Exponential ergodicity for stochastic equations of nonnegative processes with jumps

TL;DR

This paper proves exponential ergodicity in Wasserstein and total variation distances for a class of nonnegative jump processes, including complex continuous-state branching processes, under certain dissipativity and moment conditions.

Contribution

It establishes new exponential ergodicity results for stochastic equations with jumps, applicable to a broad class of processes like CBI processes with nonlinear mechanisms.

Findings

Exponential ergodicity in Wasserstein distance under comparison and dissipativity conditions.

Exponential ergodicity in total variation for subcritical CBI processes with moment conditions.

Applicable to processes in Lévy random environments.

Abstract

In this work, we study ergodicity of continuous time Markov processes on state space $\mathbb{R}_{\geq 0} := [0,\infty)$ obtained as unique strong solutions to stochastic equations with jumps. Our first main result establishes exponential ergodicity in the Wasserstein distance, provided the stochastic equation satisfies a comparison principle and the drift is dissipative. In particular, it is applicable to continuous-state branching processes with immigration (shorted as CBI processes), possibly with nonlinear branching mechanisms or in L\'evy random environments. Our second main result establishes exponential ergodicity in total variation distance for subcritical CBI processes under a first moment condition on the jump measure for branching and a $\log$-moment condition on the jump measure for immigration.