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

Francois Bolley (1) , Arnaud Guillin (2) , Florent Malrieu (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Vlasov-Fokker-Planck equation; particle approximation; chaos propagation; transportation inequality

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Francois Bolley, Arnaud Guillin, Florent Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM: Mathematical Modelling and Numerical Analysis, 2010, 44 (5), pp.867-884. ⟨10.1051/m2an/2010045⟩. ⟨hal-00392397⟩