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

TL;DR

This paper investigates the existence and uniqueness of solutions to stochastic evolution inclusions driven by random measures in separable Banach spaces, using nonlinear semigroup theory without restrictive assumptions.

Contribution

It introduces a framework for strong and mild solutions to stochastic inclusions with multi-valued operators, providing new existence, uniqueness criteria, and solution representations.

Findings

Established criteria for existence and uniqueness of solutions.

Derived a representation formula for solutions.

Applied nonlinear semigroup theory in a general Banach space setting.

Abstract

The purpose of this paper is to study stochastic evolution inclusions of the form \begin{align*} \eta(t,z) N_{\Theta}(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.