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

François Bolley; Arnaud Guillin; Florent Malrieu

doi:10.1051/m2an/2010045

All issues

Volume 44 / No 5 (September-October 2010)

ESAIM: M2AN, 44 5 (2010) 867-884

References

Free Access

Issue		ESAIM: M2AN Volume 44, Number 5, September-October 2010 Special Issue on Probabilistic methods and their applications


Page(s)		867 - 884
DOI		https://doi.org/10.1051/m2an/2010045
Published online		26 August 2010

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). [Google Scholar]
D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de probabilités XIX, 1983/84, Lecture Notes in Math. 1123, Springer, Berlin (1985) 177–206. [Google Scholar]
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. [CrossRef] [MathSciNet] [Google Scholar]
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. [CrossRef] [MathSciNet] [Google Scholar]
S.G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1–28. [CrossRef] [MathSciNet] [Google Scholar]
F. Bolley, Separability and completeness for the Wasserstein distance, in Séminaire de probabilités XLI, Lecture Notes in Math. 1934, Springer, Berlin (2008) 371–377. [Google Scholar]
F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction. ESAIM: PS (to appear). [Google Scholar]
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. [CrossRef] [Google Scholar]
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. [Google Scholar]
J.A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma 6 (2007) 75–198. [MathSciNet] [Google Scholar]
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. [MathSciNet] [Google Scholar]
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. [CrossRef] [Google Scholar]
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. [CrossRef] [Google Scholar]
P. Del Moral, Feynman-Kac formulae – Genealogical and interacting particle systems with applications, Probability and its Applications. Springer-Verlag, New York (2004). [Google Scholar]
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. [Google Scholar]
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. [Google Scholar]
P. Del Moral and E. Rio, Concentration Inequalities for Mean Field Particle Models. Preprint, http://hal.archives-ouvertes.fr/inria-00375134/en/ (2009). [Google Scholar]
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. [CrossRef] [MathSciNet] [Google Scholar]
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. [CrossRef] [MathSciNet] [Google Scholar]
R. Esposito, Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics. Arch. Rat. Mech. Anal. (to appear). [Google Scholar]
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. [CrossRef] [MathSciNet] [Google Scholar]
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. [Google Scholar]
M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs 89. American Mathematical Society, Providence (2001). [Google Scholar]
F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95 (2001) 109–132. [CrossRef] [MathSciNet] [Google Scholar]
F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540–560. [CrossRef] [MathSciNet] [Google Scholar]
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. [Google Scholar]
M. Rousset, On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824–844. [CrossRef] [MathSciNet] [Google Scholar]
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. [Google Scholar]
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. [Google Scholar]
A. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equations, in Monte Carlo and quasi-Monte Carlo methods 2004, Springer, Berlin (2006) 471–486. [Google Scholar]
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202. AMS (2009). [Google Scholar]
C. Villani, Optimal transport, old and new, Grund. der Math. Wissenschaften 338. Springer-Verlag, Berlin (2009). [Google Scholar]
L. Wu, Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stoch. Proc. Appl. 91 (2001) 205–238. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you

[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 Synthèses 10. Société Mathématique de France, Paris (2000). [Google Scholar]

[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. [Google Scholar]

[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. [CrossRef] [MathSciNet] [Google Scholar]

[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. [CrossRef] [MathSciNet] [Google Scholar]

[5] S.G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1–28. [CrossRef] [MathSciNet] [Google Scholar]

[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. [Google Scholar]

[7] F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction. ESAIM: PS (to appear). [Google Scholar]

[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. [CrossRef] [Google Scholar]

[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. [Google Scholar]

[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. [MathSciNet] [Google Scholar]

[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. [MathSciNet] [Google Scholar]

[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. [CrossRef] [Google Scholar]

[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. [CrossRef] [Google Scholar]

[14] P. Del Moral, Feynman-Kac formulae – Genealogical and interacting particle systems with applications, Probability and its Applications. Springer-Verlag, New York (2004). [Google Scholar]

[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. [Google Scholar]

[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. [Google Scholar]

[17] P. Del Moral and E. Rio, Concentration Inequalities for Mean Field Particle Models. Preprint, http://hal.archives-ouvertes.fr/inria-00375134/en/ (2009). [Google Scholar]

[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. [CrossRef] [MathSciNet] [Google Scholar]

[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. [CrossRef] [MathSciNet] [Google Scholar]

[20] R. Esposito, Y. Guo and R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics. Arch. Rat. Mech. Anal. (to appear). [Google Scholar]

[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. [CrossRef] [MathSciNet] [Google Scholar]

[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. [Google Scholar]

[23] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs 89. American Mathematical Society, Providence (2001). [Google Scholar]

[24] F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95 (2001) 109–132. [CrossRef] [MathSciNet] [Google Scholar]

[25] F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540–560. [CrossRef] [MathSciNet] [Google Scholar]

[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. [Google Scholar]

[27] M. Rousset, On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824–844. [CrossRef] [MathSciNet] [Google Scholar]

[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. [Google Scholar]

[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. [Google Scholar]

[30] A. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equations, in Monte Carlo and quasi-Monte Carlo methods 2004, Springer, Berlin (2006) 471–486. [Google Scholar]

[31] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202. AMS (2009). [Google Scholar]

[32] C. Villani, Optimal transport, old and new, Grund. der Math. Wissenschaften 338. Springer-Verlag, Berlin (2009). [Google Scholar]

[33] L. Wu, Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stoch. Proc. Appl. 91 (2001) 205–238. [CrossRef] [Google Scholar]