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Foundations of the theory of semilinear stochastic partial differential equations

Stefan Tappe

TL;DR

This review consolidates the functional-analytic foundations of semilinear SPDEs, uniting unbounded operator theory, $C_0$-semigroups, and Hilbert-space stochastic calculus to formalize solution concepts and existence results. It clarifies the relationships among strong, weak, and mild solutions for SPDEs of the form $dX_t=(A X_t+\alpha(t,X_t))dt+\sigma(t,X_t)dW_t$, proving a standard existence–uniqueness result via a Banach fixed point framework and exploring regularity of stochastic convolutions. The article further develops invariant-manifold theory for weak solutions, providing finite-dimensional reductions through parametrization and Itô calculus, thereby linking infinite-dimensional dynamics to tractable finite-dimensional models. Together, these results establish a coherent, self-contained foundation for the general theory of SPDEs and point to future directions such as martingale solutions, SPDEs with jumps, and invariance results.

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.

Foundations of the theory of semilinear stochastic partial differential equations

TL;DR

This review consolidates the functional-analytic foundations of semilinear SPDEs, uniting unbounded operator theory, -semigroups, and Hilbert-space stochastic calculus to formalize solution concepts and existence results. It clarifies the relationships among strong, weak, and mild solutions for SPDEs of the form , proving a standard existence–uniqueness result via a Banach fixed point framework and exploring regularity of stochastic convolutions. The article further develops invariant-manifold theory for weak solutions, providing finite-dimensional reductions through parametrization and Itô calculus, thereby linking infinite-dimensional dynamics to tractable finite-dimensional models. Together, these results establish a coherent, self-contained foundation for the general theory of SPDEs and point to future directions such as martingale solutions, SPDEs with jumps, and invariance results.

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.

Paper Structure

This paper contains 8 sections, 45 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Unbounded operators in Hilbert spaces
  3. Strongly continuous semigroups
  4. Stochastic processes in infinite dimension
  5. Solution concepts for SPDEs
  6. Stochastic convolution integrals
  7. Existence- and uniqueness results for SPDEs
  8. Invariant manifolds for weak solutions to SPDEs

Key Result

Proposition 2.4

Let $X$ be a normed space, let $Y$ be a Banach space, let $D \subset X$ be a dense subspace and let $\Phi : D \rightarrow Y$ be a continuous linear operator. Then there exists a unique continuous extension $\hat{\Phi} : X \rightarrow Y$, that is, a continuous linear operator with $\hat{\Phi}|_{D} =

Theorems & Definitions (96)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Definition 3.1
  • Lemma 3.2
  • ...and 86 more