Vector-valued stochastic delay equations - a weak solution and its Markovian representation

TL;DR

This paper studies stochastic delay equations in Banach spaces, establishing conditions for weak and strong solutions to be equivalent, and reformulating solutions as Markov processes to analyze their existence and continuity.

Contribution

It introduces an evolution equation approach in a Banach space to represent solutions as Markov processes, enabling new analysis of stochastic delay equations.

Findings

Weak and strong solutions are equivalent under certain conditions.

Solutions can be reformulated as Markov processes in an extended space.

Existence and continuity of solutions are established.

Abstract

A class of stochastic delay equations in Banach space $E$ driven by cylindrical Wiener process is studied. We investigate two concepts of solutions: weak and generalised strong, and give conditions under which they are equivalent. We present an evolution equation approach in a Banach space $\Ep{p}:=E\times L^p(-1,0;E)$ proving that the solutions can be reformulated as $\Ep{p}$-valued Markov processes. Based on the Markovian representation we prove the existence and continuity of the solutions. The results are applied to stochastic delay partial differential equations with an application to neutral networks and population dynamics.