Infinitely delayed stochastic evolution equations in UMD Banach spaces

TL;DR

This paper establishes existence and uniqueness for a class of infinitely delayed stochastic evolution equations in UMD Banach spaces, extending the theory of stochastic equations to include infinite delays and analytic semigroup generators.

Contribution

It introduces a framework for solving infinitely delayed stochastic evolution equations in UMD Banach spaces, building on recent advances in the theory of stochastic equations in such spaces.

Findings

Proved existence and uniqueness of solutions.

Extended stochastic evolution theory to include infinite delays.

Applied to equations with analytic semigroup generators.

Abstract

We prove an existence and uniqueness result for the infinitely delayed stochastic evolution equation $$dU(t) = &\big(AU(t) + F(t,U_t)\big) dt + B(t,U_t)dW_H(t), t\in[0,T_0]$$ where $A$ is the generator of an analytic semigroup on a UMD space $E$, $F$ and $B$ satisfy Lipschitz conditions and $\mathscr{B}$ is a weighted $L^p$ history space. This paper is based on recent work of van Neerven \emph{et al.}~which developed the theory of abstract stochastic evolution equations in UMD spaces.