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

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.

[1]  On singularities of solutions of partial differential equations in three variables , 1958 .

[2]  P. Meyer,et al.  Sur les inegalites de Sobolev logarithmiques. I , 1982 .

[3]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[4]  François Bouchut,et al.  On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials , 1995, Differential and Integral Equations.

[5]  S. Méléard.  Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models , 1996 .

[6]  Emanuele Caglioti,et al.  A Non-Maxwellian Steady Distribution for One-Dimensional Granular Media , 1998 .

[7]  S. Bobkov,et al.  Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities , 1999 .

[8]  P. Moral,et al.  On the stability of measure valued processes with applications to filtering , 1999 .

[9]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[10]  P. Moral,et al.  Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering , 2000 .

[11]  Liming Wu.  Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems , 2001 .

[12]  Florent Malrieu,et al.  Logarithmic Sobolev Inequalities for Some Nonlinear Pde's , 2001 .

[13]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[14]  S. Bobkov,et al.  Hypercontractivity of Hamilton-Jacobi equations , 2001 .

[15]  C. Villani,et al.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , 2003 .

[16]  F. Malrieu.  Convergence to equilibrium for granular media equations and their Euler schemes , 2003 .

[17]  Spatially Inhomogenous.  On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems : The linear Fokker-Planck equation , 2004 .

[18]  A. Guillin,et al.  Transportation cost-information inequalities and applications to random dynamical systems and diffusions , 2004, math/0410172.

[19]  F. Hérau,et al.  Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential , 2004 .

[20]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[21]  Frederic Herau.  Short and long time behavior of the Fokker-Planck equation in a confining potential and applications , 2005 .

[22]  C. Villani,et al.  Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces , 2005, math/0503123.

[23]  Quantitative concentration inequalities on sample path space for mean field interaction , 2005, math/0511752.

[24]  P. Cattiaux,et al.  Probabilistic approach for granular media equations in the non-uniformly convex case , 2006, math/0603541.

[25]  C. Villani,et al.  Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media , 2006 .

[26]  Mathias Rousset,et al.  On the Control of an Interacting Particle Estimation of Schrödinger Ground States , 2006, SIAM J. Math. Anal..

[27]  A. Veretennikov,et al.  On Ergodic Measures for McKean-Vlasov Stochastic Equations , 2006 .

[28]  Yan Guo,et al.  Stability of the Front under a Vlasov–Fokker–Planck Dynamics , 2007, 0708.3409.

[29]  D. Bakry,et al.  Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré , 2007, math/0703355.

[30]  G. Toscani,et al.  Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations , 2007 .

[31]  C. Villani.  Optimal Transport: Old and New , 2008 .

[32]  F. Bolley.  Separability and completeness for the Wasserstein distance , 2008 .

[33]  P. Moral,et al.  Concentration Inequalities for Mean Field Particle Models , 2011, 1211.1837.