#### 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