Stochastic Evolution Equations with L\'{e}vy Noise in the Dual of a Nuclear Space

TL;DR

This paper establishes conditions for solutions to stochastic evolution equations driven by Lévy noise in the dual of a nuclear space, developing a stochastic integration theory and analyzing solution properties.

Contribution

It introduces a theory of stochastic integration for Lévy processes in the dual of a nuclear space and provides existence, regularity, and convergence results for solutions.

Findings

Existence of weak and mild solutions under certain conditions

Solutions possess square moments and exhibit the Markov property

Established criteria for weak convergence of solution sequences

Abstract

In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) L\'{e}vy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a L\'{e}vy process taking values in the dual of a nuclear space. We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. In the final part of the paper we give sufficient conditions for the weak convergence of the solutions to a sequence of stochastic evolution equations with L\'{e}vy noises.