On uniqueness of mild solutions for dissipative stochastic evolution equations

TL;DR

This paper establishes sufficient conditions for the uniqueness of mild solutions in a broad class of semilinear stochastic evolution equations, addressing a gap in the formal reduction from mild to strong solutions.

Contribution

It provides rigorous criteria for the uniqueness of mild solutions, justifying the commonly assumed reduction from mild to strong solutions in stochastic evolution equations.

Findings

Sufficient conditions for mild solution uniqueness are identified.

The reduction from mild to strong solutions is justified under these conditions.

Applicability to a broad class of semilinear stochastic equations.

Abstract

In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.