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

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

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.

Citation: Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. Kinetic & 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. CullenW. 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. FournierM. 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. CullenW. 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. FournierM. 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.

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