^{γ,q}_{p,θ+p}(G,T)+kgk^{q}

H^{γ+1,q}p,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}_{H}γ,q

p,θ+p(G,T)≤N
kuk^{q}

H^{γ+1,q}p,θ−p(G,T)+kfk^{q}_{H}γ,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,q}p,θ+p(G,T)=EhZ T
0

X

n∈Z

e^{n(θ+p)}k(ζ_{−n}∆u(t))(e^{n}·)k^{p}_{H}γ
p

m

dti

≤NEhZ T 0

X

n∈Z

e^{n(θ+p)}

k∆(ζ_{−n}u(t))(e^{n}·)k^{p}_{H}γ
p

+k(∆ζ_{−n}u(t))(e^{n}·)k^{p}_{H}γ

p +k(ζ_{−nx}ux(t))(e^{n}·)k^{p}_{H}γ
p

m

dti
.
(Hereζ_{−nx}u_{x}is 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}_{H}γ,q

p,θ+p(G,T)≤NEhZ T 0

X

n∈Z

e^{n(θ+p)}k∆(ζ−nu(t))(e^{n}·)k^{p}_{H}γ
p

^{m}

+ku(t)k^{q}_{H}γ

p,θ−p(G)+kux(t)k^{q}_{H}γ
p,θ(G)dti

.

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

k∆uk^{q}_{H}γ,q

p,θ+p(G,T)≤NEhZ T 0

X

n∈Z

e^{n(θ+p)}k∆(ζ_{−n}u(t))(e^{n}·)k^{p}_{H}γ
p

^{m}
dti

+Nkuk^{q}

H^{γ+1,q}p,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))(e^{n}·)k^{p}_{H}γ
p

^{m}
dti

=EhZ T 0

X

n_{1},...,n_{m}∈Z

e ^{P}^{m}^{i=1}^{n}^{i}
(θ+p)

m

Y

i=1

k∆(ζ−niu(t))(e^{n}^{i}·)k^{p}_{H}γ
pdti

.

Tonelli’s theorem together with the relation
ku(c·)k^{p}_{H}γ

p =c^{pγ−d}k(c^{−2}−∆)^{γ/2}uk^{p}_{L}

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

X

n1,...,nm∈Z

e ^{P}^{m}^{i=1}^{n}^{i}

(θ+p+pγ−d)

EhZ T 0

m

Y

i=1

k(e^{−2n}^{i}−∆)^{γ/2}∆u^{(n}^{i}^{)}(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^{rs}vx^{r}x^{s}+f^{(n)})dt+ (σ^{rk}vx^{r}+g^{(n)k})dw_{t}^{k}, v(0,·) = 0,

onR^{d}^{, where}f^{(n)}=−2a^{rs}(ζ_{−n})x^{s}ux^{r}−a^{rs}(ζ_{−n})x^{r}x^{s}u+ζ_{−n}f andg^{(n)k}=−σ^{rk}(ζ_{−n})x^{r}u+

ζ_{−n}g^{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^{−2n}^{i}−∆)^{γ/2}∆u^{(n}^{i}^{)}(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_{i}f^{(n}^{i}^{)}(t)k^{p}_{L}

p

m

Y

j=1 j6=i

kΛn_{j}∆u^{(n}^{j}^{)}(t)k^{p}_{L}

pdti ,

II_{n}_{i} :=EhZ T
0

kΛnig^{(n}_{x}^{i}^{)}(t)k^{p}_{L}

p(`2) m

Y

j=1 j6=i

kΛnj∆u^{(n}^{j}^{)}(t)k^{p}_{L}_{p}dti
,

IIIn_{i}n_{j} :=EhZ T
0

kΛn_{i}g^{(n}_{x}^{i}^{)}(t)k^{p}_{L}

p(`_{2})kΛn_{j}g_{x}^{(n}^{j}^{)}(t)k^{p}_{L}

p(`_{2})
m

Y

k6=i,jk=1

kΛn_{k}∆u^{(n}^{k}^{)}(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 ^{P}^{m}^{i=1}^{n}^{i}

(θ+p+pγ−d)X^{m}

i=1

In_{i}+IIn_{i}

+ X

1≤i<j≤m

IIIn_{i}n_{j}

.

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

X

n1,...,nm∈Z

e ^{P}^{m}^{i=1}^{n}^{i}

(θ+p+pγ−d) m

X

i=1

In_{i}

= X

n1,...,nm∈Z

e ^{P}^{m}^{i=1}^{n}^{i}
(θ+p)

m

X

i=1

EhZ T 0

kf^{(n}^{i}^{)}(t, e^{n}^{i}·)k^{p}_{H}γ
p

m

Y

j=1 j6=i

k∆u^{(n}^{j}^{)}(t, e^{n}^{j}·)k^{p}_{H}γ
pdti

≤NEhZ T 0

X

n∈Z

e^{n(θ+p)}kf^{(n)}(t, e^{n}·)k^{p}_{H}γ
p

X

n∈Z

e^{n(θ+p)}k∆u^{(n)}(t, e^{n}·)k^{p}_{H}γ
p

m−1

dti

≤N(ε)EhZ T 0

X

n∈Z

e^{n(θ+p)}kf^{(n)}(t, e^{n}·)k^{p}_{H}γ
p

^{q}_{p}
dti

+εEhZ T 0

X

n∈Z

e^{n(θ+p)}k∆u^{(n)}(t, e^{n}·)k^{p}_{H}γ
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, e^{n}·)k^{p}_{H}γ

p ≤N

kux(t)k^{p}_{H}γ

p,θ(G)+ku(t)k^{p}_{H}γ

p,θ−p(G)+kf(t)k^{p}_{H}γ
p,θ+p(G)

≤N
ku(t)k^{p}

H_{p,θ−p}^{γ+1} (G)+kf(t)k^{p}_{H}γ
p,θ+p(G)

. Moreover,

X

n∈Z

e^{n(θ+p)}k∆u^{(n)}(t, e^{n}·)k^{p}_{H}γ

p ≤X

n∈Z

e^{n(θ+p)}k(∆ζ−nu(t))(e^{n}·)k^{p}_{H}γ
p

+X

n∈Z

e^{n(θ+p)}k(ζ−nxux(t))(e^{n}·)k^{p}_{H}γ
p

+X

n∈Z

e^{n(θ+p)}k(ζ−n∆u(t))(e^{n}·)k^{p}_{H}γ
p

≤N

ku(t)k^{p}_{H}γ

p,θ−p(G)+kux(t)k^{p}_{H}γ

p,θ(G)+k∆uk^{p}_{H}γ
p,θ+p(G)

≤N
ku(t)k^{p}

H^{γ+1}_{p,θ−p}(G)+k∆uk^{p}_{H}γ
p,θ+p(G)

.

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

X

n_{1},...,n_{m}∈Z

e ^{P}^{m}^{i=1}^{n}^{i}

(θ+p+pγ−d) m

X

i=1

In_{i}≤εk∆uk^{q}

H^{γ,q}p,θ+p(G,T)

+N(ε)
kfk^{q}_{H}γ,q

p,θ+p(G,T)+kuk^{q}

H^{γ+1,q}p,θ−p(G,T)

. Using similar arguments we obtain

X

n_{1},...,n_{m}∈Z

e ^{P}^{m}^{i=1}^{n}^{i}

(θ+p+pγ−d)X^{m}

i=1

IIn_{i}+ X

1≤i<j≤m

IIIn_{i}n_{j}

≤εk∆uk^{q}_{H}γ,q

p,θ+p(G,T)+N(ε)
kgk^{q}

H^{γ+1,q}_{p,θ} (G,T;`_{2})+kuk^{q}

H^{γ+1,q}p,θ−p(G,T)

, which finishes the proof.