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

François Bolley; Arnaud Guillin; Florent Malrieu

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 867-884
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Bolley, François, Guillin, Arnaud, and Malrieu, Florent. "Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 867-884. <http://eudml.org/doc/250810>.

@article{Bolley2010,
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. },
author = {Bolley, François, Guillin, Arnaud, Malrieu, Florent},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Vlasov-Fokker-Planck equation; particular approximation; concentration inequalities; transportation inequalities; stochastic particle methods; interacting particle systems; inequalities, stochastic ordering; nonlinear parabolic equations},
language = {eng},
month = {8},
number = {5},
pages = {867-884},
publisher = {EDP Sciences},
title = {Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation},
url = {http://eudml.org/doc/250810},
volume = {44},
year = {2010},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 867
EP - 884
AB - 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.
LA - eng
KW - Vlasov-Fokker-Planck equation; particular approximation; concentration inequalities; transportation inequalities; stochastic particle methods; interacting particle systems; inequalities, stochastic ordering; nonlinear parabolic equations
UR - http://eudml.org/doc/250810
ER -

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