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A note on stochastic semilinear dissipative evolution equations

Carlo Marinelli

TL;DR

This note addresses the well-posedness of stochastic semilinear dissipative evolution equations with additive noise, where the linear part $A$ generates a compact contraction semigroup and the nonlinearity $f$ is a monotone superposition operator. Existence and uniqueness of a probabilistically strong mild solution are established by approximating $f$ with Yosida regularizations, obtaining pathwise a priori estimates, and leveraging a novel compactness result for deterministic convolutions to pass to the limit. The approach avoids variational structure on $A$ and Aubin–Lions–Simon-type compactness, extending well-posedness to broader operator classes. The resulting solution $u$ decomposes into a deterministic part plus the stochastic convolution, with the remainder $v=u-S\diamond(B\cdot W)$ enjoying strong continuity in $L^1$ and weak continuity in $L^2$, and the framework accommodates quasi-monotone extensions of $A$ and $f$.

Abstract

Existence and uniqueness of mild solutions to a class of semilinear stochastic evolution equations with additive noise is proved. The linear part of the drift term is the generator of a compact semigroup of contractions, while the nonlinear part is only assumed to be the superposition operator associated to a decreasing function.

A note on stochastic semilinear dissipative evolution equations

TL;DR

This note addresses the well-posedness of stochastic semilinear dissipative evolution equations with additive noise, where the linear part generates a compact contraction semigroup and the nonlinearity is a monotone superposition operator. Existence and uniqueness of a probabilistically strong mild solution are established by approximating with Yosida regularizations, obtaining pathwise a priori estimates, and leveraging a novel compactness result for deterministic convolutions to pass to the limit. The approach avoids variational structure on and Aubin–Lions–Simon-type compactness, extending well-posedness to broader operator classes. The resulting solution decomposes into a deterministic part plus the stochastic convolution, with the remainder enjoying strong continuity in and weak continuity in , and the framework accommodates quasi-monotone extensions of and .

Abstract

Existence and uniqueness of mild solutions to a class of semilinear stochastic evolution equations with additive noise is proved. The linear part of the drift term is the generator of a compact semigroup of contractions, while the nonlinear part is only assumed to be the superposition operator associated to a decreasing function.

Paper Structure

This paper contains 10 sections, 15 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliary results
  3. Notation
  4. Convex analysis and accretive operators
  5. An energy estimate for mild solutions
  6. A uniform integrability criterion
  7. A criterion for weak sequential compactness in $L^1$
  8. Weak continuity of vector-valued functions
  9. On compactness of deterministic convolutions
  10. Main result

Key Result

Lemma 2.1

Let $S$ be a contraction semigroup on a Hilbert space $H$, $v_0 \in H$, $f \in L^1_\textnormal{loc}(\mathbb{R}_+;H)$, and $v = S v_0 + S \ast f$. Then

Theorems & Definitions (35)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 25 more