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

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 discretetime 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.

Quantitative rates of convergence to equilibrium for the degenerate linear Boltzmann equation on the torus

“…Now we want to conclude the proof of Theorem 1 using Doeblin's theorem, which states the following result, whose proof can be found, for instance, in [12,Thm 2.1].…”