Abstract Cauchy Problems in separable Banach Spaces driven by random Measures: Existence and Uniqueness

Abstract

The purpose of this paper is to study stochastic evolution inclusions of the form \begin{align*} η(t,z) N_Θ(dt \otimes z)\in dX(t)+\mathcal{A} X(t)dt, \end{align*} where $\mathcal{A}$ is a multi-valued operator acting on a separable Banach space and $N_{\Theta }$ is the counting measure induced by a point process $\Theta$. Firstly, we will set up the concepts of strong and mild solutions; then we will derive existence as well as uniqueness criteria for these kinds of solutions and give a representation formula for the solutions. The results will be formulated by means of nonlinear semigroup theory and except for separability, no assumptions on the underlying Banach space are required.

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