# American Institute of Mathematical Sciences

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April  2020, 25(4): 1279-1298. doi: 10.3934/dcdsb.2019220

## On regularity of stochastic convolutions of functional linear differential equations with memory

 a. School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China b. Department of Mathematical Sciences, School of Physical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

The author is grateful to the Tianjin Thousand Talents Plan for its financial support

Received  April 2019 Published  September 2019

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.

Citation: Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220
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