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

ESAIM: Mathematical Modelling and Numerical Analysis

Volumn 44, Issue 5, 2010, Pages 867-884

  Bolley, François ^a   Guillin, Arnaud ^b   Malrieu, Florent ^c  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITÉ BLAISE PASCAL   (France)

^c UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Concentration inequalities; Particular approximation; Transportation inequalities; Vlasov Fokker Planck equation

Indexed keywords

EID: 77956719419     PISSN: 28227840     EISSN: 28047214     Source Type: Journal    
DOI: 10.1051/m2an/2010045     Document Type: Article

References (33)

1. C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les inégalité s de Sobolev logarithmiques, Panoramas et Synthèses 10. 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. Fields 137 (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. Parma 6 (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. Iberoamericana 19 (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. Fields 140 (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, http://hal. archives-ouvertes.fr/inria-00375134/en/ (2009).
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 Monographs 89. 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. Fields 8 (2002) 163-198.
30. A. Veretennikov, On ergodic measures forMcKean-Vlasov stochastic equations, in Monte Carlo and quasi-Monte Carlo methods 2004, 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. Wissenschaften 338. 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.