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.
Highlights
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]
We review the relevant results from functional analysis about unbounded operators in Hilbert spaces and strongly continuous semigroups
Summary
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. 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.
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