Time regularity of stochastic convolutions and stochastic evolution equations in duals of nuclear spaces

TL;DR

This paper establishes conditions for the time regularity of stochastic convolutions and solutions to stochastic evolution equations in duals of nuclear spaces, which include spaces of distributions and smooth functions.

Contribution

It introduces sufficient conditions for time regularity of stochastic convolutions in duals of nuclear spaces and applies these results to stochastic evolution equations.

Findings

Established regularity conditions for stochastic convolutions in dual nuclear spaces.

Applied regularity results to stochastic evolution equations in duals of nuclear spaces.

Extended understanding of stochastic processes in infinite-dimensional distribution spaces.

Abstract

Let $\Phi$ a locally convex space and $\Psi$ be a quasi-complete, bornological, nuclear space (like spaces of smooth functions and distributions) with dual spaces $\Phi'$ and $\Psi'$. In this work we introduce sufficient conditions for time regularity properties of the $\Psi'$-valued stochastic convolution $\int_{0}^{t} \int_{U} S(t-r)'R(r,u) M(dr,du)$, $t \in [0,T]$, where $(S(t): t \geq 0)$ is a $C_{0}$-semigroup on $\Psi$, $R(r,\omega,u)$ is a suitable operator form $\Phi'$ into $\Psi'$ and $M$ is a cylindrical-martingale valued measure on $\Phi'$. Our result is latter applied to study time regularity of solutions to $\Psi'$-valued stochastic evolutions equations.