On ergodic properties of some Levy-type processes

TL;DR

This paper establishes sufficient conditions for the ergodicity of Levy-type processes with polynomial or sub-exponential tail decay, using the Foster-Lyapunov method to analyze their generators.

Contribution

It provides new ergodicity criteria for Levy-type processes with specific tail behaviors, expanding understanding of their long-term stability.

Findings

Ergodicity conditions for polynomial tail decay

Ergodicity conditions for (sub)-exponential tail decay

Application of Foster-Lyapunov approach to Levy-type processes

Abstract

In this note we prove some sufficient conditions for ergodicity of a Levy-type process, such that on the test functions the generator of the respective semigroup is of the form $$ Lf(x) = a(x)f'(x) + \int_{\mathbb{R}}{ \left( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb{I}_{|u|\leq 1} \right) \nu(x,du)}, \quad f\in C_{\infty}^{2}(\mathbb{R}). $$ Here $\nu(x,du)$ is a Levy-type kernel and $a(\cdot): \mathbb{R}\to \mathbb{R}$. We consider the case when the tails are of polynomial decay as well as the case when the decay is (sub)-exponential. For the proof the Foster-Lyapunov approach is used.