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 equation, particular approximation, concentration inequalities, transportation inequalities
@article{M2AN_2010__44_5_867_0, author = {Bolley, Fran\c{c}ois and Guillin, Arnaud and Malrieu, Florent}, title = {Trend to equilibrium and particle approximation for a weakly selfconsistent {Vlasov-Fokker-Planck} equation}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {867--884}, publisher = {EDP-Sciences}, volume = {44}, number = {5}, year = {2010}, doi = {10.1051/m2an/2010045}, mrnumber = {2731396}, zbl = {1201.82029}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2010045/} }
TY - JOUR AU - Bolley, François AU - Guillin, Arnaud AU - Malrieu, Florent TI - Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 867 EP - 884 VL - 44 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010045/ DO - 10.1051/m2an/2010045 LA - en ID - M2AN_2010__44_5_867_0 ER -
%0 Journal Article %A Bolley, François %A Guillin, Arnaud %A Malrieu, Florent %T Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 867-884 %V 44 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010045/ %R 10.1051/m2an/2010045 %G en %F M2AN_2010__44_5_867_0
Bolley, François; Guillin, Arnaud; Malrieu, Florent. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM: Modélisation mathématique et analyse numérique, Volume 44 (2010) no. 5, pp. 867-884. doi : 10.1051/m2an/2010045. http://www.numdam.org/articles/10.1051/m2an/2010045/
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