Foundations of the theory of semilinear stochastic partial differential equations

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.


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][2][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][6][7][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 Sections 2 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.

Unbounded Operators in the Hilbert Spaces
In this section, we review the relevant properties about unbounded operators. We will start with operators in Banach 2 International Journal of Stochastic Analysis 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].
Let and be Banach spaces. For a linear operator : ⊃ D( ) → , defined on some subspace D( ) of , we call D( ) the domain of . Definition 1. A linear operator : ⊃ D( ) → is called closed, if for every sequence ( ) ∈N ⊂ D( ), such that the limits = lim → ∞ ∈ and = lim → ∞ ∈ exist, one has ∈ D( ) and = .
Definition 2. A linear operator : ⊃ D( ) → is called densely defined, if its domain D( ) is dense in ; that is, D( ) = . (1) (2) The spectrum of is defined as ( ) := C \ ( ). ( Now, we will introduce the adjoint operator of a densely defined operator in a Hilbert space. Recall that for a bounded linear operator ∈ ( 1 , 2 ), mapping between two Hilbert spaces 1 and 2 , the adjoint operator is the unique bounded linear operator * ∈ ( 2 , 1 ) such that In order to extend this definition to unbounded operators, one recalls the following extension result for linear operators.

Proposition 4.
Let be a normed space, let be a Banach space, let ⊂ be a dense subspace, and let Φ : → be a continuous linear operator. Then there exists a unique continuous extensionΦ : → , that is, a continuous linear operator withΦ| = Φ. Moreover, one has ‖Φ‖ = ‖Φ‖. Now, let be a Hilbert space. We recall the representation theorem of Fréchet-Riesz. In the sequel, the space denotes the dual space of .
Let ∈ D( * ) be arbitrary. By virtue of the extension result for linear operators (Proposition 4), the operator has a unique extension to a linear functional ∈ . By the representation theorem of Fréchet-Riesz (Theorem 5), there exists a unique element ∈ with ⟨ , •⟩ = ⟨ , •⟩. This implies that Setting * := , this defines a linear operator * : ⊃ D( * ) → , and one has Definition 6. The operator * : ⊃ D( * ) → is called the adjoint operator of .
is a separable Hilbert space, too.

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 be a Banach space. (1) The family ( ) ≥0 is a called a strongly continuous semigroup (or 0 -semigroup), if the following conditions are satisfied: (2) The family ( ) ≥0 is called a norm continuous semigroup, if the following conditions are satisfied: Note that every norm continuous semigroup is also a 0semigroup. The following growth estimate (9) will often be used when dealing with SPDEs.
that is, the growth estimate (9) is satisfied with = 1.
If ( ) ≥0 is a semigroup of pseudocontractions with growth estimate (11), then ( ) ≥0 given by is a semigroup of contractions. Hence, every pseudocontractive semigroup can be transformed into a semigroup of contractions, which explains the term pseudocontractive.
(2) For all ∈ and ≥ 0, the mapping is uniformly continuous.
Definition 13. Let ( ) ≥0 be a 0 -semigroup. The infinitesimal generator (in short generator) of ( ) ≥0 is the linear operator : ⊃ D( ) → , which is defined on the domain and given by Note that the domain D( ) is indeed a subspace of . The following result gives some properties of the infinitesimal generator of a 0 -semigroup. Recall that we have provided the required concepts in Definitions 1 and 2.
We proceed with some examples of 0 -semigroups and their generators.
Example 15. For every bounded linear operator ∈ ( ) the family ( ) ≥0 given by is a norm continuous semigroup with generator . In particular, one has D( ) = .
Example 16. We consider the separable Hilbert space = 2 (R). Let ( ) ≥0 be the shift semigroup that is defined as Then ( ) ≥0 is a semigroup of contractions with generator : that is, arises as the convolution of with the density of the normal distribution (0, 2 ). Then ( ) ≥0 is a semigroup of contractions with generator Here, 2 (R ) denotes the Sobolev space and Δ the Laplace operator We proceed with some results regarding calculations with strongly continuous semigroups and their generators. (1) For every ∈ D( ), the mapping belongs to class 1 (R + ; ), and for all ≥ 0, one has ∈ D( ) and (2) For all ∈ and ≥ 0, one has ∫ 0 ∈ D( ) and (3) For all ∈ D( ) and ≥ 0, one has The following result shows that the strongly continuous semigroup ( ) ≥0 associated with generator is unique. This explains the term generator.  (1) The semigroup ( ) ≥0 is norm continuous.
(3) The domain of is given by D( ) = .
If the previous conditions are satisfied, then one has = for all ≥ 0. Now, we are interested in characterizing all linear operators which are the infinitesimal generator of some strongly continuous semigroup ( ) ≥0 . The following theorem of Hille-Yosida gives a characterization in terms of the resolvent, which we have introduced in Definition 3.
Recall that we have introduced the adjoint operator for operators in the Hilbert spaces in Definition 6.

Proposition 24.
Let be a Hilbert space and let ( ) ≥0 be a 0 -semigroup on with generator . Then the family of adjoint operators ( * ) ≥0 is a 0 -semigroup on with generator * .

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, (F ) ≥0 , P) denotes a filtered probability space satisfying the usual conditions. Let H be a separable Hilbert space and let ∈ (H) be a nuclear, selfadjoint, positive definite linear operator. In Definition 25, the distribution (0, ( − ) ) is a Gaussian measure with mean 0 and covariance operator ( − ) ; see, for example, [1, Section 2.3.2]. The operator is also called the covariance operator of the Wiener process . As is a trace class operator, we also call a trace class Wiener process.
Now, let be a -Wiener process. Then, there exist an orthonormal basis ( ) ∈N of H and a sequence ( ) ∈N ⊂ (0, ∞) with ∑ ∈N < ∞ such that International Journal of Stochastic Analysis 5 Namely, the are the eigenvalues of , and each is an eigenvector corresponding to . The space H 0 := 1/2 (H), equipped with the inner product is another separable Hilbert space, and (√ ) ∈N is an orthonormal basis. According to [1,Proposition 4.1], the sequence of stochastic processes ( ) ∈N defined as is a sequence of real-valued independent standard Wiener processes, and one has the expansion Now, let us briefly sketch the construction of the Itô integral with respect to the Wiener process . Further details can be found in [1,3]. We denote by 0 2 ( ) := 2 (H 0 , ) the space of Hilbert-Schmidt operators from H 0 into , endowed with the Hilbert-Schmidt norm which itself is a separable Hilbert space. The construction of the Itô integral is divided into three steps as follows.
Theorem 26 (Itô's formula). Let be another separable Hilbert space, let ∈ 1,2,loc (R + × ; ) be a function, and let be an -valued Itô process of the form Then ( ( , )) ≥0 is an -valued Itô process, and one has Palmost surely where one uses the notation Proof. This result is a consequence of [3, Theorem 2.9].

Solution Concepts for SPDEs
In this section, we present the concepts of strong, mild, and weak solutions to SPDEs and discuss their relations. Let be a separable Hilbert space, and let ( ) ≥0 be a = ( + ( , )) + ( , ) Here : R + × → and : R + × → 0 2 ( ) are measurable mappings. 6 International Journal of Stochastic Analysis Definition 27. Let ℎ 0 : Ω → be a F 0 -measurable random variable, and let > 0 be a strictly positive stopping time. Furthermore, let = (ℎ 0 ) be an -valued, continuous, adapted process such that and P-almost surely one has (2) is called a local weak solution to (44), if for all ∈ D( * ) the following equation is fulfilled P-almost surely: (3) is called a local mild solution to (44), if P-almost surely one has One calls the lifetime of . If ≡ ∞, then one calls 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 a F 0 -measurable random variable ℎ 0 : Ω → and a strictly positive stopping time > 0.
Proposition 29. Every local strong solution to (44) with lifetime is also a local weak solution to (44) with lifetime .
Proof. Let be a local strong solution to (44) with lifetime . Furthermore, let ∈ D( * ) be arbitrary. Then we have P-almost surely for all ≥ 0 the identities showing that is also a local weak solution to (44) with lifetime .
Proposition 30. Let be a stochastic process with 0 = ℎ 0 . Then the following statements are equivalent.
(1) The process is a local strong solution to (44) with lifetime .
(2)⇒(1): Let ∈ D( * ) be arbitrary. Then we have Palmost surely for all ≥ 0 the identities International Journal of Stochastic Analysis 7 By Proposition 7, the domain D( * ) is dense in , and hence we obtain P-almost surely Consequently, the process is also a local strong solution to (44) with lifetime .
Corollary 31. Let M ⊂ D( ) be a subset such that is continuous on M, and let be a local weak solution to (44) with lifetime such that Then is also a local strong solution to (44) with lifetime .
Proof. Since M ⊂ D( ), condition (54) implies that (46) is fulfilled. Moreover, by the continuity of on M, the sample paths of the process are P-almost surely continuous; and hence, we obtain (47). Consequently, using Proposition 30, the process is also a local strong solution to (44) with lifetime .
We recall the following technical auxiliary result without proof and refer, for example, to [3, Section 3.1].

International Journal of Stochastic Analysis
Since is a weak solution to (44), the R -valued process is an Itô process with representation By Itô's formula (Theorem 26), we obtain P-almost surely and hence This concludes the proof. (44) is also a mild solution to (44).

Proposition 35. Every weak solution to
Proof. By Proposition 24, the family ( * ) ≥0 is a 0 -semigroup with generator * . Thus, Proposition 23 yields that the family of restrictions . Now, let ≥ 0 and ∈ D(( * ) 2 ) be arbitrary. We define the function Since, by Proposition 14, the domain D( * ) is dense in , we obtain P-almost surely proving that is a mild solution to (44).
Remark 36. Now, the proof of Proposition 32 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. Let be a mild solution to (44) such that
Then is also a weak solution to (44).
Proof. Let ≥ 0 and ∈ D( * ) be arbitrary. Using Lemma 18, we obtain P-almost surely Consequently, the process is also a weak solution to (44).
Next, we provide conditions which ensure that a mild solution to (44) is also a strong solution.
Proof. By hypotheses (78) and (79), we have (46) and (47). Let ≥ 0 be arbitrary. By Lemma 18, we have Since is a mild solution to (44), we have P-almost surely and, hence, combining the latter identities, we obtain Palmost surely which implies that This proves that 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 ( ) ≥0 is norm continuous. Let be a stochastic process with 0 = ℎ 0 . Then the following statements are equivalent.
(1) The process is a strong solution to (44).
(2) The process is a weak solution to (44).
(3) The process is a mild solution to (44).

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). Let be a separable Banach space, and let ( ) ≥0 be a 0semigroup on . We start with the drift term.
Lemma 40. Let : R + → be a measurable mapping such that Then the mapping is continuous.
Proof. Let ∈ R + be arbitrary. It suffices to prove that is right-continuous and left-continuous in .
(1) Let ( ) ∈N ⊂ R + be a sequence such that ↓ . Then for every ∈ N we have By Lemma 12, the mapping is continuous. Thus, taking into account estimate (9) from Lemma 10, by Lebesgue's dominated convergence theorem we obtain (2) Let ( ) ∈N ⊂ R + be a sequence such that ↑ . Then for every ∈ N we have Proceeding as in the previous situation, by Lebesgue's dominated convergence theorem we obtain This completes the proof.

Proposition 41. Let be a progressively measurable process satisfying
Then the process defined as is continuous and adapted. International Proof. The continuity of is a consequence of Lemma 40. Moreover, is adapted, because is progressively measurable.
Now, we will deal with stochastic convolution integrals driven by the Wiener processes. Let be a separable Hilbert space, and let ( ) ≥0 be a 0 -semigroup on . Moreover, let be a trace class Wiener process on some separable Hilbert space H.
Definition 42. Let be a 0 2 ( )-valued predictable process such that One defines the stochastic convolution ⋆ as One recalls the following result concerning the regularity of stochastic convolutions.
Proposition 43. Let be a 0 2 ( )-valued predictable process such that one of the following two conditions is satisfied.
(1) There exists a constant > 1 such that

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 ( , ) be a metric space, and let Φ : → be a mapping.
(1) The mapping Φ is called a contraction, if for some constant 0 ≤ < 1 one has (2) An element ∈ is called a fixed point of Φ, if one has 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). Let be a complete metric space, and let Φ :
→ 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. Let be a complete metric space, and let Φ : → be a mapping such that for some ∈ N the mapping Φ is a contraction. Then the mapping Φ has a unique fixed point.
Proof. According to the Banach fixed point theorem (Theorem 45) the mapping Φ has a unique fixed point; that is, there exists a unique element ∈ such that Φ ( ) = . Therefore, we have showing that Φ( ) is a fixed point of Φ . Since Φ has a unique fixed point, we deduce that Φ( ) = , showing that is a fixed point of Φ.
In order to prove uniqueness, let ∈ be another fixed point of Φ; that is, we have Φ( ) = . By induction, we obtain showing that is a fixed point of Φ . Since the mapping Φ has exactly one fixed point, we obtain = .
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].
The following result shows that local Lipschitz continuity of and ensures the uniqueness of mild solutions to the SPDE (44).
Theorem 48. One supposes that for every ∈ N there exists a constant ≥ 0 such that for all ≥ 0 and all ℎ 1 , ℎ 2 ∈ with ‖ℎ 1 ‖, ‖ℎ 2 ‖ ≤ . Let ℎ 0 , 0 : Ω → be two F 0 -measurable random variables, let > 0 be a strictly positive stopping time, and let , be two local mild Proof. Defining the stopping times ( ) ∈N as we have P( → ) = 1. Let ∈ N and ≥ 0 be arbitrary, and set The mapping is nonnegative, and it is continuous by Lebesgue's dominated convergence theorem. For all ∈ [0, ] we have 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 ( ) Using Gronwall's inequality (see Lemma 47) we deduce that ≡ 0. Thus, by the continuity of the sample paths of and , we obtain and hence, by the continuity of the probability measure P, we conclude that 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 and . For this, we recall an auxiliary result which extends the Itô isometry (39).
Theorem 50. Suppose that there exists a constant ≥ 0 such that for all ≥ 0 and all ℎ 1 , ℎ 2 ∈ , and suppose that there exists a constant ≥ 0 such that for all ≥ 0 and all ℎ ∈ . Then, for every F 0 -measurable random variable ℎ 0 : Ω → , there exists a (up to indistinguishability) unique mild solution 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 ℎ 0 satisfies E[‖ℎ 0 ‖ 2 ] < ∞ for some > 1. Let ≥ 0 be arbitrary. We define the Banach space and prove that the variation of constants equation has a unique solution in the space 2 ( ). This is done in the following three steps.
Step 1.1. For ∈ 2 ( ) we define the process Φ by Then the process Φ is well defined. Indeed, by the growth estimate (9), the linear growth condition (127), and Hölder's inequality we have Furthermore, by the growth estimate (9), the linear growth condition (128), and Hölder's inequality we have The previous two estimates show that Φ is a well-defined mapping on 2 ( ).
Step 1.2. Next, we show that the mapping Φ maps 2 ( ) into itself; that is, we have Φ : 2 ( ) → 2 ( ). Indeed, let ∈ 2 ( ) be arbitrary. Defining the processes Φ and Φ as we have 16 International Journal of Stochastic Analysis By the growth estimate (9), we have By Hölder's inequality and the growth estimate (9), we have and hence, by the linear growth condition (127) and Hölder's inequality, we obtain Furthermore, by Lemma 49 and the growth estimate (9), we have and hence, by the linear growth condition (128) and Hölder's inequality, we obtain The previous three estimates show that Φ ∈ 2 ( ). Consequently, the mapping Φ maps 2 ( ) into itself.
Step 1.3. Now, we show that for some index ∈ N the mapping Φ is a contraction on 2 ( ). Let , ∈ 2 ( ), and ∈ [0, ] be arbitrary. By Hölder's inequality, the growth estimate (9), and the Lipschitz condition (125) we have Furthermore, by Lemma 49, the growth estimate (9), the Lipschitz condition (126), and Hölder's inequality we obtain Therefore, defining the constant by Hölder's inequality, we get Thus, by induction for every ∈ N, we obtain Consequently, there exists an index ∈ N such that Φ 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 ∈ 2 ( ). This fixed point is a solution to the variation of constants equation (130). Since ≥ 0 was arbitrary, there exists a process which is a solution of the variation of constants equation: Step 1.4. In order to prove that is a mild solution to (44), it remains to show that has a continuous version. By Lemma 12, the process is continuous, and by Proposition 41, the process is continuous, too. Moreover, for every ≥ 0, we have, by the linear growth condition (128), Hölder's inequality, and since ∈ 2 ( ), the following estimate: Thus, by Proposition 43 the stochastic convolution ⋆ given by has a continuous version, and consequently, the process has a continuous version, too. This continuous version is a mild solution to (44).
Step 2. Now let ℎ 0 : Ω → be an arbitrary F 0 -measurable random variable. We define the sequence (ℎ ) ∈N of F 0measurable random variables as Let ∈ N be arbitrary. Then, as ℎ 0 is bounded, we have E[‖ℎ 0 ‖ 2 ] < ∞ for all > 1. By Step 1 the SPDE = ( + ( , )) + ( , ) , 18 International Journal of Stochastic Analysis has a mild solution . We define the sequence (Ω ) ∈N ⊂ F 0 as Then, we have Ω ⊂ Ω for ≤ , Ω = ⋃ ∈N Ω , and Thus, by Theorem 48 we have (up to indistinguishability) Consequently, the process is a well-defined, continuous, and adapted process, and we have Furthermore, we obtain P-almost surely proving that is a mild solution to (44).
Remark 51. For the proof of Theorem 50, 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 ( ) ≥0 is a semigroup of pseudocontractions.
We close this section with a consequence about the existence of weak solutions.
Proof. According to Proposition 35, every weak solution 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 ≥ 0 be arbitrary. By Theorem 50 and its proof, there exists a mild solution ∈ 2 ( ) to (44). By the linear growth condition (128) and Hölder's inequality we obtain showing that condition (72) is fulfilled. Thus, by Proposition 37, the process is also a weak solution to (44).

Invariant Manifolds for Weak Solutions to SPDEs
In this section, we deal with invariant manifolds for timehomogeneous 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 , by means of a finite dimensional state process and thus to ensure larger analytical tractability. Our goal is to find conditions on the generator and the coefficients , 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, let be a separable Hilbert space.
Proof. Since 1 and 2 are open neighborhoods of ℎ, we have := 1 ∩ 2 ∩ M ̸ = 0. Thus, by Lemma 55 the mapping −1 is a -diffeomorphism. Using the chain rule, we obtain and, analogously, we prove that Definition 57. Let ℎ ∈ M be arbitrary. The tangent space of M to ℎ is the subspace (1) The elements 1 , . . . , are linearly independent in .
From now on, we assume that M is an -dimensional 2 -submanifold of .
The latter two identities prove the desired decomposition (170).
Recall that M denotes a finite dimensional 2submanifold of .
Definition 65. The submanifold M is called locally invariant for (176), if for every ℎ 0 ∈ M there exists a local weak solution to (176) with some lifetime > 0 such that ∧ ∈ M ∀ ≥ 0, P-almost surely.
In order to investigate local invariance of M, we will assume, from now on, that ∈ 1 ( ) for all ∈ N.
Lemma 66. The following statements are true.

International Journal of Stochastic Analysis
Now, let be the strong solution to (192) with initial condition 0 := ⟨ , ℎ 0 ⟩ ∈ . Since is open, there exists > 0 such that ( 0 ) ⊂ . We define the stopping time Since the process has continuous sample paths and satisfies 0 = 0 , we have > 0 and P-almost surely ∧ ∈ ∀ ≥ 0.
showing that is a local strong solution to (44) with lifetime .
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 point ℎ 0 , and we point out the papers [17][18][19][20][21][22] regarding this topic. Furthermore, we mention that Theorem 68 has been extended in [23] to SPDEs with jumps.