On Regularity Property of Retarded Ornstein-Uhlenbeck Processes in Hilbert Spaces

TL;DR

This paper investigates the regularity properties of solutions to stochastic linear functional differential equations with delays in Hilbert spaces, focusing on continuity, spatial and temporal regularity, and inequalities for stochastic convolutions.

Contribution

It introduces a class of stochastic convolutions via retarded fundamental solutions and analyzes their regularity and approximation properties, including a retarded Burkholder-Davis-Gundy inequality.

Findings

Conditions for continuous modifications of mild solutions.

Analysis of temporal and spatial regularity using Yosida approximations.

Establishment of a retarded Burkholder-Davis-Gundy inequality.

Abstract

In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we introduce a class of stochastic convolutions which naturally arise in the solutions and investigate their Yosida approximants. By means of the retarded fundamental solutions, we find conditions under which each mild solution permits a continuous modification. With the aid of Yosida approximation, we study two kinds of regularity properties, temporal and spatial ones, for the retarded solution processes. By employing a factorization method, we establish a retarded version of Burkholder-Davis-Gundy's inequality for stochastic convolutions.