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

Abstract

We provide simple and constructive proofs of Harris-type the-orems 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 geomet-ric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive esti-mates 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.