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

By using the $\mathrm{\Phi }$-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 $\mathrm{\Phi }$-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.