Continuous Dependence on Coefficients for Stochastic Evolution Equations with Multiplicative L\'evy Noise and Monotone Nonlinearity

TL;DR

This paper proves the continuous dependence of solutions to semilinear stochastic evolution equations with multiplicative Lévy noise and monotone nonlinear drift, without coercivity conditions, and explores implications like stability, approximation, and Markov properties.

Contribution

It establishes the continuous dependence of solutions on initial data and coefficients for a broad class of stochastic evolution equations without coercivity assumptions, a novel result in the field.

Findings

Solutions depend continuously on initial conditions and coefficients.

Yosida approximations converge to the solutions.

Solutions possess the Markov property.

Abstract

Semilinear stochastic evolution equations with multiplicative L\'evy noise and monotone nonlinear drift are considered. Unlike other similar works, we do not impose coercivity conditions on coefficients. We establish the continuous dependence of the mild solution with respect to initial conditions and also on coefficients which as far as we know, has not been proved before. As corollaries of the continuity result, we derive sufficient conditions for asymptotic stability of the solutions, we show that Yosida approximations converge to the solution and we prove that solutions have Markov property. Examples on stochastic partial differential equations and stochastic delay differential equations are provided to demonstrate the theory developed. The main tool in our study is an inequality which gives a pathwise bound for the norm of stochastic convolution integrals.