Kinetic and Related Models (KRM)

Hypocoercive relaxation to equilibrium for some kinetic models

Pages: 341 - 360, Volume 7, Issue 2, June 2014      doi:10.3934/krm.2014.7.341

       Abstract        References        Full Text (476.4K)       Related Articles       

Pierre Monmarché - Laboratoire de Statistiques et Probabilités, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex, France (email)

Abstract: This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincaré's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of $F_t$ (the $L^2$ norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that $F_t$ satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.

Keywords:  Ergodicity, hypocoercivity, kinetic Markov processes, PDMP.
Mathematics Subject Classification:  Primary: 60J99; Secondary: 35B40.

Received: June 2013;      Revised: February 2014;      Available Online: March 2014.