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 -

References

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  1. C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses10. Société Mathématique de France, Paris (2000).  
  2. D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de probabilités XIX, 1983/84, Lecture Notes in Math.1123, Springer, Berlin (1985) 177–206.  
  3. D. Bakry, P. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal.254 (2008) 727–759.  
  4. D. Benedetto, E. Caglioti, J.A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys.91 (1998) 979–990.  
  5. S.G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal.163 (1999) 1–28.  
  6. F. Bolley, Separability and completeness for the Wasserstein distance, in Séminaire de probabilités XLI, Lecture Notes in Math.1934, Springer, Berlin (2008) 371–377.  
  7. F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction. ESAIM: PS (to appear).  
  8. F. Bolley, C. Guillin and A. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theor. Relat. Fields137 (2007) 541–593.  
  9. F. Bouchut and J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials. Diff. Int. Eq.8 (1995) 487–514.  
  10. J.A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma6 (2007) 75–198.  
  11. J.A. Carrillo, R.J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana19 (2003) 971–1018.  
  12. J.A. Carrillo, R.J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal.179 (2006) 217–263.  
  13. P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non uniformly convex case. Probab. Theor. Relat. Fields140 (2008) 19–40.  
  14. P. Del Moral, Feynman-Kac formulae – Genealogical and interacting particle systems with applications, Probability and its Applications. Springer-Verlag, New York (2004).  
  15. P. Del Moral and A. Guionnet, On the stability of measure valued processes with applications to filtering. C. R. Acad. Sci. Paris Sér. I Math.329 (1999) 429–434.  
  16. P. Del Moral and L. Miclo, Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering, in Séminaire de Probabilités XXXIV, Lecture Notes in Math.1729, Springer, Berlin (2000) 1–145.  
  17. P. Del Moral and E. Rio, Concentration Inequalities for Mean Field Particle Models. Preprint, (2009).  URIhttp://hal.archives-ouvertes.fr/inria-00375134/en/
  18. L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math.54 (2001) 1–42.  
  19. H. Djellout, A. Guillin and L. Wu, Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab.32 (2004) 2702–2732.  
  20. R. Esposito, Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics. Arch. Rat. Mech. Anal. (to appear).  
  21. F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications. J. Funct. Anal.244 (2007) 95–118.  
  22. F. Hérau and F. Nier, Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential. Arch. Rat. Mech. Anal.2 (2004) 151–218.  
  23. M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs89. American Mathematical Society, Providence (2001).  
  24. F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl.95 (2001) 109–132.  
  25. F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab.13 (2003) 540–560.  
  26. S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, in Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Lecture Notes in Math.1627, Springer, Berlin (1996) 42–95.  
  27. M. Rousset, On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal.38 (2006) 824–844.  
  28. A. Sznitman, Topics in propagation of chaos, École d'été de Probabilités de Saint-Flour XIX–1989, Lecture Notes Math.1464, Springer, Berlin (1991) 165–251.  
  29. D. Talay, Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Mark. Proc. Rel. Fields8 (2002) 163–198.  
  30. A. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equations, in Monte Carlo and quasi-Monte Carlo methods2004, Springer, Berlin (2006) 471–486.  
  31. C. Villani, Hypocoercivity, Mem. Amer. Math. Soc.202. AMS (2009).  
  32. C. Villani, Optimal transport, old and new, Grund. der Math. Wissenschaften338. Springer-Verlag, Berlin (2009).  
  33. L. Wu, Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stoch. Proc. Appl.91 (2001) 205–238.  

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