# Foundations of the Theory of Semilinear Stochastic Partial Differential Equations

Authors: Stefan Tappe
Journal: International Journal of Stochastic Analysis
 (2013)

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## Abstract

The goal of this review article is to provide a survey about the
foundations of semilinear stochastic partial differential equations. In particular,
we provide a detailed study of the concepts of strong, weak, and mild solutions,
establish their connections, and review a standard existence and uniqueness result.
The proof of the existence result is based on a slightly extended version
of the Banach fixed point theorem.


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## Body

## 1. Introduction

Semilinear stochastic partial differential equations (SPDEs) have a broad spectrum of applications including natural sciences and economics. The goal of this review article is to provide a survey on the foundations of SPDEs, which have been presented in the monographs [1–3]. It may be beneficial for students who are already aware about stochastic calculus in finite dimensions and who wish to have survey material accompanying the aforementioned references. In particular, we review the relevant results from functional analysis about unbounded operators in Hilbert spaces and strongly continuous semigroups.A large part of this paper is devoted to a detailed study of the concepts of strong, weak, and mild solutions to SPDEs, to establish their connections and to review and prove a standard existence and uniqueness result. The proof of the existence result is based on a slightly extended version of the Banach fixed point theorem.In the last part of this paper, we study invariant manifolds for weak solutions to SPDEs. This topic does not belong to the general theory of SPDEs, but it uses and demonstrates many of the results and techniques of the previous sections. It arises from the natural desire to express the solutions of SPDEs, which generally live in an infinite dimensional state space, by means of a finite dimensional state process and thus to ensure larger analytical tractability.This paper should also serve as an introductory study to the general theory of SPDEs, and it should enable the reader to learn about further topics and generalizations in this field. Possible further directions are the study of martingale solutions (see, e.g., [1, 3]), SPDEs with jumps (see, e.g., [4] for SPDEs driven by the Lévy processes and [5–8] for SPDEs driven by Poisson random measures), and support theorems as well as further invariance results for SPDEs; see, for example, [9, 10].The remainder of this paper is organized as follows: In Sections2 and 3, we review the required results from functional analysis. In particular, we collect the relevant material about unbounded operators and strongly continuous semigroups. In Section 4 we review stochastic processes in infinite dimension. In particular, we recall the definition of a trace class Wiener process and outline the construction of the Itô integral. In Section 5 we present the solution concepts for SPDEs and study their various connections. In Section 6 we review results about the regularity of stochastic convolution integrals, which is essential for the study of mild solutions to SPDEs. In Section 7 we review a standard existence and uniqueness result. Finally, in Section 8 we deal with invariant manifolds for weak solutions to SPDEs.


## 2. Unbounded Operators in the Hilbert Spaces

In this section, we review the relevant properties about unbounded operators. We will start with operators in Banach spaces and focus on operators in Hilbert spaces later on. The reader can find the proofs of the upcoming results in any textbook about functional analysis, such as [11] or [12].LetX and Y be Banach spaces. For a linear operator A:X⊃𝒟(A)→Y, defined on some subspace 𝒟(A) of X, we call 𝒟(A) the domain of A.Definition 1.
A linear operatorA:X⊃𝒟(A)→Y is called closed, if for every sequence (xn)n∈ℕ⊂𝒟(A), such that the limits x=limn→∞xn∈X and y=limn→∞Axn∈Y exist, one has x∈𝒟(A) and Ax=y.Definition 2.
A linear operatorA:X⊃𝒟(A)→Y is called densely defined, if its domain 𝒟(A) is dense in X; that is, 𝒟(A)¯=X.Definition 3.
LetA:X⊃𝒟(A)→X be a linear operator.  (1)
Theresolvent set of A is defined as
(1)ρ(A)≔{(λ-A)-1∈L(X)λ∈ℂ:λ-A:𝒟(A)⟶Xisbijectiveand22(λ-A)-1∈L(X)}.(2)
Thespectrum ofA is defined as σ(A)≔ℂ∖ρ(A).(3)
Forλ∈ρ(A), one defines the resolventR(λ,A)∈L(X) as
(2)R(λ,A)≔(λ-A)-1.
Now, we will introduce the adjoint operator of a densely defined operator in a Hilbert space. Recall that for a bounded linear operatorT∈L(H1,H2), mapping between two Hilbert spaces H1 and H2, the adjoint operator is the unique bounded linear operator T*∈L(H2,H1) such that
(3)〈Tx,y〉H2=〈x,T*y〉H1∀x∈H1,y∈H2.
In order to extend this definition to unbounded operators, one recalls the following extension result for linear operators.Proposition 4.
LetX be a normed space, let Y be a Banach space, let D⊂X be a dense subspace, and let Φ:D→Y be a continuous linear operator. Then there exists a unique continuous extension Φ^:X→Y, that is, a continuous linear operator with Φ^|D=Φ. Moreover, one has ∥Φ^∥=∥Φ∥.Now, letH be a Hilbert space. We recall the representation theorem of Fréchet-Riesz. In the sequel, the space H′ denotes the dual space of H.Theorem 5.
For everyx′∈H′, there exists a unique element x∈H with 〈x′,•〉=〈x,•〉. In addition, one has ∥x∥=∥x′∥.LetA:H⊃𝒟(A)→H be a densely defined operator. One defines the subspace
(4)𝒟(A*)≔{y∈H:x⟼〈Ax,y〉iscontinuouson𝒟(A)y∈H:x⟼〈Ax,y〉}.
Let y∈𝒟(A*) be arbitrary. By virtue of the extension result for linear operators (Proposition 4), the operator
(5)𝒟(A)⟼ℝ,x⟼〈Ax,y〉,
has a unique extension to a linear functional z′∈H′. By the representation theorem of Fréchet-Riesz (Theorem 5), there exists a unique element z∈H with 〈z′,•〉=〈z,•〉. This implies that
(6)〈Ax,y〉=〈x,z〉∀x∈𝒟(A).
Setting A*y≔z, this defines a linear operator A*:H⊃𝒟(A*)→H, and one has
(7)〈Ax,y〉=〈x,A*y〉∀x∈𝒟(A),y∈𝒟(A*).Definition 6.
The operatorA*:H⊃𝒟(A*)→H is called the adjoint operator of A.Proposition 7.
LetA:H⊃𝒟(A)→H be densely defined and closed. Then A* is densely defined and one has A=A**.Lemma 8.
LetH be a separable Hilbert space and let A:H⊃𝒟(A)→H be a closed operator. Then the domain (𝒟(A),∥·∥𝒟(A)) endowed with the graph norm
(8)∥x∥𝒟(A)=(∥x∥2+∥Ax∥2)1/2
is a separable Hilbert space, too.


## 3. Strongly Continuous Semigroups

In this section, we present the required results about strongly continuous semigroups. Concerning the proofs of the upcoming results, the reader is referred to any textbook about functional analysis, such as [11] or [12]. Throughout this section, let X be a Banach space.Definition 9.
Let(St)t≥0 be a family of continuous linear operators St:X→X, t≥0.  (1)
The family(St)t≥0 is a called a strongly continuous semigroup (or C0-semigroup), if the following conditions are satisfied:
(i)
S0=Id,(ii)
Ss+t=SsSt for all s,t≥0,(iii)
limt→0Stx=x for all x∈X.(2)
The family(St)t≥0 is called a norm continuous semigroup, if the following conditions are satisfied:
(i)
S0=Id,(ii)
Ss+t=SsSt for all s,t≥0,(iii)
limt→0∥St-Id∥=0.Note that every norm continuous semigroup is also aC0-semigroup. The following growth estimate (9) will often be used when dealing with SPDEs.Lemma 10.
Let(St)t≥0 be a C0-semigroup. Then there are constants M≥1 and ω∈ℝ such that
(9)∥St∥≤Meωt∀t≥0.Definition 11.
Let(St)t≥0 be a C0-semigroup.  (1)
The semigroup(St)t≥0 is called a semigroup of contractions (or contractive), if
(10)∥St∥≤1∀t≥0;
that is, the growth estimate (9) is satisfied with M=1 and ω=0.(2)
The semigroup(St)t≥0 is called a semigroup of pseudocontractions (or pseudocontractive), if there exists a constant ω∈ℝ such that
(11)∥St∥≤eωt∀t≥0;
that is, the growth estimate (9) is satisfied with M=1.If(St)t≥0 is a semigroup of pseudocontractions with growth estimate (11), then (Tt)t≥0 given by
(12)Tt≔e-ωtSt,t≥0,
is a semigroup of contractions. Hence, every pseudocontractive semigroup can be transformed into a semigroup of contractions, which explains the term pseudocontractive.Lemma 12.
Let(St)t≥0 be a C0-semigroup. Then the following statements are true.(1)
The mapping(13)ℝ+×X⟶X,(t,x)⟼Stx,
is continuous.(2)
For allx∈X and T≥0, the mapping
(14)[0,T]⟶X,t⟼Stx,
is uniformly continuous.Definition 13.
Let(St)t≥0 be a C0-semigroup. The infinitesimal generator (in short generator) of (St)t≥0 is the linear operator A:X⊃𝒟(A)→X, which is defined on the domain
(15)𝒟(A)≔{x∈X:limt→0Stx-xtexists}
and given by
(16)Ax≔limt→0Stx-xt.Note that the domain𝒟(A) is indeed a subspace of X. The following result gives some properties of the infinitesimal generator of a C0-semigroup. Recall that we have provided the required concepts in Definitions 1 and 2.Proposition 14.
The infinitesimal generatorA:X⊃𝒟(A)→X of a C0-semigroup (St)t≥0 is densely defined and closed.We proceed with some examples ofC0-semigroups and their generators.Example 15.
For every bounded linear operatorA∈L(X) the family (etA)t≥0 given by
(17)etA≔∑n=0∞‍tnAnn!
is a norm continuous semigroup with generator A. In particular, one has 𝒟(A)=X.Example 16.
We consider the separable Hilbert spaceX=L2(ℝ). Let (St)t≥0 be the shift semigroup that is defined as
(18)Stf≔f(t+•),t≥0.
Then (St)t≥0 is a semigroup of contractions with generator A:L2(ℝ)⊃𝒟(A)→L2(ℝ) given by
(19)𝒟(A)={f′∈L2(ℝ)f∈L2(ℝ):fisabsolutelycontinuousandf′∈L2(ℝ)},Af=f′.Example 17.
On the separable Hilbert spaceX=L2(ℝd) we define the heat semigroup(St)t≥0 by S0≔Id and
(20)(Stf)(x)≔1(4πt)d/2∫ℝd‍exp(-|x-y|24t)f(y)dy,2222222222222222222222222222222t>0;
that is, Stf arises as the convolution of f with the density of the normal distribution N(0,2t). Then (St)t≥0 is a semigroup of contractions with generator A:L2(ℝd)⊃𝒟(A)→L2(ℝd) given by
(21)𝒟(A)=W2(ℝd),Af=Δf.
Here, W2(ℝd) denotes the Sobolev space
(22)W2(ℝd)={ℕ0df∈L2(ℝd):D(α)f∈L2(ℝd)exists∀α∈ℕ0dwith|α|≤2}
and Δ the Laplace operator
(23)Δ=∑i=1d‍∂2∂xi2.We proceed with some results regarding calculations with strongly continuous semigroups and their generators.Lemma 18.
Let(St)t≥0 be a C0-semigroup with infinitesimal generator A. Then the following statements are true.(1)
For everyx∈𝒟(A), the mapping
(24)ℝ+⟶X,t⟼Stx,
belongs to class C1(ℝ+;X), and for all t≥0, one has Stx∈𝒟(A) and
(25)ddtStx=AStx=StAx.(2)
For allx∈X and t≥0, one has ∫0t‍Ssxds∈𝒟(A) and
(26)A(∫0t‍Ssxds)=Stx-x.(3)
For allx∈𝒟(A) and t≥0, one has
(27)∫0t‍SsAxds=Stx-x.The following result shows that the strongly continuous semigroup(St)t≥0 associated with generator A is unique. This explains the term generator.Proposition 19.
TwoC0-semigroups (St)t≥0 and (Tt)t≥0 with the same infinitesimal generator A coincide; that is, one has St=Tt for all t≥0.The next result characterizes all norm continuous semigroups in terms of their generators.Proposition 20.
Let(St)t≥0 be a C0-semigroup with infinitesimal generator A. Then the following statements are equivalent.(1)
The semigroup(St)t≥0is norm continuous.(2)
The operatorAis continuous.(3)
The domain ofAis given by𝒟(A)=X. If the previous conditions are satisfied, then one has St=etA for all t≥0.Now, we are interested in characterizing all linear operatorsA which are the infinitesimal generator of some strongly continuous semigroup (St)t≥0. The following theorem of Hille-Yosida gives a characterization in terms of the resolvent, which we have introduced in Definition 3.Theorem 21 (Hille-Yosida theorem).
LetA:X⊃𝒟(A)→X be a linear operator and let M≥1, ω∈ℝ be constants. Then the following statements are equivalent.  (1)
A is the generator of a C0-semigroup (St)t≥0 with growth estimate (9).(2)
A is densely defined and closed and one has (ω,∞)⊂ρ(A) and
(28)∥R(λ,A)n∥≤M(λ-ω)-n∀λ∈(ω,∞),n∈ℕ.In particular, we obtain the following characterization of the generators of semigroups of contractions.Corollary 22.
For a linear operatorA:X⊃𝒟(A)→X the following statements are equivalent.(1)
A is the generator of a semigroup (St)t≥0 of contractions.(2)
A is densely defined and closed, and one has (0,∞)⊂ρ(A) and
(29)∥R(λ,A)∥≤1λ∀λ∈(0,∞).Proposition 23.
Let(St)t≥0 be a C0-semigroup on X with generator A. Then the family (St|𝒟(A))t≥0 is a C0-semigroup on (𝒟(A),∥·∥𝒟(A)) with generator A:𝒟(A2)⊂𝒟(A)→𝒟(A2), where the domain is given by
(30)𝒟(A2)={x∈𝒟(A):Ax∈𝒟(A)}.Recall that we have introduced the adjoint operator for operators in the Hilbert spaces in Definition6.Proposition 24.
LetH be a Hilbert space and let (St)t≥0 be a C0-semigroup on H with generator A. Then the family of adjoint operators (St*)t≥0 is a C0-semigroup on H with generator A*.


## 4. Stochastic Processes in Infinite Dimension

In this section, we recall the required foundations about stochastic processes in infinite dimension. In particular, we recall the definition of a trace class Wiener process and outline the construction of the Itô integral.In the sequel,(Ω,ℱ,(ℱt)t≥0,ℙ) denotes a filtered probability space satisfying the usual conditions. Let ℍ be a separable Hilbert space and let Q∈L(ℍ) be a nuclear, self-adjoint, positive definite linear operator.Definition 25.
Aℍ-valued, adapted, continuous process W is called a Q-Wiener process, if the following conditions are satisfied.(i)
One hasW0=0.(ii)
The random variableWt-Ws and the σ-algebra ℱs are independent for all 0≤s≤t.(iii)
One hasWt-Ws~N(0,(t-s)Q) for all 0≤s≤t.In Definition25, the distribution N(0,(t-s)Q) is a Gaussian measure with mean 0 and covariance operator (t-s)Q; see, for example, [1, Section 2.3.2]. The operator Q is also called the covariance operator of the Wiener process W. As Q is a trace class operator, we also call W a trace class Wiener process.Now, letW be a Q-Wiener process. Then, there exist an orthonormal basis (ej)j∈ℕ of ℍ and a sequence (λj)j∈ℕ⊂(0,∞) with ∑j∈ℕ‍λj<∞ such that
(31)Qu=∑j∈ℕ‍λj〈u,ej〉ℍej,u∈ℍ.
Namely, the λj are the eigenvalues of Q, and each ej is an eigenvector corresponding to λj. The space ℍ0≔Q1/2(ℍ), equipped with the inner product
(32)〈u,v〉ℍ0≔〈Q-1/2u,Q-1/2v〉ℍ,
is another separable Hilbert space, and (λjej)j∈ℕ is an orthonormal basis. According to [1, Proposition 4.1], the sequence of stochastic processes (βj)j∈ℕ defined as
(33)βj≔1λj〈W,ej〉ℍ,j∈ℕ,
is a sequence of real-valued independent standard Wiener processes, and one has the expansion
(34)W=∑j∈ℕ‍λjβjej.
Now, let us briefly sketch the construction of the Itô integral with respect to the Wiener process W. Further details can be found in [1, 3]. We denote by L20(H)≔L2(ℍ0,H) the space of Hilbert-Schmidt operators from ℍ0 into H, endowed with the Hilbert-Schmidt norm
(35)∥Φ∥L20(H)≔(∑j∈ℕ‍λj∥Φej∥2)1/2,Φ∈L20(H),
which itself is a separable Hilbert space. The construction of the Itô integral is divided into three steps as follows.(1)
For everyL(ℍ,H)-valued simple process of the form(36)X=X01{0}+∑i=1n‍Xi1(ti,ti+1],
with 0=t1<⋯<tn+1=T and ℱti-measurable random variables Xi:Ω→L(ℍ,H) for i=1,…,n, we set
(37)∫0t‍XsdWs≔∑i=1n‍Xi(Wt∧ti+1-Wt∧ti).(2)
For every predictableL20(H)-valued process X satisfying
(38)𝔼[∫0T‍∥Xs∥L20(H)2ds]<∞,
we extend the Itô integral ∫0t‍XsdWs by an extension argument for linear operators. In particular, we obtain the Itô isometry as follows:
(39)𝔼[∥∫0T‍XsdWs∥2]=𝔼[∫0T‍∥Xs∥L20(H)2ds].(3)
By localization, we extend the Itô integral∫0t‍XsdWs for every predictable L20(H)-valued process X satisfying
(40)ℙ(∫0t‍∥Φs∥L20(H)2ds<∞)=1∀t≥0. The Itô integral (∫0t‍XsdWs)t≥0 is an H-valued, continuous, local martingale, and we have the series expansion
(41)∫0t‍XsdWs=∑j∈ℕ‍∫0t‍Xsjdβsj,t≥0,
where Xj≔λjXej for each j∈ℕ. An indispensable tool for stochastic calculus in infinite dimensions is Itô's formula, which we will recall here.Theorem 26 (Itô's formula).
LetE be another separable Hilbert space, let f∈Cb1,2,loc(ℝ+×H;E) be a function, and let X be an H-valued Itô process of the form
(42)Xt=X0+∫0t‍Ysds+∫0t‍ZsdWs,t≥0.
Then (f(t,Xt))t≥0 is an E-valued Itô process, and one has ℙ-almost surely
(43)f(t,Xt)=f(0,X0)+∫0t‍(12∑j∈ℕ‍Dsf(s,Xs)+Dxf(s,Xs)Ys222222222222+12∑j∈ℕ‍Dxxf(s,Xs)(Zsj,Zsj))ds+∫0t‍Dxf(s,Xs)ZsdWs,t≥0,
where one uses the notation Zj≔λjZej for each j∈ℕ.Proof.
This result is a consequence of [3, Theorem 2.9].


## 5. Solution Concepts for SPDEs

In this section, we present the concepts of strong, mild, and weak solutions to SPDEs and discuss their relations.LetH be a separable Hilbert space, and let (St)t≥0 be a C0-semigroup on H with infinitesimal generator A. Furthermore, let W be a trace class Wiener process on some separable Hilbert space ℍ. We consider the SPDE:
(44)dXt=(AXt+α(t,Xt))dt+σ(t,Xt)dWtX0=h0.
Here α:ℝ+×H→H and σ:ℝ+×H→L20(H) are measurable mappings.Definition 27.
Leth0:Ω→H be a ℱ0-measurable random variable, and let τ>0 be a strictly positive stopping time. Furthermore, let X=X(h0) be an H-valued, continuous, adapted process such that
(45)ℙ(∫0t∧τ‍(∥Xs∥+∥α(s,Xs)∥+∥σ(s,Xs)∥L20(H)2)ds<∞)=1∀t≥0.(1)
X is called a local strong solution to (44), if
(46)Xt∧τ∈𝒟(A)∀t≥0,ℙ-almostsurely,(47)ℙ(∫0t∧τ‍∥AXs∥ds<∞)=1∀t≥0,
and ℙ-almost surely one has
(48)Xt∧τ=h0+∫0t∧τ‍(AXs+α(s,Xs))ds+∫0t∧τ‍σ(s,Xs)dWs,t≥0.(2)
X is called a local weak solution to (44), if for all ζ∈𝒟(A*) the following equation is fulfilled ℙ-almost surely:
(49)〈ζ,Xt∧τ〉=〈ζ,h0〉+∫0t∧τ‍(〈A*ζ,Xs〉+〈ζ,α(s,Xs)〉)ds+∫0t∧τ‍〈ζ,σ(s,Xs)〉dWs,t≥0.(3)
X is called a local mild solution to (44), if ℙ-almost surely one has
(50)Xt∧τ=St∧τh0+∫0t∧τ‍S(t∧τ)-sα(s,Xs)ds+∫0t∧τ‍S(t∧τ)-sσ(s,Xs)dWs,t≥0.One calls τ the lifetime of X. If τ≡∞, then one calls X a strong, weak or mild solution to (44), respectively.Remark 28.
Note that the concept of a strong solution is rather restrictive, because condition (46) has to be fulfilled.For what follows, we fix aℱ0-measurable random variable h0:Ω→H and a strictly positive stopping time τ>0.Proposition 29.
Every local strong solutionX to (44) with lifetime τ is also a local weak solution to (44) with lifetime τ.Proof.
LetX be a local strong solution to (44) with lifetime τ. Furthermore, let ζ∈𝒟(A*) be arbitrary. Then we have ℙ-almost surely for all t≥0 the identities
(51)〈ζ,Xt∧τ〉=〈ζ,h0+∫0t∧τ‍(AXs+α(s,Xs))ds+∫0t∧τ‍σ(s,Xs)dWs〉=〈ζ,h0〉+∫0t∧τ‍〈ζ,AXs+α(s,Xs)〉ds+∫0t∧τ‍〈ζ,σ(s,Xs)〉dWs=〈ζ,h0〉+∫0t∧τ‍(〈A*ζ,Xs〉+〈ζ,α(s,Xs)〉)ds+∫0t∧τ‍〈ζ,σ(s,Xs)〉dWs,
showing that X is also a local weak solution to (44) with lifetime τ.Proposition 30.
LetX be a stochastic process with X0=h0. Then the following statements are equivalent.(1)
The processXis a local strong solution to (44) with lifetimeτ.(2)
The processXis a local weak solution to (44) with lifetimeτ, and one has (46), (47).Proof.
(1)⇒(2): This implication is a direct consequence of Proposition 29.
(2)⇒(1): Let ζ∈𝒟(A*) be arbitrary. Then we have ℙ-almost surely for all t≥0 the identities
(52)〈ζ,Xt∧τ〉=〈ζ,h0〉+∫0t∧τ‍(〈A*ζ,Xs〉+〈ζ,α(s,Xs)〉)ds+∫0t∧τ‍〈ζ,σ(s,Xs)〉dWs=〈ζ,h0〉+∫0t∧τ‍〈ζ,AXs+α(s,Xs)〉ds+∫0t∧τ‍〈ζ,σ(s,Xs)〉dWs=〈ζ,h0+∫0t∧τ‍(AXs+α(s,Xs))ds+∫0t∧τ‍σ(s,Xs)dWs〉.
By Proposition 7, the domain 𝒟(A*) is dense in H, and hence we obtain ℙ-almost surely
(53)Xt∧τ=h0+∫0t∧τ‍(AXs+α(s,Xs))ds+∫0t∧τ‍σ(s,Xs)dWs,t≥0.
Consequently, the process X is also a local strong solution to (44) with lifetime τ.Corollary 31.
Letℳ⊂𝒟(A) be a subset such that A is continuous on ℳ, and let X be a local weak solution to (44) with lifetime τ such that
(54)Xt∧τ∈ℳ∀t≥0,ℙ-almostsurely.
Then X is also a local strong solution to (44) with lifetime τ.Proof.
Sinceℳ⊂𝒟(A), condition (54) implies that (46) is fulfilled. Moreover, by the continuity of A on ℳ, the sample paths of the process AX are ℙ-almost surely continuous; and hence, we obtain (47). Consequently, using Proposition 30, the process X is also a local strong solution to (44) with lifetime τ.Proposition 32.
Every strong solutionX to (44) is also a mild solution to (44).Proof.
According to Lemma8, the domain (𝒟(A),∥·∥𝒟(A)) endowed with the graph norm is a separable Hilbert space, too. Hence, by Lemma 18, for all t≥0, the function
(55)f:[0,t]×𝒟(A)⟶H,f(s,x)≔St-sx,
belongs to the class Cb1,2,loc([0,t]×𝒟(A);H) with partial derivatives
(56)Dtf(t,x)=-ASt-sx,Dxf(t,x)=St-s,Dxxf(t,x)=0.
Hence, by Itô's formula (see Theorem 26) and Lemma 18, we obtain ℙ-almost surely:
(57)Xt=f(t,Xt)=f(0,h0)+∫0t‍(Dxf(s,Xs)(AXs+α(s,Xs))Dsf(s,Xs)+Dxf(s,Xs)(AXs+α(s,Xs)))ds+∫0t‍Dxf(s,Xs)σ(s,Xs)dWs=Sth0+∫0t‍(-ASt-sXs+St-s(AXs+α(s,Xs)))ds+∫0t‍St-sσ(s,Xs)dWs=Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs.
Thus, X is also a mild solution to (44).We recall the following technical auxiliary result without proof and refer, for example, to [3, Section 3.1].Lemma 33.
LetT≥0 be arbitrary. Then the linear space
(58)UT≔lin{gζ:g∈C1([0,T];ℝ),ζ∈𝒟(A*)}
is dense in C1([0,T],𝒟(A*)), where (𝒟(A*),∥·∥𝒟(A*)) is endowed with the graph norm.Lemma 34.
LetX be a weak solution to (44). Then for all T≥0 and all f∈C1([0,T],𝒟(A*)), one has ℙ-almost surely
(59)〈f(t),Xt〉=〈f(0),h0〉+∫0t‍(〈f′(s)+A*f(s),Xs〉+〈f(s),α(s,Xs)〉)ds+∫0t‍〈f(s),σ(s,Xs)〉dWs,t∈[0,T].Proof.
By virtue of Lemma33, it suffices to prove formula (59) for all f∈UT. Let f∈UT be arbitrary. Then there are g1,…,gn∈C1([0,T];ℝ) and ζ1,…,ζn∈𝒟(A*) for some n∈ℕ such that
(60)f(t)=∑i=1n‍gi(t)ζi,t∈[0,T].
We define the function
(61)F:[0,T]×ℝn⟶ℝ,F(t,x)≔∑i=1n‍gi(t)xi.
Then, we have F∈C1,2([0,T]×ℝn;ℝ) with partial derivatives
(62)DtF(t,x)=∑i=1n‍gi′(t)xi,DxF(t,x)=〈g(t),•〉ℝn,DxxF(t,x)=0.
Since X is a weak solution to (44), the ℝn-valued process
(63)〈ζ,X〉≔〈ζi,X〉i=1,…,n,
is an Itô process with representation
(64)〈ζ,Xt〉=〈ζ,h0〉+∫0t‍(〈A*ζ,Xs〉+〈ζ,α(s,Xs)〉)ds+∫0t‍〈ζ,σ(s,Xs)〉dWs,t≥0.
By Itô's formula (Theorem 26), we obtain ℙ-almost surely
(65)〈f(t),Xt〉=〈∑i=1n‍gi(t)ζi,Xt〉=∑i=1n‍gi(t)〈ζi,Xt〉=F(t,〈ζ,Xt〉)=F(0,〈ζ,h0〉)+∫0t‍((〈A*ζ,Xs〉+〈ζ,α(s,Xs)〉)DsF(s,〈ζ,Xs〉)+DxF(s,〈ζ,Xs〉)22222×(〈A*ζ,Xs〉+〈ζ,α(s,Xs)〉))ds+∫0t‍DxF(s,〈ζ,Xs〉)〈ζ,σ(s,Xs)〉dWs=∑i=1n‍gi(0)〈ζi,h0〉+∫0t‍(∑i=1n‍gi′(t)〈ζi,Xs〉+∑i=1n‍gi(t)(〈A*ζi,Xs〉2222222+〈ζi,α(s,Xs)〉)∑i=1n‍)ds+∫0t‍(∑i=1n‍gi(s)〈ζi,σ(s,Xs)〉)dWs,22222222222222222222222t∈[0,T],
and hence
(66)〈f(t),Xt〉=〈∑i=1n‍gi(0)ζi,h0〉+∫0t‍(〈∑i=1n‍gi′(s)ζi,Xs〉+〈A*(∑i=1n‍gi(s)ζi),Xs〉+〈∑i=1n‍gi(s)ζi,α(s,Xs)〉)ds+∫0t‍〈∑i=1n‍gi(s)ζi,σ(s,Xs)〉dWs=〈f(0),h0〉+∫0t‍(〈f′(s)+A*f(s),Xs〉2222222222222222+〈f(s),α(s,Xs)〉〈f′(s)+A*f(s),Xs〉)ds+∫0t‍〈f(s),σ(s,Xs)〉dWs,t∈[0,T].
This concludes the proof.Proposition 35.
Every weak solutionX to (44) is also a mild solution to (44).Proof.
By Proposition24, the family (St*)t≥0 is a C0-semigroup with generator A*. Thus, Proposition 23 yields that the family of restrictions (St*|𝒟(A*))t≥0 is a C0-semigroup on (𝒟(A*),∥·∥𝒟(A*)) with generator A*:𝒟((A*)2)⊂𝒟(A*)→𝒟((A*)2).
Now, lett≥0 and ζ∈𝒟((A*)2) be arbitrary. We define the function
(67)f:[0,t]⟶𝒟(A*),f(s)≔St-s*ζ.
By Lemma 18 we have f∈C1([0,t];𝒟(A*)) with derivative
(68)f′(s)=-A*St-s*ζ=-A*f(s).
Using Lemma 34, we obtain ℙ-almost surely
(69)〈ζ,Xt〉=〈f(t),Xt〉=〈f(0),h0〉+∫0t‍〈f(s),α(s,Xs)〉ds+∫0t‍〈f(s),σ(s,Xs)〉dWs=〈St*ζ,h0〉+∫0t‍〈St-s*ζ,α(s,Xs)〉ds+∫0t‍〈St-s*ζ,σ(s,Xs)〉dWs=〈ζ,Sth0〉+∫0t‍〈ζ,St-sα(s,Xs)〉ds+∫0t‍〈ζ,St-sσ(s,Xs)〉dWs=〈ζ,Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs〉.
Since, by Proposition 14, the domain 𝒟((A*)2) is dense in (𝒟(A*),∥·∥𝒟(A*)), we get ℙ-almost surely for all ζ∈𝒟(A*) the identity
(70)〈ζ,Xt〉=〈ζ,Sth0+∫0t‍St-sα(s,Xs)ds22+∫0t‍St-sσ(s,Xs)dWs〉.
Since, by Proposition 14, the domain 𝒟(A*) is dense in H, we obtain ℙ-almost surely
(71)Xt=Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs,
proving that X is a mild solution to (44).Remark 36.
Now, the proof of Proposition32 is an immediate consequence of Propositions 29 and 35.We have just seen that every weak solution to (44) is also a mild solution. Under the following regularity condition (72), the converse of this statement holds true as well.Proposition 37.
LetX be a mild solution to (44) such that
(72)𝔼[∫0T‍∥σ(s,Xs)∥L20(H)2ds]<∞∀T≥0.
Then X is also a weak solution to (44).Proof.
Lett≥0 and ζ∈𝒟(A*) be arbitrary. Using Lemma 18, we obtain ℙ-almost surely
(73)∫0t‍〈A*ζ,Ssh0〉ds=〈A*ζ,∫0t‍Ssh0ds︸∈𝒟(A)〉=〈ζ,A(∫0t‍Ssh0ds)〉=〈ζ,Sth0-h0〉=〈ζ,Sth0〉-〈ζ,h0〉.
By Fubini's theorem for Bochner integrals (see [3, Section 1.1, page 21]) and Lemma 18. we obtain ℙ-almost surely
(74)∫0t‍〈A*ζ,∫0s‍Ss-uα(u,Xu)du〉ds=〈A*ζ,∫0t‍(∫0s‍Ss-uα(u,Xu)du)ds〉=〈A*ζ,∫0t‍(∫ut‍Ss-uα(u,Xu)ds)du〉=∫0t‍〈A*ζ,∫ut‍Ss-uα(u,Xu)ds〉du=∫0t‍〈A*ζ,∫0t-s‍Suα(s,Xs)du︸∈𝒟(A)〉ds=∫0t‍〈ζ,A(∫0t-s‍Suα(s,Xs)du)〉ds=∫0t‍〈ζ,St-sα(s,Xs)-α(s,Xs)〉ds=〈ζ,∫0t‍St-sα(s,Xs)ds〉-∫0t‍〈ζ,α(s,Xs)〉ds.
Due to assumption (72), we may use Fubini's theorem for stochastic integrals (see [3, Theorem 2.8]), which, together with Lemma 18, gives us ℙ-almost surely
(75)∫0t‍〈A*ζ,∫0s‍Ss-uσ(u,Xu)dWu〉ds=〈A*ζ,∫0t‍(∫0s‍Ss-uσ(u,Xu)dWu)ds〉=〈A*ζ,∫0t‍(∫ut‍Ss-uσ(u,Xu)ds)dWu〉=∫0t‍〈A*ζ,∫ut‍Ss-uσ(u,Xu)ds〉dWu=∫0t‍〈A*ζ,∫0t-s‍Suσ(s,Xs)du︸∈𝒟(A)〉dWs=∫0t‍〈ζ,A(∫0t-s‍Suσ(s,Xs)du)〉dWs=∫0t‍〈ζ,St-sσ(s,Xs)-σ(s,Xs)〉dWs=〈ζ,∫0t‍St-sσ(s,Xs)dWs〉-∫0t‍〈ζ,σ(s,Xs)〉dWs.Therefore, and since X is a mild solution to (44), we obtain ℙ-almost surely
(76)〈ζ,Xt〉=〈ζ,Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs〉=〈ζ,Sth0〉+〈ζ,∫0t‍St-sα(s,Xs)ds〉+〈ζ,∫0t‍St-sσ(s,Xs)dWs〉=〈ζ,h0〉+∫0t‍〈A*ζ,Ssh0〉ds+∫0t‍〈A*ζ,∫0s‍Ss-uα(u,Xu)du〉ds+∫0t‍〈ζ,α(s,Xs)〉ds+∫0t‍〈A*ζ,∫0s‍Ss-uσ(u,Xu)dWu〉ds+∫0t‍〈ζ,σ(s,Xs)〉dWs,
and hence(77)〈ζ,Xt〉=〈ζ,h0〉+∫0t‍〈A*ζ,Ssh0+∫0s‍Ss-uα(u,Xu)du+∫0s‍Ss-uσ(u,Xu)dWu︸=Xs〉ds+∫0t‍〈ζ,α(s,Xs)〉ds+∫0t‍〈ζ,σ(s,Xs)〉dWs=〈ζ,h0〉+∫0t‍(〈A*ζ,Xs〉+〈ζ,α(s,Xs)〉)ds+∫0t‍〈ζ,σ(s,Xs)〉dWs.Consequently, the process X is also a weak solution to (44).Next, we provide conditions which ensure that a mild solution to (44) is also a strong solution.Proposition 38.
LetX be a mild solution to (44) such that ℙ-almost surely one has
(78)Xs,α(s,Xs)∈𝒟(A),σ(s,Xs)∈L20(𝒟(A))∀s≥0,
as well as
(79)ℙ(∫0t‍(∥Xs∥𝒟(A)+∥α(s,Xs)∥𝒟(A))ds<∞)=1∀t≥0,(80)𝔼[∫0T‍∥σ(s,Xs)∥L20(𝒟(A))2ds]<∞∀T≥0.
Then X is also a strong solution to (44).Proof.
By hypotheses (78) and (79), we have (46) and (47). Let t≥0 be arbitrary. By Lemma 18, we have
(81)Sth0-h0=∫0t‍ASsh0ds.
Furthermore, by Lemma 18 and Fubini’s theorem for Bochner integrals (see [3, Section 1.1, page 21]) we have ℙ-almost surely
(82)∫0t‍(St-sα(s,Xs)-α(s,Xs))ds=∫0t‍(∫0t-s‍ASuα(s,Xs)du)ds=∫0t‍(∫ut‍ASs-uα(u,Xu)ds)du=∫0t‍(∫0s‍ASs-uα(u,Xu)du)ds=∫0t‍A(∫0s‍Ss-uα(u,Xu)du)ds.
Due to assumption (80), we may use Fubini’s theorem for stochastic integrals (see [3, Theorem 2.8]), which, together with Lemma 18, gives us ℙ-almost surely
(83)∫0t‍(St-sσ(s,Xs)-σ(s,Xs))dWs=∫0t‍(∫0t-s‍ASuσ(s,Xs)du)dWs=∫0t‍(∫ut‍ASs-uσ(u,Xu)ds)dWu=∫0t‍(∫0s‍ASs-uσ(u,Xu)dWu)ds=∫0t‍A(∫0s‍Ss-uσ(u,Xu)dWu)ds.
Since X is a mild solution to (44), we have ℙ-almost surely
(84)Xt=Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs=h0+∫0t‍α(s,Xs)ds+∫0t‍σ(s,Xs)dWs+(Sth0-h0)+∫0t‍(St-sα(s,Xs)-α(s,Xs))ds+∫0t‍(St-sσ(s,Xs)-σ(s,Xs))dWs,
and, hence, combining the latter identities, we obtain ℙ-almost surely
(85)Xt=h0+∫0t‍α(s,Xs)ds+∫0t‍σ(s,Xs)dWs+∫0t‍ASsh0ds+∫0t‍A(∫0s‍Ss-uα(u,Xu)du)ds+∫0t‍A(∫0s‍Ss-uσ(u,Xu)dWu)ds,
which implies that(86)Xt=h0+∫0t‍α(s,Xs)ds+∫0t‍σ(s,Xs)dWs+∫0t‍A(Ssh0+∫0s‍Ss-uα(u,Xu)du+∫0s‍Ss-uσ(u,Xu)dWu)︸=Xsds=h0+∫0t‍(AXs+α(s,Xs))ds+∫0t‍σ(s,Xs)dWs.This proves that X is also a strong solution to (44).The following result shows that for norm continuous semigroups, the concepts of strong, weak, and mild solutions are equivalent. In particular, this applies for finite dimensional state spaces.Proposition 39.
Suppose that the semigroup(St)t≥0 is norm continuous. Let X be a stochastic process with X0=h0. Then the following statements are equivalent.(1)
The processXis a strong solution to (44).(2)
The processXis a weak solution to (44).(3)
The processXis a mild solution to (44).Proof.
(1)⇒(2): This implication is a consequence of Proposition 29.
(2)⇒(3): This implication is a consequence of Proposition 35.
(3)⇒(1): By Proposition 20, we have A∈L(H) and St=etA, t≥0. Furthermore, the family (etA)t∈ℝ is a C0-group on H. Therefore, and since X is a mild solution to (44), we have ℙ-almost surely
(87)Xt=etAh0+∫0t‍e(t-s)Aα(s,Xs)ds+∫0t‍e(t-s)Aσ(s,Xs)dWs=etAh0+etA∫0t‍e-sAα(s,Xs)ds+etA∫0t‍e-sAσ(s,Xs)dWs,t≥0.
Let Y be the Itô process:
(88)Yt≔∫0t‍e-sAα(s,Xs)ds+∫0t‍e-sAσ(s,Xs)dWs,t≥0.
Then, we have ℙ-almost surely
(89)Xt=etA(h0+Yt),t≥0,
and, by Lemma 18, we have
(90)etAh0-h0=∫0t‍AesAh0ds.
Defining the function
(91)f:ℝ+×H⟶H,f(s,y)≔esAy,
by Lemma 18, we have f∈Cb1,2,loc(ℝ+×H;H) with partial derivatives
(92)Dsf(s,y)=AesAy,Dyf(s,y)=esA,Dyyf(s,y)=0.
By Itô's formula (Theorem 26), we get ℙ-almost surely
(93)etAYt=f(t,Yt)=f(0,0)+∫0t‍(Dsf(s,Ys)+Dyf(s,Ys)e-sAα(s,Xs))ds+∫0t‍Dyf(s,Ys)e-sAσ(s,Xs)dWs=∫0t‍(AesAYs+α(s,Xs))ds+∫0t‍σ(s,Xs)dWs.
Combining the previous identities, we obtain ℙ-almost surely
(94)Xt=etA(h0+Yt)=h0+(etAh0-h0)+etAYt=h0+∫0t‍AesAh0ds+∫0t‍(AesAYs+α(s,Xs))ds+∫0t‍σ(s,Xs)dWs=h0+∫0t‍(AesA(h0+Ys)︸=Xs+α(s,Xs))ds+∫0t‍σ(s,Xs)dWs=h0+∫0t‍(AXs+α(s,Xs))ds+∫0t‍σ(s,Xs)dWs,222222222222222222222222222222222222t≥0,
proving that X is a strong solution to (44).


## 6. Stochastic Convolution Integrals

In this section, we deal with the regularity of stochastic convolution integrals, which occur when dealing with mild solutions to SPDEs of the type (44).LetE be a separable Banach space, and let (St)t≥0 be a C0-semigroup on E. We start with the drift term.Lemma 40.
Letf:ℝ+→E be a measurable mapping such that
(95)∫0t‍∥f(s)∥ds<∞∀t≥0.
Then the mapping
(96)F:ℝ+⟶E,F(t)≔∫0t‍St-sf(s)ds,
is continuous.Proof.
Lett∈ℝ+ be arbitrary. It suffices to prove that F is right-continuous and left-continuous in t.  (1)
Let(tn)n∈ℕ⊂ℝ+ be a sequence such that tn↓t. Then for every n∈ℕ we have
(97)∥F(t)-F(tn)∥=∥∫0t‍St-sf(s)ds-∫0tn‍Stn-sf(s)ds∥=∥∫0t‍St-sf(s)ds-∫0t‍Stn-sf(s)ds-∫ttn‍Stn-sf(s)ds∥≤∫0t∥St-sf(s)-Stn-sf(s)∥ds+∫ttn∥Stn-sf(s)∥ds.
By Lemma 12, the mapping
(98)ℝ+×E⟶E,(u,x)⟼Sux,
is continuous. Thus, taking into account estimate (9) from Lemma 10, by Lebesgue's dominated convergence theorem we obtain
(99)∥F(t)-F(tn)∥⟶0forn⟶∞.(2)
Let(tn)n∈ℕ⊂ℝ+ be a sequence such that tn↑t. Then for every n∈ℕ we have
(100)∥F(t)-F(tn)∥=∥∫0t‍St-sf(s)ds-∫0tn‍Stn-sf(s)ds∥=∥∫0tn‍St-sf(s)ds-∫tnt‍St-sf(s)ds-∫0tn‍Stn-sf(s)ds∥≤∫0tn‍∥St-sf(s)-Stn-sf(s)∥ds+∫tnt‍∥St-sf(s)∥ds.
Proceeding as in the previous situation, by Lebesgue's dominated convergence theorem we obtain
(101)∥F(t)-F(tn)∥⟶0forn⟶∞.
This completes the proof.Proposition 41.
LetX be a progressively measurable process satisfying
(102)ℙ(∫0t‍∥Xs∥ds<∞)=1∀t≥0.
Then the process Y defined as
(103)Yt≔∫0t‍St-sXsds,t≥0,
is continuous and adapted.Proof.
The continuity ofY is a consequence of Lemma 40. Moreover, Y is adapted, because X is progressively measurable.Now, we will deal with stochastic convolution integrals driven by the Wiener processes. LetH be a separable Hilbert space, and let (St)t≥0 be a C0-semigroup on H. Moreover, let W be a trace class Wiener process on some separable Hilbert space ℍ.Definition 42.
LetX be a L20(H)-valued predictable process such that
(104)ℙ(∫0t‍∥Xs∥L20(H)2ds<∞)=1∀t≥0.
One defines the stochastic convolutionX⋆W as
(105)(X⋆W)t≔∫0t‍St-sXsdWs,t≥0.One recalls the following result concerning the regularity of stochastic convolutions.Proposition 43.
LetX be a L20(H)-valued predictable process such that one of the following two conditions is satisfied.(1)
There exists a constantp>1such that(106)𝔼[∫0t‍∥Xs∥L20(H)2pds]<∞∀t≥0.(2)
The semigroup(St)t≥0is a semigroup of pseudocontractions, and one has(107)𝔼[∫0t‍∥Xs∥L20(H)2ds]<∞∀t≥0.Then the stochastic convolution X⋆W has a continuous version.Proof.
See [3, Lemma 3.3].


## 7. Existence and Uniqueness Results for SPDEs

In this section, we will present results concerning existence and uniqueness of solutions to the SPDE (44).First, we recall the Banach fixed point theorem, which will be a basic result for proving the existence of mild solutions to (44).Definition 44.
Let(E,d) be a metric space, and let Φ:E→E be a mapping.  (1)
The mappingΦ is called a contraction, if for some constant 0≤L<1 one has
(108)d(Φ(x),Φ(y))≤L·d(x,y)∀x,y∈E.(2)
An elementx∈E is called a fixed point of Φ, if one has
(109)Φ(x)=x.The following result is the well-known Banach fixed point theorem. Its proof can be found, for example, in [13, Theorem 3.48].Theorem 45 (The Banach fixed point theorem).
LetE be a complete metric space, and let Φ:E→E be a contraction. Then the mapping Φ has a unique fixed point.In this text, we will use the following slight extension of the Banach fixed point theorem.Corollary 46.
LetE be a complete metric space, and let Φ:E→E be a mapping such that for some n∈ℕ the mapping Φn is a contraction. Then the mapping Φ has a unique fixed point.Proof.
According to the Banach fixed point theorem (Theorem45) the mapping Φn has a unique fixed point; that is, there exists a unique element x∈E such that Φn(x)=x. Therefore, we have
(110)Φ(x)=Φ(Φn(x))=Φn(Φ(x)),
showing that Φ(x) is a fixed point of Φn. Since Φn has a unique fixed point, we deduce that Φ(x)=x, showing that x is a fixed point of Φ.
In order to prove uniqueness, lety∈E be another fixed point of Φ; that is, we have Φ(y)=y. By induction, we obtain
(111)Φn(y)=Φn-1(Φ(y))=Φn-1(y)=⋯=Φ(y)=y,
showing that y is a fixed point of Φn. Since the mapping Φn has exactly one fixed point, we obtain x=y.An indispensable tool for proving uniqueness of mild solutions to (44) will be the following version of Gronwall's inequality; see, for example, [14, Theorem 5.1].Lemma 47 (Gronwall's inequality).
LetT≥0 be fixed, let f:[0,T]→ℝ+ be a nonnegative continuous mapping, and let β≥0 be a constant such that
(112)f(t)≤β∫0t‍f(s)ds∀t∈[0,T].
Then one has f≡0.The following result shows that local Lipschitz continuity ofα and σ ensures the uniqueness of mild solutions to the SPDE (44).Theorem 48.
One supposes that for everyn∈ℕ there exists a constant Ln≥0 such that
(113)∥α(t,h1)-α(t,h2)∥≤Ln∥h1-h2∥,∥σ(t,h1)-σ(t,h2)∥L20(H)≤Ln∥h1-h2∥,
for all t≥0 and all h1,h2∈H with ∥h1∥,∥h2∥≤n. Let h0,g0:Ω→H be two ℱ0-measurable random variables, let τ>0 be a strictly positive stopping time, and let X, Y be two local mild solutions to (44) with initial conditions h0, g0 and lifetime τ. Then one has up to indistinguishability
(114)Xτ1{h0=g0}=Yτ1{h0=g0}
(Two processes X and Y are called indistinguishable if the set {ω∈Ω:Xt(ω)≠Yt(ω) for some t∈ℝ+} is a ℙ-nullset.)Proof.
Defining the stopping times(τn)n∈ℕ as
(115)τn≔τ∧inf{t≥0:∥Xt∥≥n}∧inf{t≥0:∥Yt∥≥n},
we have ℙ(τn→τ)=1. Let n∈ℕ and T≥0 be arbitrary, and set
(116)Γ≔{h0=g0}∈ℱ0.
The mapping
(117)f:[0,T]⟶ℝ,f(t)≔𝔼[1Γ∥Xt∧τn-Yt∧τn∥2],
is nonnegative, and it is continuous by Lebesgue's dominated convergence theorem. For all t∈[0,T] we have
(118)f(t)=𝔼[1Γ∥Xt∧τn-Yt∧τn∥2]≤3𝔼[1Γ∥St∧τn(h0-g0)∥2]︸=0+3𝔼[1Γ∥∫0t∧τn‍S(t∧τn)-s(α(s,Xs)-α(s,Ys))ds∥2]+3𝔼[1Γ∥∫0t∧τn‍S(t∧τn)-s(σ(s,Xs)-σ(s,Ys))dWs∥2]=3𝔼[∥∫0t∧τn‍1ΓS(t∧τn)-s(α(s,Xs)-α(s,Ys))ds∥2]+3𝔼[∥∫0t∧τn‍1ΓS(t∧τn)-s(σ(s,Xs)-σ(s,Ys))dWs∥2],
and hence, by the Cauchy-Schwarz inequality, the Itô isometry (39), the growth estimate (9) from Lemma 10, and the local Lipschitz conditions (113) we obtain
(119)f(t)≤3T𝔼[∫0t∧τn‍∥1ΓS(t∧τn)-s(α(s,Xs)-α(s,Ys))∥2ds]+3𝔼[∫0t∧τn‍∥1ΓS(t∧τn)-s(σ(s,Xs)-σ(s,Ys))∥L20(H)2ds]≤3T(MeωT)2𝔼[∫0t∧τn‍1Γ∥α(s,Xs)-α(s,Ys)∥2ds]+3(MeωT)2𝔼[∫0t∧τn‍1Γ∥σ(s,Xs)-σ(s,Ys)∥L20(H)2ds]≤3(T+1)(MeωT)2Ln2∫0t‍𝔼[1Γ∥Xs∧τn-Ys∧τn∥2]ds=3(T+1)(MeωT)2Ln2∫0t‍f(s)ds.
Using Gronwall's inequality (see Lemma 47) we deduce that f≡0. Thus, by the continuity of the sample paths of X and Y, we obtain
(120)ℙ(⋂t≥0‍{Xt∧τn1Γ=Yt∧τn1Γ})=1∀n∈ℕ,
and hence, by the continuity of the probability measure ℙ, we conclude that
(121)ℙ(⋂t≥0‍{Xt∧τ1Γ=Yt∧τ1Γ})=ℙ(⋂n∈ℕ‍⋂t≥0‍{Xt∧τn1Γ=Yt∧τn1Γ})=limn→∞ℙ(⋂t≥0‍{Xt∧τn1Γ=Yt∧τn1Γ})=1,
which completes the proof.The local Lipschitz conditions (113) are, in general, not sufficient in order to ensure the existence of mild solutions to the SPDE (44). Now, we will prove that the existence of mild solutions follows from global Lipschitz and linear growth conditions on α and σ. For this, we recall an auxiliary result which extends the Itô isometry (39).Lemma 49.
LetT≥0 be arbitrary, and let X=(Xt)t∈[0,T] be a L20(H)-valued, predictable process such that
(122)𝔼[∫0T‍∥Xs∥L20(H)2ds]<∞.
Then, for every p≥1 one has
(123)𝔼[∥∫0T‍XsdWs∥2p]≤Cp𝔼[∫0T‍∥Xs∥L20(H)2ds]p,
where the constant Cp>0 is given by
(124)Cp=(p(2p-1))p(2p2p-1)2p2.Proof.
See [3, Lemma 3.1].Theorem 50.
Suppose that there exists a constantL≥0 such that
(125)∥α(t,h1)-α(t,h2)∥≤L∥h1-h2∥,(126)∥σ(t,h1)-σ(t,h2)∥L20(H)≤L∥h1-h2∥,
for all t≥0 and all h1,h2∈H, and suppose that there exists a constant K≥0 such that
(127)∥α(t,h)∥≤K(1+∥h∥),(128)∥σ(t,h)∥L20(H)≤K(1+∥h∥),
for all t≥0 and all h∈H. Then, for every ℱ0-measurable random variable h0:Ω→H, there exists a (up to indistinguishability) unique mild solution X to (44).Proof.
The uniqueness of mild solutions to (44) is a direct consequence of Theorem 48, and hence, we may concentrate on the existence proof, which we divide into the following several steps.
Step  1. First, we suppose that the initial condition h0 satisfies 𝔼[∥h0∥2p]<∞ for some p>1. Let T≥0 be arbitrary. We define the Banach space
(129)LT2p(H)≔L2p(Ω×[0,T],𝒫T,ℙ⊗dt;H),
and prove that the variation of constants equation
(130)Xt=Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs,t∈[0,T],
has a unique solution in the space LT2p(H). This is done in the following three steps.
Step  1.1. For X∈LT2p(H) we define the process ΦX by
(131)(ΦX)t=Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs,t∈[0,T].
Then the process ΦX is well defined. Indeed, by the growth estimate (9), the linear growth condition (127), and Hölder's inequality we have
(132)𝔼[∫0T‍∥St-sα(s,Xs)∥ds]≤MeωT𝔼[∫0T‍∥α(s,Xs)∥ds]≤MeωTK𝔼[∫0T‍(1+∥Xs∥)ds]=MeωTK(T+𝔼[∫0T‍∥Xs∥ds])≤MeωTK([∫0T‍∥Xs∥2pds]1/2pT+T1-1/2p222222222×𝔼[∫0T‍∥Xs∥2pds]1/2p)<∞.
Furthermore, by the growth estimate (9), the linear growth condition (128), and Hölder's inequality we have
(133)𝔼[∫0T‍∥St-sσ(s,Xs)∥L20(H)2ds]≤(MeωT)2𝔼[∫0T‍∥σ(s,Xs)∥L20(H)2ds]≤(MeωTK)2𝔼[∫0T‍(1+∥Xs∥)2ds]≤2(MeωTK)2𝔼[∫0T‍(1+∥Xs∥2)ds]=2(MeωTK)2(T+𝔼[∫0T‍∥Xs∥2ds])≤2(MeωTK)2(T+T1-1/p𝔼[∫0T‍∥Xs∥2p]1/p)<∞.
The previous two estimates show that Φ is a well-defined mapping on LT2p(H).
Step  1.2. Next, we show that the mapping Φ maps LT2p(H) into itself; that is, we have Φ:LT2p(H)→LT2p(H). Indeed, let X∈LT2p(H) be arbitrary. Defining the processes ΦαX and ΦσX as
(134)(ΦαX)t≔∫0t‍St-sα(s,Xs)ds,t∈[0,T],(ΦσX)t≔∫0t‍St-sσ(s,Xs)dWs,t∈[0,T],
we have
(135)(ΦX)t=Sth0+(ΦαX)t+(ΦσX)t,t∈[0,T].
By the growth estimate (9), we have
(136)𝔼[∫0T‍∥Sth0∥2pdt]≤(MeωT)2pT𝔼[∥h0∥2p]<∞.
By Hölder's inequality and the growth estimate (9), we have
(137)𝔼[∫0T‍∥(ΦαX)t∥2pdt]=𝔼[∫0T∥∫0t‍St-sα(s,Xs)ds∥2pdt]≤t2p-1𝔼[∫0T‍∫0t‍∥St-sα(s,Xs)∥2pdsdt]≤T2p-1(MeωT)2p𝔼[∫0T‍∫0t‍∥α(s,Xs)∥2pdsdt],
and hence, by the linear growth condition (127) and Hölder's inequality, we obtain
(138)𝔼[∫0T‍∥(ΦαX)t∥2pdt]≤T2p-1(MeωTK)2p𝔼[∫0T‍∫0t‍(1+∥Xs∥)2pdsdt]≤T2p-1(MeωTK)2p22p-1𝔼[∫0T‍∫0t‍(1+∥Xs∥2p)dsdt]≤(2T)2p-1(MeωTK)2p(T22+T𝔼[∫0T‍∥Xs∥2pds])<∞.
Furthermore, by Lemma 49 and the growth estimate (9), we have
(139)𝔼[∫0T‍∥(ΦσX)t∥2pdt]=𝔼[∫0T‍∥∫0t‍St-sσ(s,Xs)dWs∥2pdt]=∫0T‍𝔼[∥∫0t‍St-sσ(s,Xs)dWs∥2p]dt≤Cp∫0T‍𝔼[∫0t‍∥St-sσ(s,Xs)∥L20(H)2ds]pdt≤Cp(MeωT)2p∫0T‍𝔼[∫0t‍∥σ(s,Xs)∥L20(H)2ds]pdt,
and hence, by the linear growth condition (128) and Hölder's inequality, we obtain
(140)𝔼[∫0T‍∥(ΦσX)t∥2pdt]≤Cp(MeωT)2ptp-1𝔼[∫0T‍∫0t‍∥σ(s,Xs)∥L20(H)2pdsdt]≤Cp(MeωTK)2pTp-1𝔼[∫0T‍∫0t‍(1+∥Xs∥)2pdsdt]≤Cp(MeωTK)2pTp-122p-1𝔼[∫0T‍∫0t‍(1+∥Xs∥2p)dsdt]≤Cp(MeωTK)2p2p(2T)p-1×(T22+T𝔼[∫0T‍∥Xs∥2pds])<∞.
The previous three estimates show that ΦX∈L2p(H). Consequently, the mapping Φ maps LT2p(H) into itself.
Step  1.3. Now, we show that for some index n∈ℕ the mapping Φn is a contraction on LT2p(H). Let X,Y∈LT2p(H), and t∈[0,T] be arbitrary. By Hölder's inequality, the growth estimate (9), and the Lipschitz condition (125) we have
(141)𝔼[∥(ΦαX)t-(ΦαY)t∥2p]=𝔼[∥∫0t‍St-sα(s,Xs)ds-∫0t‍St-sα(s,Ys)ds∥2p]=𝔼[∥∫0t‍St-s(α(s,Xs)-α(s,Ys))ds∥2p]≤t2p-1𝔼[∫0t‍∥St-s(α(s,Xs)-α(s,Ys))∥2pds]≤T2p-1(MeωT)2p𝔼[∫0t‍∥α(s,Xs)-α(s,Ys)∥2pds]≤T2p-1(MeωTL)2p∫0t‍𝔼[∥Xs-Ys∥2p]ds.
Furthermore, by Lemma 49, the growth estimate (9), the Lipschitz condition (126), and Hölder's inequality we obtain
(142)𝔼[∥(ΦσX)t-(ΦσY)t∥2p]=𝔼[∥∫0t‍St-sσ(s,Xs)dWs-∫0t‍St-sσ(s,Ys)dWs∥2p]=𝔼[∥∫0t‍St-s(σ(s,Xs)-σ(s,Ys))dWs∥2p]≤Cp𝔼[∫0t‍∥St-s(σ(s,Xs)-σ(s,Ys))∥L20(H)2ds]p≤Cp(MeωT)2p𝔼[∫0t‍∥σ(s,Xs)-σ(s,Ys)∥L20(H)2ds]p≤Cp(MeωTL)2p∫0t‍𝔼[∥Xs-Ys∥2p]ds.
Therefore, defining the constant
(143)C≔22p-1(T2p-1(MeωTL)2p+Cp(MeωTL)2p),
by Hölder's inequality, we get
(144)𝔼[∥(ΦX)t-(ΦY)t∥2p]≤22p-1(𝔼[∥(ΦαX)t-(ΦαY)t∥2p]+𝔼[∥(ΦσX)t-(ΦσY)t∥2p])≤C∫0t‍𝔼[∥Xs-Ys∥2p]ds.
Thus, by induction for every n∈ℕ, we obtain
(145)∥ΦnX-ΦnY∥LT2p(H)=(∫0T‍𝔼[∥(ΦnX)t1-(ΦnY)t1∥2p]dt1)1/2p≤(C∫0T(∫0t1‍𝔼[-(Φn-1Y)t2∥2p∥(Φn-1X)t2-(Φn-1Y)t2∥2p]dt2∫0t1‍𝔼[∥(Φn-1X)t2)dt1∫0T(∫0t1‍𝔼[-(Φn-1Y)t2∥2p∥(Φn-1X)t2)1/2p≤⋯≤(Cn∫0T‍∫0t1‍⋯∫0tn-1‍(∫0tn‍𝔼[∥Xs-Ys∥2p]ds)22003222222×dtn⋯dt2dt1Cn∫0T‍∫0t1‍⋯∫0tn-1‍(∫0tn‍𝔼[∥Xs-Ys∥2p]ds))1/2p≤(Cn(∫0T‍∫0t1‍⋯∫0tn-1‍1dtn⋯dt2dt1)︸=Tn/n!2222222×𝔼[∫0T‍∥Xs-Ys∥2pds](∫0T‍∫0t1‍⋯∫0tn-1‍1dtn…dt2dt1)︸=Tn/n!)1/2p=((CT)nn!)1/2p︸→0forn→∞∥X-Y∥LT2p(H).
Consequently, there exists an index n∈ℕ such that Φn is a contraction, and hence, according to the extension of the Banach fixed point theorem (see Corollary 46), the mapping Φ has a unique fixed point X∈LT2p(H). This fixed point X is a solution to the variation of constants equation (130). Since T≥0 was arbitrary, there exists a process X which is a solution of the variation of constants equation:
(146)Xt=Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs,2222222222222222222222222222222222t≥0.
Step  1.4. In order to prove that X is a mild solution to (44), it remains to show that X has a continuous version. By Lemma 12, the process
(147)t⟼Sth0,t≥0,
is continuous, and by Proposition 41, the process
(148)∫0t‍St-sα(s,Xs)ds,t≥0,
is continuous, too. Moreover, for every T≥0, we have, by the linear growth condition (128), Hölder's inequality, and since X∈LT2p(H), the following estimate:
(149)𝔼[∫0T‍∥σ(s,Xs)∥L20(H)2pds]≤K2p𝔼[∫0T‍(1+∥Xs∥)2pds]≤K2p22p-1𝔼[∫0T‍(1+∥Xs∥2p)ds]=K(2K)2p-1(T+𝔼[∫0T‍∥Xs∥2pds])<∞.
Thus, by Proposition 43 the stochastic convolution σ⋆W given by
(150)(σ⋆W)t=∫0t‍St-sσ(s,Xs)dWs,t≥0,
has a continuous version, and consequently, the process X has a continuous version, too. This continuous version is a mild solution to (44).
Step  2. Now let h0:Ω→H be an arbitrary ℱ0-measurable random variable. We define the sequence (hn)n∈ℕ of ℱ0-measurable random variables as
(151)h0n≔h01{∥h0∥≤n},n∈ℕ.
Let n∈ℕ be arbitrary. Then, as h0n is bounded, we have 𝔼[∥h0n∥2p]<∞ for all p>1. By Step 1 the SPDE
(152)dXtn=(AXtn+α(t,Xtn))dt+σ(t,Xtn)dWt,X0n=h0n,
has a mild solution Xn. We define the sequence (Ωn)n∈ℕ⊂ℱ0 as
(153)Ωn≔{∥h0∥≤n},n∈ℕ.
Then, we have Ωn⊂Ωm for n≤m, Ω=⋃n∈ℕ‍Ωn, and
(154)Ωn⊂{h0n=h0m}⊂{h0n=h0}∀n≤m.
Thus, by Theorem 48 we have (up to indistinguishability)
(155)Xn1Ωn=Xm1Ωn∀n≤m.
Consequently, the process
(156)X≔limn→∞Xn1Ωn
is a well-defined, continuous, and adapted process, and we have
(157)Xn1Ωn=Xm1Ωn=X1Ωn∀n≤m.
Furthermore, we obtain ℙ-almost surely
(158)Xt=limn→∞Xtn1Ωn=limn→∞1Ωn(Sth0n+∫0t‍St-sα(s,Xsn)ds2222222222+∫0t‍St-sσ(s,Xsn)dWs)=limn→∞(St(1Ωnh0n)+∫0t‍1ΩnSt-sα(s,Xsn)ds22222222+∫0t‍1ΩnSt-sσ(s,Xsn)dWs)=limn→∞(St(1Ωnh0)+∫0t‍1ΩnSt-sα(s,Xs)ds22222222+∫0t‍1ΩnSt-sσ(s,Xs)dWs)=limn→∞1Ωn(Sth0+∫0t‍St-sα(s,Xs)ds2222222222+∫0t‍St-sσ(s,Xs)dWs)=Sth0+∫0t‍St-sα(s,Xs)ds+∫0t‍St-sσ(s,Xs)dWs,t≥0,
proving that X is a mild solution to (44).Remark 51.
For the proof of Theorem50, we have used Corollary 46, which is a slight extension of the Banach fixed point theorem. Such an idea has been applied, for example, in [15].Remark 52.
A recent method for proving existence and uniqueness of mild solutions to the SPDE (44) is the method of the moving frame presented in [6]; see also [8]. It allows to reduce SPDE problems to the study of SDEs in infinite dimension. In order to apply this method, we need that the semigroup (St)t≥0 is a semigroup of pseudocontractions.We close this section with a consequence about the existence of weak solutions.Corollary 53.
Suppose that conditions (125)–(128) are fulfilled. Let h0:Ω→H be a ℱ0-measurable random variable such that 𝔼[∥h0∥2p]<∞ for some p>1. Then there exists a (up to indistinguishability) unique weak solution X to (44).Proof.
According to Proposition35, every weak solution X to (44) is also a mild solution to (44). Therefore, the uniqueness of weak solutions to (44) is a consequence of Theorem 48.
It remains to prove the existence of a weak solution to (44). Let T≥0 be arbitrary. By Theorem 50 and its proof, there exists a mild solution X∈LT2p(H) to (44). By the linear growth condition (128) and Hölder's inequality we obtain
(159)𝔼[∫0T‍∥σ(s,Xs)∥L20(H)2]≤K2𝔼[∫0T‍(1+∥Xs∥)2ds]≤2K2𝔼[∫0T‍(1+∥Xs∥2)ds]=2K2(T+𝔼[∫0T∥Xs∥2ds])≤2K2(T+T1-1/p𝔼[∫0T‍∥Xs∥2pds]1/p)<∞,
showing that condition (72) is fulfilled. Thus, by Proposition 37, the process X is also a weak solution to (44).


## 8. Invariant Manifolds for Weak Solutions to SPDEs

In this section, we deal with invariant manifolds for time-homogeneous SPDEs of the type (44). This topic arises from the natural desire to express the solutions of the SPDE (44), which generally live in the infinite dimensional Hilbert space H, by means of a finite dimensional state process and thus to ensure larger analytical tractability. Our goal is to find conditions on the generator A and the coefficients α, σ such that for every starting point of a finite dimensional submanifold the solution process stays on the submanifold.We start with the required preliminaries about finite dimensional submanifolds in Hilbert spaces. In the sequel, letH be a separable Hilbert space.Definition 54.
Letm,k∈ℕ be positive integers. A subset ℳ⊂H is called an m-dimensional Ck-submanifold of H, if for every h∈ℳ there exist an open neighborhood U⊂H of h, an open set V⊂ℝm, and a mapping ϕ∈C2(V;H) such that(1)
the mappingϕ:V→U∩ℳ is a homeomorphism;(2)
for ally∈V the mapping Dϕ(y) is injective.The mapping ϕ is called a parametrization of ℳ around h.In what follows, letℳ be an m-dimensional Ck-submanifold of H.Lemma 55.
Letϕi:Vi→Ui∩ℳ, i=1,2 be two parametrizations with W≔U1∩U2∩ℳ≠∅. Then the mapping
(160)ϕ1-1∘ϕ2:ϕ2-1(W)⟶ϕ1-1(W)
is a Ck-diffeomorphism.Proof.
See [16, Lemma 6.1.1].Corollary 56.
Leth∈ℳ be arbitrary, and let ϕi:Vi→Ui∩ℳ, i=1,2 be two parametrizations of ℳ around h. Then one has
(161)Dϕ1(y1)(ℝm)=Dϕ2(y2)(ℝm),
where yi=ϕi-1(h) for i=1,2.Proof.
SinceU1 and U2 are open neighborhoods of h, we have W≔U1∩U2∩ℳ≠∅. Thus, by Lemma 55 the mapping
(162)ϕ1-1∘ϕ2:ϕ2-1(W)⟶ϕ1-1(W)
is a Ck-diffeomorphism. Using the chain rule, we obtain
(163)Dϕ2(y2)(ℝm)=D(ϕ1∘(ϕ1-1∘ϕ2))(y2)(ℝm)=Dϕ1(y1)D(ϕ1-1∘ϕ2)(y2)(ℝm)⊂Dϕ1(y1)(ℝm),
and, analogously, we prove that Dϕ1(y1)(ℝm)⊂Dϕ2(y2)(ℝm).Definition 57.
Leth∈ℳ be arbitrary. The tangent space of ℳ to h is the subspace
(164)Thℳ≔Dϕ(y)(ℝm),
where y=ϕ-1(h) and ϕ:V→U∩ℳ denotes a parametrization of ℳ around h.Remark 58.
Note that, according to Corollary56, the Definition 57 of the tangent space Thℳ does not depend on the choice of the parametrization ϕ:V→U∩ℳ.Proposition 59.
Leth∈ℳ be arbitrary, and let ϕ:V→U∩ℳ be a parametrization of ℳ around h. Then there exist an open set V0⊂V, an open neighborhood U0⊂U of h, and a mapping ϕ^∈Cbk(ℝm;H) with ϕ|V0=ϕ^|V0 such that ϕ|V0:V0→U0∩ℳ is a parametrization of ℳ around h, too.Proof.
See [16, Remark 6.1.1].Remark 60.
By Proposition59 we may assume that any parametrization ϕ:V→U∩ℳ has an extension ϕ∈Cbk(ℝm;H).Proposition 61.
LetD⊂H be a dense subset. For every h0∈ℳ there exist ζ1,…,ζm∈D and a parametrization ϕ:V→U∩ℳ around h0 such that
(165)ϕ(〈ζ,h〉)=h∀h∈U∩ℳ,
where one uses the notation 〈ζ,h〉≔(〈ζ1,h〉,…,〈ζm,h〉)∈ℝm.Proof.
See [16, Proposition 6.1.2].Proposition 62.
Letϕ:V→U∩ℳ be a parametrization as in Proposition 61. Then the following statements are true.(1)
The elementsζ1,…,ζmare linearly independent inH.(2)
For everyh∈U∩ℳ, one has the direct sum decomposition(166)H=Thℳ⊕〈ζ1,…,ζm〉⊥.(3)
For everyh∈U∩ℳthe mapping(167)Πh=Dϕ(y)(〈ζ,•〉):H⟶Thℳ,wherey=〈ζ,h〉,is the corresponding projection according to (166) fromHontoThℳ, that is, we have(168)Πh∈L(H),Πh2=Πh,ran(Πh)=Thℳ,ker(Πh)=〈ζ1,…,ζm〉⊥.Proof.
See [16, Lemma 6.1.3].From now on, we assume thatℳ is an m-dimensional C2-submanifold of H.Proposition 63.
Letϕ:V→U∩ℳ be a parametrization as in Proposition 61. Furthermore, let σ∈C1(H) be a mapping such that
(169)σ(h)∈Thℳ∀h∈U∩ℳ.
Then, for every h∈U∩ℳ the direct sum decomposition of Dσ(h)σ(h) according to (166) is given by
(170)Dσ(h)σ(h)=Dϕ(y)(〈ζ,Dσ(h)σ(h)〉)+D2ϕ(y)(〈ζ,σ(h)〉,〈ζ,σ(h)〉),
where y=ϕ-1(h).Proof.
SinceV is an open subset of ℝm, there exists ϵ>0 such that
(171)y+tDϕ(y)-1σ(h)∈V∀t∈(-ϵ,ϵ).
Therefore, the curve
(172)c:(-ϵ,ϵ)⟶U∩ℳ,c(t)≔ϕ(y+tDϕ(y)-1σ(h)),
is well defined, and we have c∈C1((-ϵ,ϵ);H) with c(0)=h and c′(0)=σ(h). Hence, we have
(173)ddtσ(c(t))|t=0=Dσ(h)σ(h).
Moreover, by condition (169) and Proposition 62, we have
(174)ddtσ(c(t))|t=0=ddtΠc(t)σ(c(t))|t=0=ddtDϕ(〈ζ,c(t)〉)(〈ζ,σ(c(t))〉)|t=0=Dϕ(y)(〈ζ,Dσ(h)σ(h)〉)+D2ϕ(y)(〈ζ,σ(h)〉,〈ζ,σ(h)〉).
The latter two identities prove the desired decomposition (170).After these preliminaries, we will study invariant manifolds for time-homogeneous SPDEs of the form(175)dXt=(AXt+α(Xt))dt+σ(Xt)dWtX0=h0,
with measurable mappings α:H→H and σ:H→L20(H). As in the previous sections, the operator A is the infinitesimal generator of a C0-semigroup (St)t≥0 on H. Note that, by (41), the SPDE (175) can be rewritten equivalently as
(176)dXt=(AXt+α(Xt))dt+∑j∈ℕ‍σj(Xt)dβtj,X0=h0,
where (βj)j∈ℕ denotes the sequence of real-valued independent standard Wiener processes defined in (33) and the mappings σj:H→H, j∈ℕ are given by σj=λjσej.For the rest of this section, we assume that there exist a constantL≥0 such that
(177)∥α(h1)-α(h2)∥≤L∥h1-h2∥,h1,h2∈H,
and a sequence (κj)j∈ℕ⊂ℝ+ with ∑j∈ℕ‍κj2<∞ such that for every j∈ℕ we have
(178)∥σj(h1)-σj(h2)∥≤κj∥h1-h2∥,h1,h2∈H,(179)∥σj(h)∥≤κj(1+∥h∥),h∈H.Proposition 64.
For everyh0∈H there exists a (up to indistinguishability) unique weak solution to (176).Proof.
By (178), for all h1,h2∈H we have
(180)∥σ(h1)-σ(h2)∥L20(H)=(∑j∈ℕ‍∥σj(h1)-σj(h2)∥2)1/2≤(∑j∈ℕ‍κj2)1/2∥h1-h2∥.
Moreover, by (177), for every h∈H we have
(181)∥α(h)∥≤∥α(h)-α(0)∥+∥α(0)∥≤L∥h∥+∥α(0)∥≤max{L,∥α(0)∥}(1+∥h∥),
and, by (179) we obtain
(182)∥σ(h)∥L20(H)=(∑j∈ℕ‍∥σj(h)∥2)1/2≤(∑j∈ℕ‍κj2)1/2(1+∥h∥).
Therefore, conditions (125)–(128) are fulfilled, and hence, applying Corollary 53 completes the proof.Recall thatℳ denotes a finite dimensional C2-submanifold of H.Definition 65.
The submanifoldℳ is called locally invariant for (176), if for every h0∈ℳ there exists a local weak solution X to (176) with some lifetime τ>0 such that
(183)Xt∧τ∈ℳ∀t≥0,ℙ-almostsurely.In order to investigate local invariance ofℳ, we will assume, from now on, that σj∈C1(H) for all j∈ℕ.Lemma 66.
The following statements are true.(1)
For everyh∈H one has
(184)∑j∈ℕ‍∥Dσj(h)σj(h)∥<∞.(2)
The mapping(185)H⟶H,h⟼∑j∈ℕ‍Dσj(h)σj(h),is continuous.Proof.
By (178) and (179), for every h∈H we have
(186)∑j∈ℕ‍∥Dσj(h)σj(h)∥≤∑j∈ℕ‍∥Dσj(h)∥∥σj(h)∥≤(1+∥h∥)∑j∈ℕ‍κj2<∞,
showing (184). Moreover, for every j∈ℕ the mapping
(187)H⟼H,Dσj(h)σj(h),
is continuous, because for all h1,h2∈H we have
(188)∥Dσj(h1)σj(h1)-Dσj(h2)σj(h2)∥≤∥Dσj(h1)σj(h1)-Dσj(h1)σj(h2)∥+∥Dσj(h1)σj(h2)-Dσj(h2)σj(h2)∥≤∥Dσj(h1)∥∥σj(h1)-σj(h2)∥+∥Dσj(h1)-Dσj(h2)∥∥σj(h2)∥.
Let ν be the counting measure on (ℕ,𝔓(ℕ)), which is given by ν({j})=1 for all j∈ℕ. Then we have
(189)∑j∈ℕ‍Dσj(h)σj(h)=∫ℕ‍Dσj(h)σj(h)ν(dj).
Hence, because of the estimate
(190)∥Dσj(h)σj(h)∥≤(1+∥h∥)κj2,h∈H,j∈ℕ,
the continuity of the mapping (185) is a consequence of Lebesgue's dominated convergence theorem.For a mappingϕ∈Cb2(ℝm;H) and elements ζ1,…,ζm∈𝒟(A*) we define the mappings αϕ,ζ:ℝm→ℝm and σϕ,ζj:ℝm→ℝm, j∈ℕ as
(191)αϕ,ζ(y)≔〈A*ζ,ϕ(y)〉+〈ζ,α(ϕ(y))〉,σϕ,ζj(y)≔〈ζ,σj(ϕ(y))〉.Proposition 67.
Letϕ∈Cb2(ℝm;H) and ζ1,…,ζm∈𝒟(A*) be arbitrary. Then, for every y0∈ℝm there exists a (up to indistinguishability) unique strong solution to the SDE:
(192)dYt=αϕ,ζ(Yt)dt+∑j∈ℕ‍σϕ,ζj(Yt)dβtj,Y0=y0.Proof.
By virtue of the assumptionϕ∈Cb2(ℝm;H) and (177)–(179), there exist a constant L~≥0 such that
(193)∥αϕ,ζ(y1)-αϕ,ζ(y2)∥ℝm≤L~∥y1-y2∥ℝm,y1,y2∈ℝm,
and a sequence (κ~j)j∈ℕ⊂ℝ+ with ∑j∈ℕ‍κ~j2<∞ such that for every j∈ℕ we have
(194)∥σϕ,ζj(y1)-σϕ,ζj(y2)∥ℝm≤κ~j∥y1-y2∥ℝm,y1,y2∈ℝm,∥σϕ,ζj(y)∥ℝm≤κ~j(1+∥y∥ℝm),y∈ℝm.
Therefore, by Proposition 64, for every y0∈ℝm there exists a (up to indistinguishability) unique weak solution to (192), which, according to Proposition 39 is also a strong solution to (192). The uniqueness of strong solutions to (192) is a consequence of Proposition 39 and Theorem 48.Now, we are ready to formulate and prove our main result of this section.Theorem 68.
The following statements are equivalent.(1)
The submanifoldℳ is locally invariant for (176).(2)
One has(195)ℳ⊂𝒟(A),(196)σj(h)∈Thℳ∀h∈ℳ,allj∈ℕ,(197)Ah+α(h)-12∑j∈ℕ‍Dσj(h)σj(h)∈Thℳ∀h∈ℳ.(3)
The operatorA is continuous on ℳ, and for each h0∈ℳ there exists a local strong solution X to (176) with some lifetime τ>0 such that
(198)Xt∧τ∈ℳ∀t≥0,ℙ-almostsurely.Proof.
(1)⇒(2): Let h∈ℳ be arbitrary. By Proposition 61 and Remark 60 there exist elements ζ1,…,ζm∈𝒟(A*) and a parametrization ϕ:V→U∩ℳ around h such that the inverse ϕ-1:U∩ℳ→V is given by ϕ-1=〈ζ,•〉, and ϕ has an extension ϕ∈Cb2(ℝm;H). Since the submanifold ℳ is locally invariant for (176), there exists a local weak solution X to (176) with initial condition h and some lifetime ϱ>0 such that
(199)Xt∧ϱ∈ℳ∀t≥0,ℙ-almostsurely.
Since U is an open neighborhood of h, there exists ϵ>0 such that Bϵ(h)¯⊂U, where Bϵ(h) denotes the open ball:
(200)Bϵ(h)={g∈H:∥g-h∥<ϵ}.
We define the stopping time
(201)τ≔ϱ∧inf{t≥0:Xt∉Bϵ(h)}.
Since the process X has continuous sample paths and satisfies X0=h, we have τ>0 and ℙ-almost surely
(202)Xt∧τ∈U∩ℳ∀t≥0.
Defining the ℝm-valued process Y≔〈ζ,X〉 we have ℙ-almost surely
(203)Yt∧τ∈V∀t≥0.
Moreover, since X is a weak solution to (176) with initial condition h, setting y≔〈ζ,h〉∈V we have ℙ-almost surely
(204)Yt∧τ=〈ζ,h〉+∫0t∧τ‍(〈A*ζ,Xs〉+〈ζ,α(Xs)〉)ds+∑j∈ℕ‍∫0t∧τ‍〈ζ,σj(Xs)〉dβsj=〈ζ,h〉+∫0t∧τ‍αϕ,ζ(〈ζ,Xs〉)ds+∑j∈ℕ‍∫0t∧τ‍σϕ,ζj(〈ζ,Xs〉)dβsj=y+∫0t∧τ‍αϕ,ζ(Ys)ds+∑j∈ℕ‍∫0t∧τ‍σϕ,ζj(Ys)dβsj,t≥0,
showing that Y is a local strong solution to (192) with initial condition y. By Itô's formula (Theorem 26) we obtain ℙ-almost surely
(205)Xt∧τ=ϕ(Yt∧τ)=h+∫0t∧τ‍(12∑j∈ℕ‍Dϕ(Ys)αϕ,ζ(Ys)2222222222+12∑j∈ℕ‍D2ϕ(Ys)(σϕ,ζj(Ys),σϕ,ζj(Ys)))ds+∑j∈ℕ‍∫0t∧τ‍Dϕ(Ys)σϕ,ζj(Ys)dβsj,t≥0.
Now, let ξ∈𝒟(A*) be arbitrary. Then we have ℙ-almost surely
(206)〈ξ,Xt∧τ〉=〈ξ,h〉+∫0t∧τ‍〈+12∑j∈ℕ‍D2ϕ(Ys)(σϕ,ζj(Ys),σϕ,ζj(Ys))ξ,Dϕ(Ys)αϕ,ζ(Ys)+12∑j∈ℕ‍D2ϕ(Ys)(σϕ,ζj(Ys),σϕ,ζj(Ys))〉ds+∑j∈ℕ‍∫0t∧τ‍〈ξ,Dϕ(Ys)σϕ,ζj(Ys)〉dβsj,t≥0.
On the other hand, since X is a local weak solution to (176) with initial condition h and lifetime τ, we have ℙ-almost surely for all t≥0 the identity
(207)〈ξ,Xt∧τ〉=〈ξ,h〉+∫0t∧τ‍(〈A*ξ,Xs〉+〈ξ,α(Xs)〉)ds+∑j∈ℕ‍∫0t∧τ‍〈ξ,σj(Xs)〉dβsj.
Combining (206) and (207) yields up to indistinguishability
(208)B+M=0,
where the processes B and M are defined as
(209)Bt≔∫0t∧τ‍(〈A*ξ,Xs〉+〈(σϕ,ζj(Ys),σϕ,ζj(Ys))ξ,α(Xs)-Dϕ(Ys)αϕ,ζ(Ys)-12∑j∈ℕ‍D2ϕ(Ys)×(σϕ,ζj(Ys),σϕ,ζj(Ys))〉)ds,222222222222222222222222222222222t≥0,Mt≔∑j∈ℕ‍∫0t∧τ‍〈ξ,σj(Xs)-Dϕ(Ys)σϕ,ζj(Ys)〉dβsj,22222222222222222222222222222222t≥0.
The process B+M is a continuous semimartingale with canonical decomposition (208). Since the canonical decomposition of a continuous semimartingale is unique up to indistinguishability, we deduce that B=M=0 up to indistinguishability. Using the Itô isometry (39) we obtain ℙ-almost surely
(210)∫0t∧τ‍(-12∑j∈ℕ‍D2ϕ(Ys)(σϕ,ζj(Ys),σϕ,ζj(Ys))〉〈A*ξ,Xs〉222222+〈-12∑j∈ℕ‍D2ϕ(Ys)(σϕ,ζj(Ys),σϕ,ζj(Ys))ξ,α(Xs)-Dϕ(Ys)αϕ,ζ(Ys)2222222222-12∑j∈ℕ‍D2ϕ(Ys)(σϕ,ζj(Ys),σϕ,ζj(Ys))〉)ds=0,t≥0,∫0t∧τ‍∑j∈ℕ‍|〈ξ,σj(Xs)-Dϕ(Ys)σϕ,ζj(Ys)〉|2ds=0,2222222222222222222222222222222222222t≥0.
By the continuity of the processes X and Y we obtain for all ξ∈𝒟(A*) the following identities:
(211)〈A*ξ,h〉+〈12∑j∈ℕ‍D2ϕ(y)(σϕ,ζj(y),σϕ,ζj(y))ξ,α(h)-Dϕ(y)αϕ,ζ(y)-12∑j∈ℕ‍D2ϕ(y)(σϕ,ζj(y),σϕ,ζj(y))〉=0,〈ξ,σj(h)-Dϕ(y)σϕ,ζj(y)〉=0,j∈ℕ.
Consequently, the mapping ξ↦〈A*ξ,h〉 is continuous on 𝒟(A*), and hence we have h∈𝒟(A**) by the definition of the domain provided in (4). By Proposition 7 we have A=A**, and thus we obtain h∈𝒟(A), proving (195). By Proposition 7, the domain 𝒟(A*) is dense in H, and thus
(212)σj(h)=Dϕ(y)σϕ,ζj(y)∈Thℳ,j∈ℕ,
showing (196). Moreover, for all ξ∈𝒟(A*) we have
(213)〈-12∑j∈ℕ‍D2ϕ(y)(σϕ,ζj(y),σϕ,ζj(y))ξ,Ah+α(h)-Dϕ(y)αϕ,ζ(y)-12∑j∈ℕ‍D2ϕ(y)(σϕ,ζj(y),σϕ,ζj(y))〉=0.
Since the domain 𝒟(A*) is dense in H, together with Proposition 63, we obtain
(214)Ah+α(h)-12∑j∈ℕ‍Dσj(h)σj(h)=Ah+α(h)-12∑j∈ℕ‍((σϕ,ζj(y),σϕ,ζj(y))Dϕ(y)(〈ζ,Dσj(h)σj(h)〉)2222222222222222222+D2ϕ(y)(σϕ,ζj(y),σϕ,ζj(y)))=Dϕ(y)αϕ,ζ(y)-12∑j∈ℕ‍Dϕ(y)(〈ζ,Dσj(h)σj(h)〉)=Dϕ(y)(αϕ,ζ(y)-12∑j∈ℕ‍〈ζ,Dσj(h)σj(h)〉)∈Thℳ,
which proves (197).
(2)⇒(1): Let h0∈ℳ be arbitrary. By Proposition 61 and Remark 60 there exist ζ1,…,ζm∈𝒟(A*) and a parametrization ϕ:V→U∩ℳ around h0 such that the inverse ϕ-1:U∩ℳ→V is given by ϕ-1=〈ζ,•〉, and ϕ has an extension ϕ∈Cb2(ℝm;H). Let h∈U∩ℳ be arbitrary, and set y≔〈ζ,h〉∈V. By relations (195), (197) and Proposition 62, we obtain
(215)Ah+α(h)-12∑j∈ℕ‍Dσj(h)σj(h)=Dϕ(y)(〈ζ,Ah+α(h)-12∑j∈ℕ‍Dσj(h)σj(h)〉),
and thus
(216)Ah=Dϕ(y)(12∑j∈ℕ‍〈A*ζ,h〉+〈ζ,α(h)-12∑j∈ℕ‍Dσj(h)σj(h)〉)-α(h)+12∑j∈ℕ‍Dσj(h)σj(h).
Together with Lemma 66, this proves the continuity of A on U∩ℳ. Since h0∈ℳ was arbitrary, this proves that A is continuous on ℳ.
Furthermore, by (196) and Proposition 62 we have
(217)σj(h)=Dϕ(y)σϕ,ζj(h)foreveryj∈ℕ.
Moreover, by (195), (197), and Propositions 62 and 63, we obtain
(218)Ah+α(h)-12∑j∈ℕ‍Dσj(h)σj(h)=Dϕ(y)(〈-12∑j∈ℕ‍Dσj(h)σj(h)ζ,Ah+α(h)22222222222222-12∑j∈ℕ‍Dσj(h)σj(h)〉)=Dϕ(y)(〈A*ζ,h〉+〈ζ,α(h)〉)-12∑j∈ℕ‍Dϕ(y)〈ζ,Dσj(h)σj(h)〉=Dϕ(y)αϕ,ζ(y)+12∑j∈ℕ‍(D2ϕ(y)(σϕ,ζj(y),σϕ,ζj(y))22222222222-Dσj(h)σj(h)(σϕ,ζj(y),σϕ,ζj(y))).
This gives us
(219)Ah+α(h)=Dϕ(y)αϕ,ζ(y)+12∑j∈ℕ‍D2ϕ(y)(σϕ,ζj(y),σϕ,ζj(y)).
Now, let Y be the strong solution to (192) with initial condition y0≔〈ζ,h0〉∈V. Since V is open, there exists ϵ>0 such that Bϵ(y0)¯⊂V. We define the stopping time
(220)τ≔inf{t≥0:Yt∉Bϵ(y0)}.
Since the process Y has continuous sample paths and satisfies Y0=y0, we have τ>0 and ℙ-almost surely
(221)Yt∧τ∈V∀t≥0.
Therefore, defining the H-valued process X≔ϕ(Y), we have ℙ-almost surely
(222)Xt∧τ∈U∩ℳ∀t≥0.
Moreover, using Itô's formula (Theorem 26) and incorporating (217), (219), we obtain ℙ-almost surely
(223)Xt∧τ=ϕ(y0)+∫0t∧τ‍((σϕ,ζj(Ys),σϕ,ζj(Ys))Dϕ(Ys)αϕ,ζ(Ys)+12∑j∈ℕ‍D2ϕ(Ys)ϕ(Ys)×(σϕ,ζj(Ys),σϕ,ζj(Ys)))ds+∑j∈ℕ‍∫0t∧τ‍Dϕ(Ys)σϕ,ζj(Ys)dβsj=ϕ(y0)+∫0t∧τ‍(Aϕ(Ys)+α(ϕ(Ys)))ds+∑j∈ℕ‍∫0t∧τ‍σj(ϕ(Ys))dβsj=h0+∫0t∧τ‍(AXs+α(Xs))ds+∑j∈ℕ‍∫0t∧τ‍σj(Xs)dβsj,t≥0,
showing that X is a local strong solution to (44) with lifetime τ.
(3)⇒(1): This implication is a direct consequence of Proposition 29.The results from this section are closely related to the existence of finite dimensional realizations, that is, the existence of invariant manifolds for each starting pointh0, and we point out the papers [17–22] regarding this topic. Furthermore, we mention that Theorem 68 has been extended in [23] to SPDEs with jumps.



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