On quantitative hypocoercivity estimates based on Harris-type theorems

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