International Journal of Stochastic AnalysisVolume 2013 (2013), Article ID 798549, 25 pageshttp://dx.doi.org/10.1155/2013/798549

Review Article

Foundations of the Theory of Semilinear Stochastic Partial Differential Equations

Stefan Tappe

Leibniz Universität Hannover, Institut für Mathematische Stochastik, Welfengarten 1, 30167 Hannover, Germany

Received 28 May 2013; Accepted 16 August 2013

Academic Editor: Hong-Kun Xu

Copyright © 2013 Stefan Tappe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 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.

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].

Let and be Banach spaces. For a linear operator , defined on some subspace of , we call the domain of .

Definition 1. A linear operator is called closed, if for every sequence , such that the limits and exist, one has and .

Definition 2. A linear operator is called densely defined, if its domain is dense in ; that is, .

Definition 3. Let be a linear operator. (1)The resolvent set of is defined as (2)The spectrum of is defined as .(3)For , one defines the resolvent as Now, we will introduce the adjoint operator of a densely defined operator in a Hilbert space. Recall that for a bounded linear operator , mapping between two Hilbert spaces and , the adjoint operator is the unique bounded linear operator 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 be a continuous linear operator. Then there exists a unique continuous extension , that is, a continuous linear operator with . Moreover, one has .

Now, let be a Hilbert space. We recall the representation theorem of Fréchet-Riesz. In the sequel, the space denotes the dual space of .

Theorem 5. For every , there exists a unique element with . In addition, one has .

Let be a densely defined operator. One defines the subspace Let 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 , and one has

Definition 6. The operator is called the adjoint operator of .

Proposition 7. Let be densely defined and closed. Then is densely defined and one has .

Lemma 8. Let be a separable Hilbert space and let be a closed operator. Then the domain endowed with the graph norm 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 be a Banach space.

Definition 9. Let be a family of continuous linear operators , . (1)The family is a called a strongly continuous semigroup (or -semigroup), if the following conditions are satisfied: (i),(ii) for all ,(iii) for all . (2)The family is called a norm continuous semigroup, if the following conditions are satisfied: (i),(ii) for all ,(iii).

Note that every norm continuous semigroup is also a -semigroup. The following growth estimate (9) will often be used when dealing with SPDEs.

Lemma 10. Let be a -semigroup. Then there are constants and such that

Definition 11. Let be a -semigroup. (1)The semigroup is called a semigroup of contractions (or contractive), if that is, the growth estimate (9) is satisfied with and .(2)The semigroup is called a semigroup of pseudocontractions (or pseudocontractive), if there exists a constant such that that is, the growth estimate (9) is satisfied with .

If is a semigroup of pseudocontractions with growth estimate (11), then given by 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 be a -semigroup. Then the following statements are true. (1)The mapping is continuous.(2)For all and , the mapping is uniformly continuous.

Definition 13. Let be a -semigroup. The infinitesimal generator (in short generator) of is the linear operator , which is defined on the domain and given by

Note that the domain is indeed a subspace of . The following result gives some properties of the infinitesimal generator of a -semigroup. Recall that we have provided the required concepts in Definitions 1 and 2.

Proposition 14. The infinitesimal generator of a -semigroup is densely defined and closed.

We proceed with some examples of -semigroups and their generators.

Example 15. For every bounded linear operator the family given by is a norm continuous semigroup with generator . In particular, one has .

Example 16. We consider the separable Hilbert space . Let be the shift semigroup that is defined as Then is a semigroup of contractions with generator given by

Example 17. On the separable Hilbert space we define the heat semigroup by and that is, arises as the convolution of with the density of the normal distribution . Then is a semigroup of contractions with generator given by Here, denotes the Sobolev space and the Laplace operator

We proceed with some results regarding calculations with strongly continuous semigroups and their generators.

Lemma 18. Let be a -semigroup with infinitesimal generator . Then the following statements are true. (1)For every , the mapping belongs to class , and for all , one has and (2)For all and , one has and (3)For all and , one has

The following result shows that the strongly continuous semigroup associated with generator is unique. This explains the term generator.

Proposition 19. Two -semigroups and with the same infinitesimal generator coincide; that is, one has for all .

The next result characterizes all norm continuous semigroups in terms of their generators.

Proposition 20. Let be a -semigroup with infinitesimal generator . Then the following statements are equivalent. (1)The semigroup is norm continuous.(2)The operator is continuous.(3)The domain of is given by . If the previous conditions are satisfied, then one has for all .

Now, we are interested in characterizing all linear operators which are the infinitesimal generator of some strongly continuous semigroup . 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). Let be a linear operator and let , be constants. Then the following statements are equivalent. (1) is the generator of a -semigroup with growth estimate (9).(2) is densely defined and closed and one has and

In particular, we obtain the following characterization of the generators of semigroups of contractions.

Corollary 22. For a linear operator the following statements are equivalent. (1) is the generator of a semigroup of contractions.(2) is densely defined and closed, and one has and

Proposition 23. Let be a -semigroup on with generator . Then the family is a -semigroup on with generator , where the domain is given by

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 be a -semigroup on with generator . Then the family of adjoint operators is a -semigroup on with generator .

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, denotes a filtered probability space satisfying the usual conditions. Let be a separable Hilbert space and let be a nuclear, self-adjoint, positive definite linear operator.

Definition 25. A -valued, adapted, continuous process is called a -Wiener process, if the following conditions are satisfied.(i)One has .(ii)The random variable and the -algebra are independent for all .(iii)One has for all .

In Definition 25, the distribution is a Gaussian measure with mean and covariance operator ; see, for example, [1, Section ]. 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 of and a sequence with such that Namely, the are the eigenvalues of , and each is an eigenvector corresponding to . The space , equipped with the inner product is another separable Hilbert space, and is an orthonormal basis. According to [1, Proposition 4.1], the sequence of stochastic processes 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 the space of Hilbert-Schmidt operators from 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.(1)For every -valued simple process of the form with and -measurable random variables for , we set (2)For every predictable -valued process satisfying we extend the Itô integral by an extension argument for linear operators. In particular, we obtain the Itô isometry as follows: (3)By localization, we extend the Itô integral for every predictable -valued process satisfying The Itô integral is an -valued, continuous, local martingale, and we have the series expansion where for each . An indispensable tool for stochastic calculus in infinite dimensions is Itô's formula, which we will recall here.

Theorem 26 (Itô's formula). Let be another separable Hilbert space, let be a function, and let be an -valued Itô process of the form Then is an -valued Itô process, and one has -almost surely where one uses the notation for each .

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.

Let be a separable Hilbert space, and let be a -semigroup on with infinitesimal generator . Furthermore, let be a trace class Wiener process on some separable Hilbert space . We consider the SPDE: Here and are measurable mappings.

Definition 27. Let be a -measurable random variable, and let be a strictly positive stopping time. Furthermore, let be an -valued, continuous, adapted process such that (1) is called a local strong solution to (44), if and -almost surely one has (2) is called a local weak solution to (44), if for all the following equation is fulfilled -almost surely: (3) is called a local mild solution to (44), if -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 -measurable random variable and a strictly positive stopping time .

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 be arbitrary. Then we have -almost surely for all the identities showing that is also a local weak solution to (44) with lifetime .

Proposition 30. Let be a stochastic process with . Then the following statements are equivalent. (1)The process is a local strong solution to (44) with lifetime .(2)The process is a local weak solution to (44) with lifetime , and one has (46), (47).

Proof. (1)(2): This implication is a direct consequence of Proposition 29.(2)(1): Let be arbitrary. Then we have -almost surely for all the identities By Proposition 7, the domain is dense in , and hence we obtain -almost surely Consequently, the process is also a local strong solution to (44) with lifetime .

Corollary 31. Let be a subset such that is continuous on , 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 , condition (54) implies that (46) is fulfilled. Moreover, by the continuity of on , the sample paths of the process are -almost surely continuous; and hence, we obtain (47). Consequently, using Proposition 30, the process is also a local strong solution to (44) with lifetime .

Proposition 32. Every strong solution to (44) is also a mild solution to (44).

Proof. According to Lemma 8, the domain endowed with the graph norm is a separable Hilbert space, too. Hence, by Lemma 18, for all , the function belongs to the class with partial derivatives Hence, by Itô's formula (see Theorem 26) and Lemma 18, we obtain -almost surely: Thus, 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. Let be arbitrary. Then the linear space is dense in , where is endowed with the graph norm.

Lemma 34. Let be a weak solution to (44). Then for all and all , one has -almost surely

Proof. By virtue of Lemma 33, it suffices to prove formula (59) for all . Let be arbitrary. Then there are and for some such that We define the function Then, we have with partial derivatives Since is a weak solution to (44), the -valued process is an Itô process with representation By Itô's formula (Theorem 26), we obtain -almost surely and hence This concludes the proof.

Proposition 35. Every weak solution to (44) is also a mild solution to (44).

Proof. By Proposition 24, the family is a -semigroup with generator . Thus, Proposition 23 yields that the family of restrictions is a -semigroup on with generator .Now, let and be arbitrary. We define the function By Lemma 18 we have with derivative Using Lemma 34, we obtain -almost surely Since, by Proposition 14, the domain is dense in , we get -almost surely for all the identity Since, by Proposition 14, the domain is dense in , we obtain -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 and be arbitrary. Using Lemma 18, we obtain -almost surely By Fubini's theorem for Bochner integrals (see [3, Section 1.1, page 21]) and Lemma 18. we obtain -almost surely 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 -almost surely Therefore, and since is a mild solution to (44), we obtain -almost surely and henceConsequently, 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.

Proposition 38. Let be a mild solution to (44) such that -almost surely one has as well as Then is also a strong solution to (44).

Proof. By hypotheses (78) and (79), we have (46) and (47). Let be arbitrary. By Lemma 18, we have Furthermore, by Lemma 18 and Fubini’s theorem for Bochner integrals (see [3, Section 1.1, page 21]) we have -almost surely 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 -almost surely Since is a mild solution to (44), we have -almost surely and, hence, combining the latter identities, we obtain -almost surely which implies thatThis 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 is norm continuous. Let be a stochastic process with . 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).

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 and , . Furthermore, the family is a -group on . Therefore, and since is a mild solution to (44), we have -almost surely Let be the Itô process: Then, we have -almost surely and, by Lemma 18, we have Defining the function by Lemma 18, we have with partial derivatives By Itô's formula (Theorem 26), we get -almost surely Combining the previous identities, we obtain -almost surely proving that 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).

Let be a separable Banach space, and let be a -semigroup on . We start with the drift term.

Lemma 40. Let be a measurable mapping such that Then the mapping is continuous.

Proof. Let be arbitrary. It suffices to prove that is right-continuous and left-continuous in . (1)Let be a sequence such that . Then for every 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 be a sequence such that . Then for every 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.

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 be a -semigroup on . Moreover, let be a trace class Wiener process on some separable Hilbert space .

Definition 42. Let be a -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 -valued predictable process such that one of the following two conditions is satisfied. (1)There exists a constant such that (2)The semigroup is a semigroup of pseudocontractions, and one has Then the stochastic convolution 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 be a metric space, and let be a mapping. (1)The mapping is called a contraction, if for some constant one has (2)An element is called a fixed point of , 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 be a contraction. Then the mapping 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 be a mapping such that for some the mapping is a contraction. Then the mapping has a unique fixed point.

Proof. According to the Banach fixed point theorem (Theorem 45) the mapping has a unique fixed point; that is, there exists a unique element such that . Therefore, we have showing that is a fixed point of . Since has a unique fixed point, we deduce that , showing that is a fixed point of .In order to prove uniqueness, let be another fixed point of ; that is, we have . By induction, we obtain showing that is a fixed point of . Since the mapping 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].

Lemma 47 (Gronwall's inequality). Let be fixed, let be a nonnegative continuous mapping, and let be a constant such that Then one has .

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 there exists a constant such that for all and all with . Let be two -measurable random variables, let be a strictly positive stopping time, and let , be two local mild solutions to (44) with initial conditions , and lifetime . Then one has up to indistinguishability (Two processes and are called indistinguishable if the set for some is a -nullset.)

Proof. Defining the stopping times as we have . Let and be arbitrary, and set The mapping is nonnegative, and it is continuous by Lebesgue's dominated convergence theorem. For all 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 . Thus, by the continuity of the sample paths of and , we obtain and hence, by the continuity of the probability measure , 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).

Lemma 49. Let be arbitrary, and let be a -valued, predictable process such that Then, for every one has where the constant is given by

Proof. See [3, Lemma 3.1].

Theorem 50. Suppose that there exists a constant such that for all and all , and suppose that there exists a constant such that for all and all . Then, for every -measurable random variable , 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 satisfies for some . Let be arbitrary. We define the Banach space and prove that the variation of constants equation has a unique solution in the space . This is done in the following three steps.Step  1.1. For we define the process by Then the process 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 is a well-defined mapping on . Step  1.2. Next, we show that the mapping maps into itself; that is, we have . Indeed, let be arbitrary. Defining the processes and as we have 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 . Consequently, the mapping maps into itself.Step  1.3. Now, we show that for some index the mapping is a contraction on . Let , and 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 , we obtain Consequently, there exists an index such that is a contraction, and hence, according to the extension of the Banach fixed point theorem (see Corollary 46), the mapping has a unique fixed point . This fixed point is a solution to the variation of constants equation (130). Since 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 , we have, by the linear growth condition (128), Hölder's inequality, and since , 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 be an arbitrary -measurable random variable. We define the sequence of -measurable random variables as Let be arbitrary. Then, as is bounded, we have for all . By Step 1 the SPDE has a mild solution . We define the sequence as Then, we have for , , 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 -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 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 be a -measurable random variable such that for some . Then there exists a (up to indistinguishability) unique weak solution to (44).

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 be arbitrary. By Theorem 50 and its proof, there exists a mild solution 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).

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 , 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.

Definition 54. Let be positive integers. A subset is called an -dimensional -submanifold of , if for every there exist an open neighborhood of , an open set , and a mapping such that(1)the mapping is a homeomorphism;(2)for all the mapping is injective. The mapping is called a parametrization of around .

In what follows, let be an -dimensional -submanifold of .

Lemma 55. Let , be two parametrizations with . Then the mapping is a -diffeomorphism.

Proof. See [16, Lemma ].

Corollary 56. Let be arbitrary, and let , be two parametrizations of around . Then one has where for .

Proof. Since and are open neighborhoods of , we have . Thus, by Lemma 55 the mapping is a -diffeomorphism. Using the chain rule, we obtain and, analogously, we prove that .

Definition 57. Let be arbitrary. The tangent space of to is the subspace where and denotes a parametrization of around .

Remark 58. Note that, according to Corollary 56, the Definition 57 of the tangent space does not depend on the choice of the parametrization .

Proposition 59. Let be arbitrary, and let be a parametrization of around . Then there exist an open set , an open neighborhood of , and a mapping with such that is a parametrization of around , too.

Proof. See [16, Remark ].

Remark 60. By Proposition 59 we may assume that any parametrization has an extension .

Proposition 61. Let be a dense subset. For every there exist and a parametrization around such that where one uses the notation .

Proof. See [16, Proposition ]. 

Proposition 62. Let be a parametrization as in Proposition 61. Then the following statements are true. (1)The elements are linearly independent in .(2)For every , one has the direct sum decomposition (3)For every the mapping  is the corresponding projection according to (166) from onto , that is, we have

Proof. See [16, Lemma ].

From now on, we assume that is an -dimensional -submanifold of .

Proposition 63. Let be a parametrization as in Proposition 61. Furthermore, let be a mapping such that Then, for every the direct sum decomposition of according to (166) is given by where .

Proof. Since is an open subset of , there exists such that Therefore, the curve is well defined, and we have with and . Hence, we have Moreover, by condition (169) and Proposition 62, we have The latter two identities prove the desired decomposition (170).

After these preliminaries, we will study invariant manifolds for time-homogeneous SPDEs of the form with measurable mappings and . As in the previous sections, the operator is the infinitesimal generator of a -semigroup on . Note that, by (41), the SPDE (175) can be rewritten equivalently as where denotes the sequence of real-valued independent standard Wiener processes defined in (33) and the mappings , are given by .

For the rest of this section, we assume that there exist a constant such that and a sequence with such that for every we have

Proposition 64. For every there exists a (up to indistinguishability) unique weak solution to (176).

Proof. By (178), for all we have Moreover, by (177), for every we have and, by (179) we obtain Therefore, conditions (125)–(128) are fulfilled, and hence, applying Corollary 53 completes the proof.

Recall that denotes a finite dimensional -submanifold of .

Definition 65. The submanifold is called locally invariant for (176), if for every there exists a local weak solution to (176) with some lifetime such that

In order to investigate local invariance of , we will assume, from now on, that for all .

Lemma 66. The following statements are true. (1)For every one has (2)The mapping is continuous.

Proof. By (178) and (179), for every we have showing (184). Moreover, for every the mapping is continuous, because for all we have Let be the counting measure on , which is given by for all . Then we have Hence, because of the estimate the continuity of the mapping (185) is a consequence of Lebesgue's dominated convergence theorem.

For a mapping and elements we define the mappings and , as

Proposition 67. Let and be arbitrary. Then, for every there exists a (up to indistinguishability) unique strong solution to the SDE:

Proof. By virtue of the assumption and (177)–(179), there exist a constant such that and a sequence with such that for every we have Therefore, by Proposition 64, for every 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 submanifold is locally invariant for (176).(2)One has (3)The operator is continuous on , and for each there exists a local strong solution to (176) with some lifetime such that

Proof. (1)(2): Let be arbitrary. By Proposition 61 and Remark 60 there exist elements and a parametrization around such that the inverse is given by , and has an extension . Since the submanifold is locally invariant for (176), there exists a local weak solution to (176) with initial condition and some lifetime such that Since is an open neighborhood of , there exists such that , where denotes the open ball: We define the stopping time Since the process has continuous sample paths and satisfies , we have and -almost surely Defining the -valued process we have -almost surely Moreover, since is a weak solution to (176) with initial condition , setting we have -almost surely showing that is a local strong solution to (192) with initial condition . By Itô's formula (Theorem 26) we obtain -almost surely Now, let be arbitrary. Then we have -almost surely On the other hand, since is a local weak solution to (176) with initial condition and lifetime , we have -almost surely for all the identity Combining (206) and (207) yields up to indistinguishability where the processes and are defined as The process is a continuous semimartingale with canonical decomposition (208). Since the canonical decomposition of a continuous semimartingale is unique up to indistinguishability, we deduce that up to indistinguishability. Using the Itô isometry (39) we obtain -almost surely By the continuity of the processes and we obtain for all the following identities: Consequently, the mapping is continuous on , and hence we have by the definition of the domain provided in (4). By Proposition 7 we have , and thus we obtain , proving (195). By Proposition 7, the domain is dense in , and thus showing (196). Moreover, for all we have Since the domain is dense in , together with Proposition 63, we obtain which proves (197).(2)(1): Let be arbitrary. By Proposition 61 and Remark 60 there exist and a parametrization around such that the inverse is given by , and has an extension . Let be arbitrary, and set . By relations (195), (197) and Proposition 62, we obtain and thus Together with Lemma 66, this proves the continuity of on . Since was arbitrary, this proves that is continuous on .Furthermore, by (196) and Proposition 62 we have Moreover, by (195), (197), and Propositions 62 and 63, we obtain This gives us Now, let be the strong solution to (192) with initial condition . Since is open, there exists such that . We define the stopping time Since the process has continuous sample paths and satisfies , we have and -almost surely Therefore, defining the -valued process , we have -almost surely Moreover, using Itô's formula (Theorem 26) and incorporating (217), (219), we obtain -almost surely showing that is a local strong solution to (44) with lifetime .(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 point , 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.

Acknowledgments

The author is grateful to Daniel Gaigall, Georg Grafendorfer, Florian Modler, and Thomas Salfeld for the valuable comments and discussions.

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