Asymptotic and spectral properties of exponentially \phi-ergodic Markov processes

TL;DR

This paper establishes new relations between ergodic rates, L_p convergence, and tail probabilities for hitting times in Markov processes, providing conditions applicable even to time-irreversible processes like Levy-driven Ornstein-Uhlenbeck processes.

Contribution

It introduces exponential -coupling conditions that connect local mixing and recurrence to convergence rates and spectral properties, extending ergodic theory to broader classes of Markov processes.

Findings

Derived new relations between ergodic rate and tail probabilities.

Established sufficient conditions for spectral gap in Levy-driven Ornstein-Uhlenbeck processes.

Extended applicability to time-irreversible Markov processes.

Abstract

New relations between ergodic rate, L_p convergence rates, and asymptotic behavior of tail probabilities for hitting times of a time homogeneous Markov process are established. For L_p convergence rates and related spectral and functional properties (spectral gap and Poincare inequality) sufficient conditions are given in the terms of an exponential \phi-coupling. This provides sufficient conditions for L_p convergence rates in the terms of appropriate combination of `local mixing' and `recurrence' conditions on the initial process, typical in the ergodic theory of Markov processes. The range of application of the approach includes time-irreversible processes. In particular, sufficient conditions for spectral gap property for Levy driven Ornstein-Uhlenbeck process are established.