Trends to Equilibrium in Total Variation Distance

TL;DR

This paper explores various functional inequality methods to analyze the convergence speed in total variation distance for ergodic diffusion processes, introducing new inequalities and dynamic approaches for improved bounds.

Contribution

It introduces new $ extit{I}_ ext{psi}$-inequalities characterized by measure-capacity conditions and develops a dynamic approach to enhance convergence bounds.

Findings

Established upper bounds using Pinsker's inequality and functional inequalities.

Introduced $ extit{I}_ ext{psi}$-inequalities for measure-capacity characterization.

Proposed a dynamic method to improve convergence bounds.

Abstract

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.