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. 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