Long time behavior of Markov processes and beyond

TL;DR

This paper reviews recent advances in understanding the long-term behavior of Markov processes, highlighting their connections to PDEs, physics, and biology, and discussing mathematical tools like coupling and functional inequalities.

Contribution

It summarizes recent progress and classical methods for analyzing the convergence to equilibrium of Markov processes across various scientific fields.

Findings

Use of propagation of chaos to study convergence

Application of coupling techniques for quantitative rates

Functional inequalities providing convergence estimates

Abstract

This note provides several recent progresses in the study of long time behavior of Markov processes. The examples presented below are related to other scientific fields as PDE's, physics or biology. The involved mathematical tools as propagation of chaos, coupling, functional inequalities, provide a good picture of the classical methods that furnish quantitative rates of convergence to equilibrium.