Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T11:25:57.667Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  26 August 2010

François Bolley ,
Arnaud Guillin  and
Florent Malrieu
François Bolley
Affiliation:
Ceremade, UMR CNRS 7534, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France. [email protected]
Arnaud Guillin
Affiliation:
UMR CNRS 6620, Laboratoire de Mathématiques, Université Blaise Pascal, avenue des Landais, 63177 Aubière Cedex, France. [email protected]
Florent Malrieu
Affiliation:
UMR CNRS 6625, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France. [email protected]
Get access

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.

Keywords

Vlasov-Fokker-Planck equationparticular approximationconcentration inequalitiestransportation inequalities
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 867 - 884
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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 Synthèses 10. Société Mathématique de France, Paris (2000).
D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de probabilités XIX, 1983/84, Lecture Notes in Math. 1123, Springer, Berlin (1985) 177–206.
Bakry, D., Cattiaux, P. and Guillin, A., Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 (2008) 727–759. CrossRef
Benedetto, D., Caglioti, E., Carrillo, J.A. and Pulvirenti, M., A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91 (1998) 979–990. CrossRef
Bobkov, S.G. and Götze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1–28. CrossRef
F. Bolley, Separability and completeness for the Wasserstein distance, in Séminaire de probabilités XLI, Lecture Notes in Math. 1934, Springer, Berlin (2008) 371–377.
F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction. ESAIM: PS (to appear).
Bolley, F., Guillin, C. and Villani, A., Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theor. Relat. Fields 137 (2007) 541–593. CrossRef
Bouchut, F. and Dolbeault, J., 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.
Carrillo, J.A. and Toscani, G., Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma 6 (2007) 75–198.
Carrillo, J.A., McCann, R.J. and Villani, C., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (2003) 971–1018. CrossRef
Carrillo, J.A., McCann, R.J. and Villani, C., Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179 (2006) 217–263. CrossRef
Cattiaux, P., Guillin, A. and Malrieu, F., Probabilistic approach for granular media equations in the non uniformly convex case. Probab. Theor. Relat. Fields 140 (2008) 19–40. CrossRef
P. Del Moral, Feynman-Kac formulae – Genealogical and interacting particle systems with applications, Probability and its Applications. Springer-Verlag, New York (2004).
Del Moral, P. and Guionnet, A., On the stability of measure valued processes with applications to filtering. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 429–434. CrossRef
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.
P. Del Moral and E. Rio, Concentration Inequalities for Mean Field Particle Models. Preprint, http://hal.archives-ouvertes.fr/inria-00375134/en/ (2009).
Desvillettes, L. and Villani, C., 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. 3.0.CO;2-Q>CrossRef
Djellout, H., Guillin, A. and Transportation, L. Wu cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 (2004) 2702–2732.
R. Esposito, Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics. Arch. Rat. Mech. Anal. (to appear).
Hérau, F., Short and long time behavior of the Fokker-Planck equation in a confining potential and applications. J. Funct. Anal. 244 (2007) 95–118. CrossRef
Hérau, F. and Nier, F., Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential. Arch. Rat. Mech. Anal. 2 (2004) 151–218. CrossRef
M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs 89. American Mathematical Society, Providence (2001).
Malrieu, F., Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95 (2001) 109–132. CrossRef
Malrieu, F., Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540–560. CrossRef
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.
Rousset, M., On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824–844. CrossRef
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.
Talay, D., 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.
A. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equations, in Monte Carlo and quasi-Monte Carlo methods 2004, Springer, Berlin (2006) 471–486.
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202. AMS (2009).
C. Villani, Optimal transport, old and new, Grund. der Math. Wissenschaften 338. Springer-Verlag, Berlin (2009).
Large, L. Wu and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stoch. Proc. Appl. 91 (2001) 205–238.