# On Regularity of Stochastic Convolutions of Functional Linear Differential Equations with Memory

## TL;DR

This paper investigates the regularity of stochastic convolutions in linear stochastic retarded functional differential equations with unbounded operators, developing new estimates and resolvent constructions for Volterra-type equations.

## Contribution

It introduces novel estimates for fundamental solutions and constructs resolvent operators for Volterra-type integrodifferential equations, advancing understanding of stochastic convolution regularity.

## Key findings

- Established estimates on fundamental solutions for delay equations
- Constructed resolvent operators for Volterra-type equations
- Proved regularity properties of stochastic convolutions

## Abstract

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $C_0$-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.