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

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.