Eva Locherbach (University of Cergy-Pontoise)
Titre: Ergodicity and speed of convergence to equilibrium for
diffusion processes


Abstract : We discuss the long time behaviour of diffusion processes
(or more general Markov processes). We start by introducing the basic
concept of Harris recurrence and establish the link with ergodic
theory. We recall classical recurrence conditions from the theory of
Markov chains (Doeblin condition, Dobrushin condition, local Doeblin
condition). Then we turn to the study of one dimensional diffusions
where hitting time moments

determine the speed of convergence. We recall Kac's moments formula
for hitting times and caracterize the speed of convergence to
equilibrium under Vertennikov's drift conditions both in the ergodic
theorem and for the total variation distance.

In higher dimensional models we show how to use coupling techniques in
order to introduce regeneration times that allow to mimick the one
dimensional case.