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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

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