Trends to Equilibrium in Total Variation Distance

Patrick Cattiaux, Arnaud Guillin

Published 2007-03-15Version 1

This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound "\`{a} la Pinsker" enabling us to study our problem firstly via usual functional inequalities (Poincar\'{e} inequality, weak Poincar\'{e},...) and truncation procedure, and secondly through the introduction of new functional inequalities $\Ipsi$. These $\Ipsi$-inequalities are characterized through measure-capacity conditions and $F$-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds.