Ergodic convergence rates for time-changed symmetric L\'{e}vy processes in dimension one

TL;DR

This paper establishes lower bounds for ergodic convergence rates of time-changed symmetric Lévy processes in one dimension, providing explicit conditions for exponential and strong ergodicity using harmonic functions and reversible measures.

Contribution

It introduces new lower bounds for convergence rates and explicit conditions for ergodicity in time-changed symmetric Lévy processes, expanding understanding of their long-term behavior.

Findings

Derived lower bounds for spectral gaps and convergence rates

Provided explicit conditions for exponential ergodicity

Presented examples illustrating the theoretical results

Abstract

We obtain the lower bounds for ergodic convergence rates, including spectral gaps and convergence rates in strong ergodicity for time-changed symmetric L\'{e}vy processes by using harmonic function and reversible measure. As direct applications, explicit sufficient conditions for exponential and strong ergodicity are given. Some examples are also presented.