On quantitative hypocoercivity estimates based on Harris-type theorems

TL;DR

This paper reviews recent advances in quantitative convergence analysis of kinetic equations using Harris-type theorems, emphasizing explicit rates and applicability to non-explicit steady states.

Contribution

It provides a comprehensive overview of Harris-type theorems applied to kinetic equations, highlighting new quantitative results and methodological guidelines.

Findings

Explicit convergence rates for kinetic equations are obtained.

Harris-type theorems relax assumptions on steady state existence.

Applications include gas theory and biological models.

Abstract

This review concerns recent results on the quantitative study of convergence towards the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the $L^1$ (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (both for geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to the kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.