Time regularity of stochastic convolutions and stochastic evolution equations in duals of nuclear 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 ${\Phi }^{\prime }$ and ${\Psi }^{\prime }$. In this work we introduce sufficient conditions for time regularity properties of the ${\Psi }^{\prime }$-valued stochastic convolution ${\int }_0^t{\int }_{U}S(t-r{)}^{\prime }R(r,u)M(dr,du)$, $t\in [0,T]$, where $(S(t):t\ge 0)$ is a $C_0$-semigroup on $\Psi$, $R(r,\omega ,u)$ is a suitable operator form ${\Phi }^{\prime }$ into ${\Psi }^{\prime }$ and $M$ is a cylindrical-martingale valued measure on ${\Phi }^{\prime }$. Our result is latter applied to study time regularity of solutions to ${\Psi }^{\prime }$-valued stochastic evolutions equations.

References in corpus (4)

• Stochastic Integration and Stochastic PDEs Driven by Jumps on the Dual of a Nuclear Space
• Existence of Continuous and Càdlàg Versions for Cylindrical Processes in the Dual of a Nuclear Space
• Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space
• Stochastic Evolution Equations with Lévy Noise in the Dual of a Nuclear Space

Cited by in corpus (1)

• Convergence Uniform on Compacts in Probability with Applications to Stochastic Analysis in Duals of Nuclear Spaces