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Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation

Published online by Cambridge University Press:  26 August 2010

François Bolley ,
Arnaud Guillin  and
Florent Malrieu
François Bolley
Affiliation:
Ceremade, UMR CNRS 7534, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France. bolley@ceremade.dauphine.fr
Arnaud Guillin
Affiliation:
UMR CNRS 6620, Laboratoire de Mathématiques, Université Blaise Pascal, avenue des Landais, 63177 Aubière Cedex, France. guillin@math.univ-bpclermont.fr
Florent Malrieu
Affiliation:
UMR CNRS 6625, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France. florent.malrieu@univ-rennes1.fr
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Abstract

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time.

Keywords

Vlasov-Fokker-Planck equationparticular approximationconcentration inequalitiestransportation inequalities

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