Lyapunov-type Conditions for Non-strong Ergodicity of Markov Processes

TL;DR

This paper establishes Lyapunov-type criteria to determine when certain Markov processes are not strongly ergodic, with applications to diffusion and Ornstein-Uhlenbeck processes driven by symmetric alpha-stable noise.

Contribution

It introduces new Lyapunov conditions for non-strong ergodicity applicable to a range of Markov processes, including those driven by stable noise.

Findings

Non-strong ergodicity criteria for diffusion processes on manifolds.

Analysis of Ornstein-Uhlenbeck processes with symmetric alpha-stable noise.

Independence of ergodicity from alpha in alpha-stable driven SDEs.

Abstract

We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed including diffusion processes on Riemannian manifolds and Ornstein-Uhlenbeck processes driven by symmetric $\alpha$-stable processes. For SDE driven by $\alpha$-stable process ($\alpha\in (0,2]$) with polynomial drift, the strong ergodicity or not is independent on $\alpha$.