1Introduction RaphaelKruse StigLarsson Optimalregularityforsemilinearstochasticpartialdifferentialequationswithmultiplicativenoise

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.17(2012), no. 65, 1–19.

ISSN:1083-6489 DOI:10.1214/EJP.v17-2240

Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise

Raphael Kruse

Stig Larsson

Abstract

This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic parabolic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions and certain linear growth bounds. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is elementary and makes use of existing results on the regularity of the solution, in particular, the Hölder continuity with a non-optimal exponent.

Keywords:SPDE; Hölder continuity; temporal and spatial regularity; multiplicative noise; Lip- schitz nonlinearities; linear growth bound.

AMS MSC 2010:35B65; 35R60; 60H15.

Submitted to EJP on October 10, 2011, final version accepted on July 14, 2012.

SupersedesarXiv:1109.6487.

1 Introduction

Consider the following semilinear stochastic partial differential equation (SPDE) dX(t) + [AX(t) +F(X(t))] dt=G(X(t)) dW(t), for0≤t≤T,

X(0) =X0, ^(1.1)

where the mild solution X takes values in a Hilbert space H. The linear operator A:D(A) ⊂ H → H is self-adjoint, positive definite with compact inverse and −A is the generator of an analytic semigroupE(t) = e^−tAonH. For example, let−Abe the Laplacian with homogeneous Dirichlet boundary conditions andH = L²(D)for some bounded domainD ⊂R^d with smooth boundary∂Dor a convex domain with polygonal boundary. The nonlinear operatorsF andGare assumed to be globally Lipschitz con- tinuous in the appropriate sense andW: [0, T]×Ω→U denotes a standard Q-Wiener process on a probability space(Ω,F,P)with values in some Hilbert spaceU.

∗Preprint http://math.uni-bielefeld.de/sfb701/files/preprints/sfb11031.pdf appeared in 2010.

Supported by CRC 701 ’Spectral Structures and Topological Methods in Mathematics’, DFG-IGK 1132

’Stochastics and Real World Models’, the Swedish Research Council (VR) and by the Swedish Foundation for Strategic Research (SSF) through GMMC, the Gothenburg Mathematical Modelling Centre.

†Bielefeld University, Germany. E-mail:[email protected]

‡Chalmers University of Technology and University of Gothenburg, Sweden. E-mail:[email protected]

Our aim is to study the spatial and temporal regularity properties of the unique mild solutionX: [0, T]×Ω→H. The spatial regularity is measured in terms of the domains H˙^r := D(A^r2), r ≥ 0, of fractional powers of the operator A. If−A is the Laplacian, these domains coincide with standard Sobolev spaces, for example, H˙¹ = H₀¹(D) or H˙²=H₀¹(D)∩H²(D)(c.f. [10, Th. 6.4] or [16, Ch. 3]). The regularity in time is expressed by the Hölder exponent.

Assuming only that the semigroupE(t)is analytic and thatFandGsatisfy appropri- ate global Lipschitz conditions onHone may show that (1.1) has a unique mild solution X and that for everyγ ∈[0,1)the solution X maps intoH˙^γ ⊂H and is ^γ₂-Hölder con- tinuous with respect to the norm E

k · k^p_H¹_p

, p ∈ [2,∞), see, for example, [6] and [7].

The border caseγ = 1is of special interest in numerical analysis. For example, if one is analyzing an approximation scheme based on a finite element method, the spatial regularity determines the order of convergence. Hence, a suboptimal regularity result leads to a suboptimal estimate of the order of convergence (c.f. [16]).

The border case can be handled by making the additional assumption on the semi- groupE(t) = e^−tAthat the generator is self-adjoint with compact inverse. Under this assumption the optimal regularity of stochastic convolutions of the form

W_A^Φ(t) = Z t

0

E(t−σ)Φ(σ) dW(σ),

is studied in [6, Prop. 6.18] and [3]. HereΦis a stochastically square integrable (p= 2) process with values in the set of Hilbert-Schmidt operators. If, forr≥0, the processΦis regular enough so that the processt7→A^r2Φ(t)is still stochastically square integrable, then the convolution is a stochastic process, which is square integrable with values in H˙^r+1. There exist some generalizations of this result, for instance, to Banach space valued integrands [5], to the casep >2[18], and to Lévy noise [2].

The recent paper [7] extends this type of higher regularity result to the nonlinear problem (1.1) by introducing an appropriate linear growth assumption for G on the spaceH˙^rfor some r ∈[0,1)(see (2.3) below). It is shown thatX maps intoH˙^r+γ for γ ∈ [0,1). The border case γ = 1is not included because no additional assumption is made on the analytic semigroup.

The purpose of the present paper is to fill this gap. We therefore assume that the semigroup is generated by a self-adjoint operator with compact inverse and we com- plement the global Lipschitz assumptions forF, GonH by a linear growth bound forG onH˙^r. Our main results are presented in Theorems 3.1, 4.1, and 4.2. The proofs are based on a very careful use of the smoothing property of the semigroup E(t) = e^−tA (see Lemma 3.2), and on the Hölder continuity ofX with a suboptimal exponent (see Lemmas 3.4 and 3.5).

Our regularity result for the mild solution of (1.1) coincides with the optimal regu- larity property of the stochastic convolution but with the restrictionr <1. In this sense we understand our result to be optimal.

Evolution equations of the form (1.1) are also studied by other authors. We refer to [6, 9, 14, 19] and the references therein. Further related results are [20], where conditions for spatialC^∞-regularity are given, and [17], which provides conditions for the existence of strong solutions to (1.1).

This paper consists of four additional sections. In the next section we give a more precise formulation of our assumptions. In Section 3 we are concerned with the spatial regularity of the mild solution. The proof is divided into several lemmas, which contain the key ideas of proof. The lemmas are also useful in the proof of the temporal Hölder continuity in Section 4. The proof of continuity in the border case requires an additional

argument in form of Lebesgue’s dominated convergence theorem. This technique is also developed in Section 4. The last section briefly reviews our results in the special case of additive noise and gives an example in which the spatial regularity results are indeed optimal.

2 Preliminaries

In this section we present the general form of the SPDE we are interested in. After introducing some notation we state our assumptions and cite the result on existence, uniqueness and regularity of a mild solution from [7].

By H we denote a separable Hilbert space (H,(·,·),k · k). Further, let A:D(A) ⊂ H → H be a densely defined, linear, self-adjoint, positive definite operator, which is not necessarily bounded but with compact inverse. Hence, there exists an increasing sequence of real numbers (λ_n)_n≥1 and an orthonormal basis (e_n)_n≥1 in H such that Ae_n=λ_ne_nand

0< λ1≤λ2≤. . .≤λn(→ ∞).

The domain ofAis characterized by D(A) =n

x∈H :

∞

X

n=1

λ²_n(x, e_n)²<∞o .

Thus, −A is the generator of an analytic semigroup of contractions, which is denoted byE(t) = e^−At.

By W: [0, T]×Ω → U we denote a Q-Wiener process with values in a separable Hilbert space(U,(·,·)U,k · kU). While our underlying probability space is(Ω,F, P), we assume that the Wiener process is adapted to a normal filtration(Ft)_t∈[0,T]withFt⊂ F for all t ∈ [0, T]. The covariance operator Q: U → U is linear, bounded, self-adjoint, positive semidefinite and trace-class, that is

Tr(Q) =

∞

X

m=1

(em, Qem)U <∞

for an arbitrary orthonormal basis(em)m∈NofU.

We study the regularity properties of a stochastic processX: [0, T]×Ω→H,T >0, which is the mild solution to the stochastic partial differential equation (1.1). Thus,X satisfies the equation

X(t) =E(t)X0− Z t

0

E(t−σ)F(X(σ)) dσ+ Z t

0

E(t−σ)G(X(σ)) dW(σ) (2.1) for all0≤t≤T.

In order to formulate our assumptions and main result we introduce the notion of fractional powers of the linear operatorA. For anyr∈Rthe operatorA^r2 is given by

A^r2x=

∞

X

n=1

λ

r

n2xnen

for all

x∈D(A^r2) =n x=

∞

X

n=1

xnen : (xn)n≥1⊂R^withkxk²_r:=kA^r2xk²=

∞

X

n=1

λ^r_nx²_n<∞o .

By definingH˙^r:=D(A^r2)together with the normkxkrforr∈R^,H˙^rbecomes a Hilbert space.

Instead of defining H˙^−r forr >0 as above one can also work with the dual space ( ˙H^r)⁰. But, as it is shown, for example in [8, Th. B.8], our spacesH˙^−rare isometrically isomorphic to( ˙H^r)⁰forr >0. Therefore, the results in our paper are independent of the wayH˙^−ris defined. We prefer to work with the spacesH˙^rforr∈R, since the identical spectral structure for allr∈Rallows for a simple extension of the operatorA^−r2 seen as a mapping fromH toH˙^rto a mapping fromH˙^−rtoH.

As usual [6, 12] we introduce the separable Hilbert space U0 := Q¹²(U) with the inner product(u0, v0)U₀ := (Q⁻¹²u0, Q⁻¹²v0)U withQ⁻¹² denoting the pseudoinverse. The diffusion operatorGmapsHintoL⁰₂, whereL⁰₂denotes the space of all Hilbert-Schmidt operatorsΦ :U0→H with norm

kΦk²_L0 2 :=

∞

X

m=1

kΦψmk².

Here (ψm)_m≥1 is an arbitrary orthonormal basis of U0 (for details see, for example, Proposition 2.3.4 in [12]). Further,L⁰_2,r denotes the set of all Hilbert-Schmidt operators Φ :U0→H˙^rtogether with the normkΦk_L0

2,r :=kA^r2Φk_L0 2.

Let r ∈ [0,1), p ∈ [2,∞) be given. As in [7, 13] we make the following additional assumptions.

Assumption 2.1. There exists a constantCsuch that kG(x)−G(y)k_L⁰

2 ≤Ckx−yk ∀x, y∈H (2.2) and we have thatG( ˙H^r)⊂L⁰_2,r and

kG(x)k_L0

2,r ≤C(1 +kxk_r) ∀x∈H˙^r. (2.3) Assumption 2.2. The nonlinearityF mapsH intoH˙^−1+r. Furthermore, there exists a constantCsuch that

kF(x)−F(y)k_−1+r≤Ckx−yk ∀x, y∈H. (2.4) Assumption 2.3. The initial valueX0: Ω→H˙^r+1is anF0-measurable random variable withE

kX0k^p_r+1

<∞.

Under the above conditions Theorem1in [7] states that for everyγ ∈[r, r+ 1)and T >0there exists an up to modification unique mild solutionX: [0, T]×Ω→H˙^γto (1.1) of the form (2.1), which satisfies

sup

t∈[0,T]

E

kX(t)k^p_γ

<∞.

Moreover, the solution process is continuous with respect to E k · k^p_γ

1

p and fulfills

sup

t₁,t₂∈[0,T],t16=t2

E[kX(t1)−X(t2)k^p_s]¹_p

|t1−t₂|^{min(12,γ−s2 )} <∞ for everys∈[0, γ].

The aim of this paper is to show that these results on the spatial regularity and the temporal Hölder continuity also hold withγ =r+ 1. Forr = 0we also prove that the solution process remains continuous with respect to the norm E

k · k^p₁¹_p .

Remarks 2.4. 1. Actually, Theorem1in [7] assumes that F:H →H is globally Lips- chitz, which is slightly stronger than Assumption 2.2. That Assumption 2.2 is sufficient can be proved by just following the given proof line by line and making the appropriate changes where everF comes into play.

2. The linear growth bound (2.3)follows from(2.2)whenr= 0.

3. Assumption 2.3 can be relaxed to X0: Ω → H being an F0-measurable random variable withE[kX0k^p] < ∞. But, as it is known from deterministic PDE theory, this will lead to a singularity att= 0.

4. The framework is quite general. More explicit examples and a detailed discussion of Assumption 2.1 can be found in [7]. We also refer to the discussion in [13] for further examples, references and a related result for temporal regularity.

3 Spatial regularity

In this section we deal with the spatial regularity of the mild solution. Our result is given by the following theorem. For a more convenient notation we setk · kL^p(Ω;H) :=

(E[k · k^p_H])^1p for any Hilbert spaceH. Also, if applied to an operator, the norm k · k is understood as the operator norm for bounded, linear operators fromH toH.

Theorem 3.1(Spatial regularity). Letr ∈[0,1),p∈ [2,∞). Given the assumptions of Section 2 the unique mild solutionX in (2.1)satisfies

P

X(t)∈/H˙^r+1

= 0

for allt ∈[0, T]. In particular, there exists a constantC > 0 depending onp, r, A, F, G,T and the Hölder continuity constant ofX with respect to the normk · kL^p(Ω;H)such that

sup

t∈[0,T]

E X(t)

p r+1

_p¹

≤ E X0

p r+1

¹_p

+ C

1 + sup

t∈[0,T]

E X(t)

p r

¹_p .

Before we prove the theorem we introduce several useful lemmas. The first states some well known facts on the semigroups(E(t))_t∈[0,T]. The parts(i),(ii) and(iv) hold true for analytic semigroups in general, while we use an orthonormal eigenbasis of the generator for the proof of(iii). For a proof of(i) and (ii) we refer to [11, Ch. 2.6, Th. 6.13]. Since parts(iii),(iv) are not readily found in the literature, we provide proofs here.

Lemma 3.2. Let the infinitesimal generator −Aof the semigroup(E(t))_[0,∞) be self- adjoint with compact inverse. Then the following properties hold true:

(i) For anyµ≥0there exists a constantC=C(µ)such that kA^µE(t)k ≤Ct^−µ fort >0.

(ii) For any0≤ν≤1there exists a constantC=C(ν)such that kA^−ν(E(t)−I)k ≤Ct^νfort≥0.

(iii) For any0≤ρ≤1there exists a constantC=C(ρ)such that Z τ2

τ₁

kA^ρ2E(τ₂−σ)xk²dσ≤C(τ₂−τ₁)^1−ρkxk² for allx∈H,0≤τ₁< τ₂. (iv) For any0≤ρ≤1there exists a constantC=C(ρ)such that

A^ρ

Z τ₂

τ₁

E(τ2−σ)xdσ

≤C(τ2−τ1)^1−ρkxkfor allx∈H,0≤τ1< τ2.

Proof. For the proof of(iii) we use the expansion ofx∈ H in terms of the eigenbasis (e_n)_n≥1of the operatorA. By Parseval’s identity we get

Z τ2

τ₁

A^ρ2E(τ₂−σ)x

2dσ= Z τ2

τ₁

∞

X

n=1

A^ρ2E(τ₂−σ)(x, e_n)e_n

2

dσ

=

∞

X

n=1

Z τ₂

τ1

(x, en)²λ^ρ_ne^{−2λn(τ2−σ)}dσ

=1 2

∞

X

n=1

(x, en)²λ^ρ−1_n 1−e^{−2λn(τ2−τ1)} .

By the boundedness of the functionx7→x^ρ−1(1−e^−x)forx∈[0,∞)andρ∈[0,1]there exists a constantC=C(ρ)>0such that

Z τ₂

τ₁

A^ρ2E(τ2−σ)x

2dσ≤C(ρ)(τ2−τ1)^1−ρ

∞

X

n=1

(x, en)²,

which completes the proof of(iii).

The following proof of(iv) also works for analytic semigroups in general. By [11, Ch. 1.2, Th. 2.4 (ii)] we first note that

A^ρ

Z τ2

τ1

E(τ₂−σ)xdσ =

A^ρ−1A

Z τ2−τ1

0

E(σ)xdσ

=

A^ρ−1 E(τ₂−τ₁)−I x

. Then,(iv) follows from(ii).

The next lemma is a special case of [6, Lem. 7.2] and will be needed to estimate stochastic integrals.

Lemma 3.3. For anyp≥2,0≤τ₁< τ₂≤T, and for anyL⁰₂-valued predictable process Φ(t),t∈[τ₁, τ₂], which satisfies

EhZ τ₂ τ1

Φ(σ)

2

L⁰₂dσ^p₂i

<∞, we have

Eh

Z τ2

τ₁

Φ(σ) dW(σ)

pi

≤C(p)EhZ τ2

τ₁

Φ(σ)

2

L⁰₂dσ^p₂i .

Here the constant can be chosen to be C(p) =p

2(p−1)^p₂ p p−1

p(^p₂−1)

.

The following two lemmas contain our main idea of proof and yield the key esti- mates. The applied technique is already known in the literature for Bochner integrals consisting of a convolution with an analytic semigroup, for example in [4, Prop. 3] and [15, p. 157].

Lemma 3.4. Lets∈[0, r+ 1],p≥2, andY be a predictable stochastic process on[0, T] which maps intoH˙^rwithsup_σ∈[0,T]kA^r2Y(σ)k_Lp(Ω;H)<∞. Then there exists a constant C=C(p, r, s, A, G)such that, for allτ₁, τ₂∈[0, T]withτ₁< τ₂,

EhZ τ₂ τ1

A^s2E(τ2−σ)G(Y(τ2))

2

L⁰₂dσ^p₂i¹_p

≤C

1 + sup

σ∈[0,T]

A^r2Y(σ) _Lp(Ω;H)

(τ₂−τ₁)^{min(12,1+r−s2 )}. (3.1)

If, in addition, for someδ > ^r₂ there existsC_δ such that

kY(t1)−Y(t2)k_Lp(Ω;H)≤Cδ|t2−t1|^δ for allt1, t2∈[0, T], then we also have, withC=C(p, s, G, Cδ), that

EhZ τ2

τ₁

A^s2E(τ₂−σ) G(Y(σ))−G(Y(τ₂))

2

L⁰₂ dσ^p₂i¹_p

≤ C

√1 + 2δ−s(τ2−τ1)^1+2δ−s2 . (3.2) In particular, withC=C(T, δ, p, r, s, A, G, Cδ)it holds that

Z τ2

τ1

A^s2E(τ2−σ)G(Y(σ)) dW(σ) _Lp(Ω;H)

≤C

1 + sup

σ∈[0,T]

A^r2Y(σ) _Lp(Ω;H)

(τ₂−τ₁)^{min(12,1+r−s2 )}. (3.3)

Proof. First note that, for0 ≤τ₁ < τ₂ ≤T fixed, the mapping[τ₁, τ₂]3 σ7→A^s2E(τ₂− σ)G(Y(σ)) is a predictable L⁰₂-valued process. Hence, Lemma 3.3 is applicable and gives

Z τ2

τ1

A^s2E(τ2−σ)G(Y(σ)) dW(σ) _Lp(Ω;H)

≤C(p)

Z τ2

τ₁

A^s2E(τ₂−σ)G(Y(σ))

2

L⁰₂ dσ¹_{2 Lp(Ω;}

R)

≤C(p)

Z τ₂

τ1

A^s2E(τ2−σ)G(Y(τ2))

2

L⁰₂ dσ¹_{2 Lp(Ω;}

R)

+C(p)

Z τ2

τ₁

A^s2E(τ₂−σ) G(Y(σ))−G(Y(τ₂))

2

L⁰₂ dσ¹_{2 Lp(Ω;}

R)

=:S1+S2.

In the second step we just used the triangle inequality. Now we deal with both sum- mands separately. In the first term S1 the time inG(Y(τ2)) is fixed. We also notice thatη :=s−r−max(0, s−r)≤0and, hence, A^η2 is a bounded linear operator onH. Furthermore, sinces∈[0, r+ 1]we haveρ:= max(0, s−r)∈[0,1]and Lemma 3.2(iii) is applicable. By writings=η+ρ+r, we get

Z τ2

τ₁

A^s2E(τ₂−σ)G(Y(τ₂))

2 L⁰₂ dσ

= Z τ₂

τ1

∞

X

m=1

A^s2E(τ2−σ)G(Y(τ2))ϕm

2 dσ

≤

∞

X

m=1

Z τ₂

τ₁

A^η2

2

A^ρ2E(τ2−σ)A^r2G(Y(τ2))ϕm

2dσ

≤C(s, r) A^η2

2

A^r2G(Y(τ₂))

2

L⁰₂(τ₂−τ₁)min(1,1+r−s),

where(ϕm)m≥1 denotes an orthonormal basis of U0. We also used that 1−ρ = 1− max(0, s−r) = min(1,1 +r−s). Finally, by Assumption 2.1 we conclude

S₁≤C(p, r, s, A, G)

1 + sup

σ∈[0,T]

A^r2Y(σ) _Lp(Ω;H)

(τ₂−τ₁)^{min(12,1+r−s2 )}.

This proves (3.1). ForS₂ we first make use of the fact thatkBΦk_L⁰

2 ≤ kBkkΦk_L⁰

2 and then apply Lemma 3.2(i)followed by (2.2) to get

S2≤C(p, s, G)

Z τ₂

τ₁

(τ2−σ)^−skY(σ)−Y(τ2)k²dσ¹_{2 Lp(Ω;}

R)

=C(p, s, G)

Z τ₂

τ1

(τ2−σ)^−skY(σ)−Y(τ2)k²dσ

L^p/2(Ω;R)

¹₂

≤C(p, s, G)Z τ2

τ₁

(τ₂−σ)^−skY(σ)−Y(τ₂)k²_Lp(Ω;H)dσ¹₂ .

By the Hölder continuity ofY we arrive at S2≤C(p, s, G, Cδ)Z τ₂

τ1

(τ2−σ)^−s+2δdσ¹₂

≤ C(p, s, G, C_δ)

√1 + 2δ−s (τ₂−τ₁)^1+2δ−s2 .

This shows (3.2). Combination of the estimates for S₁ and S₂ yields (3.3) by using (τ₂−τ₁)^2δ−r≤T^2δ−r.

Lemma 3.5. Lets∈[0, r+ 1],p≥2, andY be a stochastic process on[0, T]which maps intoH withsup_σ∈[0,T]kY(σ)k_Lp(Ω;H) <∞. Then there exists a constantC =C(r, s, F) such that, for allτ1, τ2∈[0, T]withτ1< τ2,

A^s2

Z τ2

τ1

E(τ2−σ)F(Y(τ2)) dσ _Lp(Ω;H)

≤C

1 + sup

σ∈[0,T]

kY(σ)k_Lp(Ω;H)

(τ₂−τ₁)^1+r−s2 . (3.4)

If, in addition, for someδ >0there existsCδ such that

kY(t₁)−Y(t₂)k_Lp(Ω;H)≤C_δ|t₂−t₁|^δ for allt₁, t₂∈[0, T], then we also have, withC=C(r, s, F, Cδ), that

A^s2

Z τ₂

τ1

E(τ2−σ) F(Y(τ2))−F(Y(σ)) dσ

_Lp(Ω;H)

≤ C

1 +r−s+ 2δ(τ₂−τ₁)^1+r−s+2δ2 . (3.5) In particular, withC=C(T, δ, r, s, F, Cδ)it holds that

A^s2

Z τ2

τ₁

E(τ₂−σ)F(Y(σ)) dσ _Lp(Ω;H)

≤C

1 + sup

σ∈[0,T]

kY(σ)k_Lp(Ω;H)

(τ2−τ1)^1+r−s2 . (3.6) Proof. As in the previous lemma the main idea is to use the Hölder continuity ofY to estimate the left-hand side in (3.6). We have

A^s2

Z τ2

τ1

E(τ2−σ)F(Y(σ)) dσ _Lp(Ω;H)

≤ A^s2

Z τ2

τ₁

E(τ₂−σ)F(Y(τ₂)) dσ _Lp(Ω;H)

+ A^s2

Z τ₂

τ₁

E(τ2−σ) F(Y(τ2))−F(Y(σ)) dσ

_Lp(Ω;H).

Therefore, if we show (3.4) and (3.5) then (3.6) follows immediately by using(τ₂−τ1)^δ≤ T^δ.

For (3.4) first note that the random variableA^−1+r2 F(X(τ2))takes values inH almost surely. Hence, we can apply Lemma 3.2(iv). Together with Assumption 2.2 this yields

A^2s

Z τ2

τ₁

E(τ₂−σ)F(Y(τ₂)) dσ _Lp(Ω;H)

≤ A^s+1−r2

Z τ₂

τ₁

E(τ2−σ)A^−1+r2 F(Y(τ2)) dσ _Lp(Ω;H)

≤C(r, s)(τ2−τ1)^1+r−s2

A^−1+r2 F(Y(τ2)) _Lp(Ω;H)

≤C(r, s, F)

1 + sup

σ∈[0,T]

kY(σ)k_Lp(Ω;H)

(τ₂−τ₁)^1+r−s2 .

Finally, again by Lemma 3.2 and Assumption 2.2, we show (3.5):

A^s2

Z τ₂

τ₁

E(τ2−σ) F(Y(τ2))−F(Y(σ)) dσ

_Lp(Ω;H)

≤ Z τ₂

τ1

A^s+1−r2 E(τ2−σ)A^−1+r2 (F(Y(τ2))−F(Y(σ)))

_Lp(Ω;H)dσ

≤C(r, s, F) Z τ2

τ1

(τ2−σ)^r−s−12 kY(τ2)−Y(σ)k_Lp(Ω;H) dσ

≤C(r, s, F, C_δ) Z τ₂

τ₁

(τ₂−σ)^{r−s−1+2δ2} dσ=2C(r, s, F, C_δ)

1 +r−s+ 2δ(τ₂−τ₁)^1+r−s+2δ2 . This completes the proof.

Now we are well prepared for the proof of Theorem 3.1.

Proof of Theorem 3.1. By taking norms in (2.1) we get, fort∈[0, T], E

kX(t)k^p_r+1¹_p

=kA^r+12 X(t)kL^p(Ω;H)

≤ kA^r+12 E(t)X0k_Lp(Ω;H)

+ A^r+12

Z t

0

E(t−σ)F(X(σ)) dσ _Lp(Ω;H)

+ A^r+12

Z t

0

E(t−σ)G(X(σ)) dW(σ) _Lp(Ω;H)

=:I+II+III.

The first term is well-known from deterministic theory and can be estimated by kA^r+12 E(t)X0k_Lp(Ω;H)≤ kA^r+12 X0k_Lp(Ω;H)<∞,

sinceX0: Ω→H˙^r+1by Assumption 2.3.

We recall that, by Theorem1in [7], the mild solutionXis anH˙^r-valued predictable stochastic process which isδ-Hölder continuous for any0 < δ < ¹₂ with respect to the normk · k_Lp(Ω;H). We choose δ := ^r+1₄ so that0 ≤ ^r₂ < δ < ¹₂. Hence, we can apply Lemmas 3.4 and 3.5 withY =X.

For the second term we apply (3.6) withτ1 = 0,τ2 =t, s=r+ 1andY =X. This yields

II≤C

1 + sup

σ∈[0,T]

kX(σ)k_Lp(Ω;H)

<∞.

For the last term we apply (3.3) with the same parameters as above:

III≤C

1 + sup

σ∈[0,T]

A^r2X(σ) _Lp(Ω;H)

<∞.

Note thatsup_σ∈[0,T]kX(σ)kL^p(Ω;H)≤ kA^−r2ksup_σ∈[0,T]kA^r2X(σ)kL^p(Ω;H)is finite because of Theorem1in [7].

4 Regularity in time

This section is devoted to the temporal regularity of the mild solution. Our main result is summarized in the following theorem. For the border cases=r+ 1we refer to Theorem 4.2 below.

Theorem 4.1(Temporal regularity). Letr ∈[0,1), p∈[2,∞). Under the assumptions of Section 2 the unique mild solutionX to (1.1)is Hölder continuous with respect to

E[k · k^p_s]¹_p

and satisfies

sup

t1,t2∈[0,T],t16=t2

E[kX(t₁)−X(t₂)k^p_s]¹_p

|t1−t₂|^{min(12,1+r−s2 )} <∞ (4.1) for everys∈[0, r+ 1).

Proof. Let0≤t1< t2≤T be arbitrary. By using the mild formulation (2.1) we get E[kX(t1)−X(t₂)k^p_s]¹_p

=

A^s2(X(t₁)−X(t₂)) L^p(Ω;H)

≤

A^s2(E(t₁)−E(t₂))X₀ L^p(Ω;H)

+ A^s2

Z t2

t1

E(t2−σ)F(X(σ)) dσ _Lp(Ω;H)

+ A^s2

Z t1

0

(E(t₂−σ)−E(t₁−σ))F(X(σ)) dσ _Lp(Ω;H)

+ A^s2

Z t₂

t1

E(t2−σ)G(X(σ)) dW(σ) _Lp(Ω;H)

+ A^s2

Z t1

0

(E(t2−σ)−E(t1−σ))G(X(σ)) dW(σ) _Lp(Ω;H)

=:T1+T2+T3+T4+T5. (4.2)

We estimate the five terms separately. The termT1is estimated by T₁=

A^s−r−12 (I−E(t₂−t₁))A^r+12 E(t₁)X_{0 Lp(Ω;H)}

≤C

A^r+12 X₀

_Lp(Ω;H)(t₂−t₁)^1+r−s2 , where we used Lemma 3.2(ii) and Assumption 2.3.

As in the proof of Theorem 3.1 we choose the Hölder exponent δ := ^r+1₄ so that

r

2 < δ <¹₂ and we can apply Lemmas 3.4 and 3.5 withY =X. The termT₂coincides with (3.6) and we have

T₂≤C

1 + sup

σ∈[0,T]

kX(σ)k_Lp(Ω;H)

(t₂−t₁)^1+r−s2 .

For the third term we also apply Lemma 3.2(ii) before we use (3.6):

T₃=

A^s−r−12 (E(t₂−t₁)−I)A^r+12 Z t1

0

E(t₁−σ)F(X(σ)) dσ _Lp(Ω;H)

≤C(t2−t1)^1+r−s2 A^r+12

Z t₁

0

E(t1−σ)F(X(σ)) dσ _Lp(Ω;H)

≤C

1 + sup

σ∈[0,T]

kX(σ)k_Lp(Ω;H)

(t2−t1)^1+r−s2 .

The fourth term is estimated analogously by using (3.3) instead of (3.6). We get T4≤C

1 + sup

σ∈[0,T]

A^r2X(σ) _Lp(Ω;H)

(t2−t1)^{min(12,1+r−s2 )}.

Finally, for the last term we use Lemma 3.3 first. Since, for 0 ≤ t1 < t2 ≤ T fixed, the function[0, t1]3σ7→A^s2(E(t2−σ)−E(t1−σ))G(X(σ))is a predictable stochastic process Lemma 3.3 can be applied. Then, by using Lemma 3.2(ii) withν = ^1+r−s₂ and Lemma 3.4 withs=r+ 1we get

T5≤C

Z t1

0

A^s−r−12 (E(t2−t1)−I)A^r+12 E(t1−σ)G(X(σ))

2

L⁰₂dσ¹_{2 Lp(Ω;}

R)

≤C(t2−t1)^1+r−s2

Z t₁

0

A^r+12 E(t1−σ)G(X(t1))

2

L⁰₂dσ¹_{2 Lp(Ω;}

R)

+

Z t1

0

A^r+12 E(t₁−σ) G(X(σ))−G(X(t₁))

2

L⁰₂dσ¹_{2 Lp(Ω;}

R)

≤C(t₂−t₁)^1+r−s2

1 + sup

σ∈[0,T]

A^r2X(σ) _Lp(Ω;H)

.

Altogether, this proves (4.1) and the Hölder continuity of X with respect to the norm kA^s2 · k_Lp(Ω;H)for alls∈[0, r+ 1).

The temporal regularity ofXwith respect to the norm E[k · k^p_r+1¹_p

is more involved.

For the caser= 0we can prove the following result.

Theorem 4.2. Letr= 0andp∈[2,∞). Under the assumptions of Section 2 the unique mild solutionX to(1.1)is continuous with respect to E[k · k^p₁]¹_p

.

Before we begin the proof we analyze the continuity properties of the semigroup in the deterministic context.

Lemma 4.3. Let0≤τ1< τ2≤T. Then we have (i)

τ₂−τlim1→0

Z τ₂

τ1

A¹²E(τ2−σ)x

2dσ= 0 for allx∈H,

(ii)

τ₂−τlim₁→0

A

Z τ₂

τ₁

E(τ2−σ)xdσ

= 0 for allx∈H.

Proof. As in the proof of Lemma 3.2 we use the orthogonal expansion of x ∈ H with respect to the eigenbasis(e_n)_n≥1of the operatorA. Thus, for(i)we get, as in the proof of Lemma 3.2(iii),

Z τ₂

τ₁

A¹²E(τ2−σ)x

2

dσ= 1 2

∞

X

n=1

(x, en)²

1−e^{2λn(τ2−τ1)} .

We apply Lebesgue’s dominated convergence theorem. Note that the sum is dominated by ¹₂kxk²for allτ₂−τ₁≥0. Moreover, for everyn≥1we have

τ₂−τlim1→0 1−e^{2λn(τ2−τ1)}

(x, en)²= 0.

Hence, Lebesgue’s theorem gives us (i). The same argument also yields the second case, since

A

Z τ₂

τ1

E(τ2−σ)xdσ

2

=

∞

X

n=1

(x, en)² 1− e^{λn(τ2−τ1)2} .

The proof is complete.

Proof of Theorem 4.2. We must show thatlim_t2−t1→0+kX(t₂)−X(t₁)k_Lp(Ω; ˙H¹)= 0with either t₁ or t₂ fixed. As already demonstrated in the proof of Lemma 4.3 we use Lebesgue’s dominated convergence theorem. Let0 ≤t1 < t2 ≤T. We consider again the termsTi,i= 1, . . . ,5, in (4.2) but now withs= 1.

ForT1continuity follows immediately: For almost everyω∈Ωwe get thatX0(ω)∈ H˙¹. Thus, for every fixedω∈Ωwith this property we have

t2−tlim1→0k(E(t2)−E(t1))A¹²X0(ω)k= 0 by the strong continuity of the semigroup. We also have that

k(E(t2)−E(t₁))A¹²X₀(ω)k ≤ kA¹²X₀(ω)k,

where the latter is an element ofL^p(Ω;R) as a function ofω ∈ Ωby Assumption 2.3.

Hence, Lebesgue’s theorem is applicable and yieldslim_t2−t1→0T₁= 0.

In order to treat the right and left limits simultaneously in the remaining terms, we compute the limits ast1→t3andt2→t3for fixed but arbitraryt3∈[t1, t2].

In the case ofT2we get T₂≤

A

Z t₂

t₁

E(t₂−σ)A⁻¹² F(X(σ))−F(X(t₂)) dσ

_Lp(Ω;H)

+ A

Z t₂

t₁

E(t2−σ)A⁻¹² F(X(t2))−F(X(t3)) dσ

_Lp(Ω;H)

+ A

Z t₂

t1

E(t2−σ)A⁻¹²F(X(t3)) dσ

_Lp(Ω;H).

(4.3)

Because of (3.5), where we can chooses=r+ 1 = 1andδ= ¹₄ >0, the limitt₂−t₁→0 of the first summand is0. For the second summand in (4.3) we apply Lemma 3.2 (iv) withρ= 1, and Assumption 2.2 withr= 0. Then we derive

A

Z t2

t₁

E(t₂−σ)A⁻¹² F(X(t₂))−F(X(t₃)) dσ

_Lp(Ω;H)

≤C

A⁻¹²(F(X(t2))−F(X(t3))) _Lp(Ω;H)

≤CkX(t₂)−X(t₃)k_Lp(Ω;H)

and the limitt₂→t₃of this term vanishes by (4.1) withs= 0.

For the last summand in (4.3) we again apply Lemma 3.2(iv) withρ= 1and obtain, for almost everyω∈Ω,

A

Z t₂

t₁

E(t2−σ)A⁻¹²F(X(t3, ω)) dσ

≤CkA⁻¹²F(X(t3, ω))k

≤C 1 +kX(t3, ω)k ,

which belongs toL^p(Ω;R)for allt₃∈[0, T]. By Lemma 4.3(ii) it also holds that

t1lim→t3 t2→t3

A

Z t₂

t₁

E(t2−σ)A⁻¹²F(X(t3, ω)) dσ = 0

for almost allω∈Ω. Then Lebesgue’s dominated convergence theorem yields that this term vanishes, which completes the proof forT2.

Next, we take care ofT₃, which is estimated by T3≤

A¹²

Z t1

0

E(t2−σ)−E(t1−σ)

F(X(σ))−F(X(t1)) dσ

_Lp(Ω;H)

+ A¹²

Z t₁

0

E(t2−σ)−E(t1−σ)

F(X(t1))−F(X(t3)) dσ

_Lp(Ω;H)

+ A¹²

Z t1

0

E(t₂−σ)−E(t₁−σ)

F(X(t₃)) dσ

_Lp(Ω;H).

(4.4)

For the first summand in (4.4) we get by Lemma 3.2(ii)

A¹²

Z t1

0

E(t₂−σ)−E(t₁−σ)

F(X(σ))−F(X(t₁)) dσ

_Lp(Ω;H)

≤ Z t₁

0

A^−η2 E(t2−t1)−I

A^1+η2 E(t1−σ) F(X(σ))−F(X(t1))

_Lp(Ω;H)dσ

≤C(t2−t1)^η2 Z t₁

0

(t1−σ)^−2+η2

A⁻¹² F(X(σ))−F(X(t1))

_Lp(Ω;H)dσ,

(4.5)

whereη ∈(0,2]. We continue the estimate by applying Assumption 2.2 and the Hölder continuity ofXwith exponent ¹₂ with respect to the normk · k_Lp(Ω;H)as it was shown in (4.1) withs= 0. This gives

A

Z t1

0

E(t₂−σ)−E(t₁−σ)

A⁻¹² F(X(σ))−F(X(t₁)) dσ

_Lp(Ω;H)

≤C(t2−t1)^η2 Z t₁

0

(t1−σ)^−2+η−12 dσ=C 2 1−ηt

1−η 2

1 (t2−t1)^η2. Therefore, in the limitt₂−t₁→0this term is zero as long asη∈(0,1).

For the second summand in (4.4) we apply Lemma 3.2(iv) withρ= 1and get

A¹²

Z t₁

0

E(t2−σ)−E(t1−σ)

F(X(t1)) dσ _Lp(Ω;H)

= A

Z t1

0

E(t₁−σ) E(t₂−t₁)−I

A⁻¹²F(X(t₁)) dσ _Lp(Ω;H)

≤C

E(t2−t1)−I

A⁻¹²F(X(t1)) _Lp(Ω;H)

≤C

E(t₂−t₁)−I

A⁻¹² F(X(t₁))−F(X(t₃) _Lp(Ω;H)

+C

E(t2−t1)−I

A⁻¹²F(X(t3))

_Lp(Ω;H). By Assumption 2.2 and (4.1) it holds true that

E(t2−t1)−I

A⁻¹² F(X(t1))−F(X(t3) _Lp(Ω;H)

≤C

X(t1)−X(t3)

_Lp(Ω;H)≤C|t1−t3|¹².