Subgeometric rates of convergence of f-ergodic strong Markov processes

TL;DR

This paper establishes conditions under which strong Markov processes converge to their stationary distribution at subgeometric rates, using supermartingale properties and drift inequalities, with applications to various stochastic models.

Contribution

It introduces a new supermartingale-based condition for f-ergodicity at subgeometric rates and provides equivalent formulations involving drift inequalities, extending to applications in several stochastic processes.

Findings

Derived conditions for subgeometric convergence rates.

Established equivalences with drift inequalities and resolvent kernel properties.

Applied results to stochastic differential equations and damping Hamiltonian systems.

Abstract

We provide a condition for f-ergodicity of strong Markov processes at a subgeometric rate. This condition is couched in terms of a supermartingale property for a functional of the Markov process. Equivalent formulations in terms of a drift inequality on the extended generator and on the resolvent kernel are given. Results related to (f,r)-regularity and to moderate deviation principle for integral (bounded) functional are also derived. Applications to specific processes are considered, including elliptic stochastic differential equation, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian system and storage models.