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

Let Phi be a locally convex space and let 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 integral(t)(0)integral(S)(U)(t - r)'R(r, u)M(dr, du), t is an element of [0, T], where (S(t)' : t >= 0) is the dual semigroup to a C-0-semigroup (S(t) : t >= 0) 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.