Well-posedness for Stochastic Evolution Equations with Monotone Non-linearity and Multiplicative Poisson Noise in $L^p$

TL;DR

This paper establishes existence, uniqueness, and stability conditions for solutions to stochastic evolution equations with monotone nonlinearities and Poisson noise in an $L^p$ setting, using an Itô-type inequality.

Contribution

It introduces a new approach to prove well-posedness and stability for stochastic equations with Lévy noise and monotone drift in $L^p$, expanding the theoretical framework.

Findings

Proved existence and uniqueness of mild solutions in $L^p$

Derived a sufficient condition for exponential stability

Developed an Itô-type inequality for stochastic convolution integrals

Abstract

Semilinear stochastic evolution equations with L\'evy noise and monotone nonlinear drift are considered. The existence and uniqueness of the mild solutions in $L^p$ for these equations is proved and a sufficient condition for exponential asymptotic stability of the solutions is derived. The main tool in our study is an It\^o type inequality for the $p$th power of stochastic convolution integrals in Hilbert spaces.