A Maximal Inequality for $p$th Power of Stochastic Convolution Integrals

TL;DR

This paper establishes a pathwise inequality for the $p$th power of stochastic convolution integrals in Hilbert spaces, enabling new results on existence, uniqueness, and stability of solutions to stochastic evolution equations with Lévy noise.

Contribution

It introduces a stronger, pathwise inequality for stochastic convolution integrals, improving upon previous expectation-based inequalities and applying it to semilinear stochastic evolution equations.

Findings

Proved a pathwise inequality for stochastic convolution integrals.

Established existence and uniqueness of solutions in $L^p$ for equations with Lévy noise.

Derived conditions for exponential stability of solutions.

Abstract

An inequality for the $p$th power of the norm of a stochastic convolution integral in a Hilbert space is proved. The inequality is stronger than analogues inequalities in the Literature in the sense that it is pathwise and not in expectation. An application of this inequality is provided for the semilinear stochastic evolution equations with L\'evy noise and monotone nonlinear drift. 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.