$\Phi$-Entropy Inequality and Invariant Probability Measure for SDEs with Jump

TL;DR

This paper establishes $\

Contribution

It introduces a $\

Findings

Proves a $\

Demonstrates the existence of invariant measures for certain SDEs

Provides sharp conditions for invariant measure existence

Abstract

By using the $\Phi$-entropy inequality derived in \cite{Wu, Ch} for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump L\'evy processes. The semigroup $\Phi$-entropy inequality for SDEs driven by Poisson point processes as well as a sharp result on the existence of invariant probability measures are also presented.