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

Let Φ a locally convex space and Ψ be a quasi-complete, bornological, nuclear space (like spaces of smooth functions and distributions) with dual spaces Φ ′ and Ψ ′ . In this work we introduce sufficient conditions for time regularity properties of the Ψ ′ -valued stochastic convolution R t 0 R U S ( t − r ) ′ R ( r, u ) M ( dr, du ), t ∈ [0 , T ], where ( S ( t ) : t ≥ 0) is a C 0 -semigroup on Ψ, R ( r, ω, u ) is a suitable operator form Φ ′ into Ψ ′ and M is a cylindrical-martingale valued measure on Φ ′ . Our result is latter applied to study time regularity of solutions to Ψ ′ -valued stochastic evolutions equations. 2020 Mathematics Subject Classification: 60G17, 60H05, 60H15, 60G20.