On the equivalence of solutions for a class of stochastic evolution equations in a Banach space

TL;DR

This paper investigates the conditions under which different solution concepts for stochastic evolution equations in Banach spaces are equivalent, and applies these results to establish existence and uniqueness for delayed stochastic equations.

Contribution

It establishes equivalence conditions for generalized strong, weak, and mild solutions in Banach spaces and applies these to delay equations with additive noise.

Findings

Equivalence of solution concepts under specific conditions

Existence and uniqueness of weak solutions for stochastic delay equations

Examples include stochastic transport and McKendrick equations with delay

Abstract

We study a class of stochastic evolution equations in a Banach space $E$ driven by cylindrical Wiener process. Three different concept of solutions: generalised strong, weak and mild are defined and the conditions under which they are equivalent are given. We apply this result to prove existence, uniqueness and continuity of weak solutions to stochastic delay equation with additive noise. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic McKendrick equation with delay.