Geodesic
Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation
ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 867-884

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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.

DOI : 10.1051/m2an/2010045
Classification : 65C35, 35K55, 65C05, 82C22, 26D10, 60E15
Keywords: Vlasov-Fokker-Planck equation, particular approximation, concentration inequalities, transportation inequalities 
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     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: Mathematical Modelling and Numerical Analysis },
     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://geodesic.mathdoc.fr/articles/10.1051/m2an/2010045/}
}
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Bolley, François; Guillin, Arnaud; Malrieu, Florent. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 867-884. doi: 10.1051/m2an/2010045

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