American Institute of Mathematical Sciences

Advanced
April  2019, 12(2): 269-302. doi: 10.3934/krm.2019012

Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D

Maxime Hauray 1, and  Samir Salem 2,

1. 

I2M, AMU, Centrale Marseille CNRS, Marseille, France
2. 

CEREMADE, Université Paris Dauphine, Paris, France

Received  August 2016 Revised  March 2018 Published  November 2018

We consider a particle system in 1D, interacting via repulsive or attractive Coulomb forces. We prove the trajectorial propagation of molecular chaos towards a nonlinear SDE associated to the Vlasov-Poisson-Fokker-Planck equation. We obtain a quantitative estimate of convergence in the mean in MKW metric of order one, with an optimal convergence rate of order $N^{-1/2}$. We also prove some exponential concentration inequalities of the associated empirical measures. A key argument is a weak-strong stability estimate on the (nonlinear) VPFP equation, that we are able to adapt for the particle system in some sense.

Keywords: Vlasov-Poisson equation, Propagation of chaos, concentration inequalities, quantitative estimates, weak-strong stability.
Mathematics Subject Classification: Primary: 35Q83, 35Q84, 60K35; Secondary: 65C35.
Citation: Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. Kinetic and Related Models, 2019, 12 (2) : 269-302. doi: 10.3934/krm.2019012
References:
[1]

R. J. Berman and M. Onnheim, Propagation of chaos, Wasserstein gradient flows and toric Kähler-Einstein metrics, Anal. PDE, 11 (2018), 1343-1380, arXiv: 1501.07820, 2015. doi: 10.2140/apde.2018.11.1343.

[2]

R. Catellier and M. Gubinelli, Averaging along irregular curves and regularisation of odes, Stochastic Process. Appl., 126 (2016), 2323-2366, arXiv: 1205.1735. doi: 10.1016/j.spa.2016.02.002.

[3]

E. Cépa and D. Lépingle, Diffusing particles with electrostatic repulsion, Probab, Theory Related Fields, 107 (1997), 429-449.  doi: 10.1007/s004400050092.

[4]

Y.-P. Choi and S. Salem, Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones, M3AS, 28 (2018), 223-258.  doi: 10.1142/S0218202518500070.

[5]

M. Cullen, W. Gangbo and G. Pisante, The semigeostrophic equations discretized in reference and dual variables, Arch. Ration. Mech. Anal., 185 (2007), 341-363.  doi: 10.1007/s00205-006-0040-6.

[6]

A. M. Davie, Uniqueness of solutions of stochastic differential equations, Int. Math. Res. Not., 2007 (2007), Art. ID rnm124, 26 pp. doi: 10.1093/imrn/rnm124.

[7]

E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations, Electron. J. Probab., 22, 2017, Paper No. 48, 42 pp, arXiv: 1606.01088. doi: 10.1214/17-EJP65.

[8]

W. Feller, An introduction to probability theory and its applications, Second edition John Wiley & Sons, Inc., New York-London-Sydney, 1971.

[9]

A. Figalli, Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients, J. Funct. Anal., 254 (2008), 109-153.  doi: 10.1016/j.jfa.2007.09.020.

[10]

N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probab. Theory Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7.

[11]

N. Fournier and M. Hauray, Propagation of chaos for the Landau equation with moderately soft potentials, Ann. Probab., 44 (2016), 3581-3660.  doi: 10.1214/15-AOP1056.

[12]

N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2D viscous Vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.  doi: 10.4171/JEMS/465.

[13]

M. Hauray, Mean field limit for the one dimensional Vlasov-Poisson equation, Séminaire Laurent Schwarz, Palaiseau, 2014, arXiv: 1309.2531.

[14]

M. Hauray and S. Mischler, On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157.  doi: 10.1016/j.jfa.2014.02.030.

[15]

P.-E. Jabin and Z. Wang, Mean field limit and propagation of chaos for vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627, arXiv: 1511.03769. doi: 10.1016/j.jfa.2016.09.014.

[16]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.

[17]

H. P. McKean, Jr. Propagation of chaos for a class of non-linear parabolic equations, In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pages 41-57.

[18]

K. R. Parthasaraty, Probability measures on metric spaces, Academic Press, New York, 1967.

[19]

A.-S. Sznitman, Topics in propagation of chaos, In École d'Été de Probabilités de Saint-Flour XIX-1989, volume 1464, chapter Lecture Notes in Math., pages 165-251. Springer, Berlin, 1991. doi: 10.1007/BFb0085169.

[20]

A.-S. Sznitman, Équations de type de Boltzmann, spatialement homogènes, Z. Wahrsch. Verw. Gebiete, 66 (1984), 559-592.  doi: 10.1007/BF00531891.

[21]

M. Trocheris, On the derivation of the one-dimensional Vlasov equation, Transport Theory Statist. Phys., 15 (1986), 597-628.  doi: 10.1080/00411458608212706.

[22]

A. Y. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR, Sb., 39 (1981), 387-403. 

show all references

References:
[1]

R. J. Berman and M. Onnheim, Propagation of chaos, Wasserstein gradient flows and toric Kähler-Einstein metrics, Anal. PDE, 11 (2018), 1343-1380, arXiv: 1501.07820, 2015. doi: 10.2140/apde.2018.11.1343.

[2]

R. Catellier and M. Gubinelli, Averaging along irregular curves and regularisation of odes, Stochastic Process. Appl., 126 (2016), 2323-2366, arXiv: 1205.1735. doi: 10.1016/j.spa.2016.02.002.

[3]

E. Cépa and D. Lépingle, Diffusing particles with electrostatic repulsion, Probab, Theory Related Fields, 107 (1997), 429-449.  doi: 10.1007/s004400050092.

[4]

Y.-P. Choi and S. Salem, Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones, M3AS, 28 (2018), 223-258.  doi: 10.1142/S0218202518500070.

[5]

M. Cullen, W. Gangbo and G. Pisante, The semigeostrophic equations discretized in reference and dual variables, Arch. Ration. Mech. Anal., 185 (2007), 341-363.  doi: 10.1007/s00205-006-0040-6.

[6]

A. M. Davie, Uniqueness of solutions of stochastic differential equations, Int. Math. Res. Not., 2007 (2007), Art. ID rnm124, 26 pp. doi: 10.1093/imrn/rnm124.

[7]

E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations, Electron. J. Probab., 22, 2017, Paper No. 48, 42 pp, arXiv: 1606.01088. doi: 10.1214/17-EJP65.

[8]

W. Feller, An introduction to probability theory and its applications, Second edition John Wiley & Sons, Inc., New York-London-Sydney, 1971.

[9]

A. Figalli, Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients, J. Funct. Anal., 254 (2008), 109-153.  doi: 10.1016/j.jfa.2007.09.020.

[10]

N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probab. Theory Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7.

[11]

N. Fournier and M. Hauray, Propagation of chaos for the Landau equation with moderately soft potentials, Ann. Probab., 44 (2016), 3581-3660.  doi: 10.1214/15-AOP1056.

[12]

N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2D viscous Vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.  doi: 10.4171/JEMS/465.

[13]

M. Hauray, Mean field limit for the one dimensional Vlasov-Poisson equation, Séminaire Laurent Schwarz, Palaiseau, 2014, arXiv: 1309.2531.

[14]

M. Hauray and S. Mischler, On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157.  doi: 10.1016/j.jfa.2014.02.030.

[15]

P.-E. Jabin and Z. Wang, Mean field limit and propagation of chaos for vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627, arXiv: 1511.03769. doi: 10.1016/j.jfa.2016.09.014.

[16]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.

[17]

H. P. McKean, Jr. Propagation of chaos for a class of non-linear parabolic equations, In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pages 41-57.

[18]

K. R. Parthasaraty, Probability measures on metric spaces, Academic Press, New York, 1967.

[19]

A.-S. Sznitman, Topics in propagation of chaos, In École d'Été de Probabilités de Saint-Flour XIX-1989, volume 1464, chapter Lecture Notes in Math., pages 165-251. Springer, Berlin, 1991. doi: 10.1007/BFb0085169.

[20]

A.-S. Sznitman, Équations de type de Boltzmann, spatialement homogènes, Z. Wahrsch. Verw. Gebiete, 66 (1984), 559-592.  doi: 10.1007/BF00531891.

[21]

M. Trocheris, On the derivation of the one-dimensional Vlasov equation, Transport Theory Statist. Phys., 15 (1986), 597-628.  doi: 10.1080/00411458608212706.

[22]

A. Y. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR, Sb., 39 (1981), 387-403. 

[1]

Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6835-6853. doi: 10.3934/dcds.2016097
[2]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[3]

Xianglong Duan. Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data. Kinetic and Related Models, 2022, 15 (1) : 119-146. doi: 10.3934/krm.2021049
[4]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic and Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018
[5]

Megan Griffin-Pickering, Mikaela Iacobelli. Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems. Kinetic and Related Models, 2021, 14 (4) : 571-597. doi: 10.3934/krm.2021016
[6]

Trinh T. Nguyen. Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria. Kinetic and Related Models, 2020, 13 (6) : 1193-1218. doi: 10.3934/krm.2020043
[7]

Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13
[8]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic and Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[9]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181
[10]

Wenjing Zhao. Weak-strong uniqueness of incompressible magneto-viscoelastic flows. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2907-2917. doi: 10.3934/cpaa.2020127
[11]

Etienne Emmrich, Robert Lasarzik. Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4617-4635. doi: 10.3934/dcds.2018202
[12]

Xuecheng Wang. Decay estimates for the $ 3D $ relativistic and non-relativistic Vlasov-Poisson systems. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022019
[13]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic and Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[14]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic and Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729
[15]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic and Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[16]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic and Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[17]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic and Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[18]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic and Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
[19]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic and Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
[20]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic and Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004

2021 Impact Factor: 1.398

Tools

Metrics

  • PDF downloads (236)
  • HTML views (154)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy