Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

TL;DR

This paper presents simple, constructive proofs of Harris-type theorems for stochastic semigroups, covering both exponential and subexponential convergence to equilibrium, with new estimates and unified methods.

Contribution

It introduces a unified, semigroup-based approach for Harris-type theorems applicable to both geometric and subgeometric convergence, including new estimates and simplified proofs.

Findings

Constructive estimates for subgeometric convergence.

Unified proof techniques for exponential and subexponential cases.

New simple proofs for geometric convergence.

Abstract

We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.