On quantitative hypocoercivity estimates based on Harris-type theorems

Abstract

This review concerns recent results on the quantitative study of convergence towards equilibrium for spatially in-homogeneous linear 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 L 1 (or total variation) setting for a wide 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 at presenting Harris-type theorems, providing a guideline on how to apply these techniques on kinetic equations at hand, presenting an exposition of recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, as well as giving some perspectives on potential extensions to non-linear equations. Contents