Exponential ergodicity for stochastic equations of nonnegative processes with jumps

Abstract

We study the long-time behavior for continuous-time Markov processes on the state space R 0 := [0, ), which arise as unique strong solutions to stochastic equations with jumps.We establish, under a global dissipativity condition combined with a comparison principle, exponential ergodicity in various Wasserstein distances on R 0 .Our main emphasis lies on the derivation of these estimates under minimal moment conditions to be imposed on the associated Lvy measures of the noises.We apply our method to continuous-state branching processes with immigration (shorted as CBI processes), to nonlinear CBI processes, and finally to CBI processes in Lvy random environments.