Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.