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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 |
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.
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.
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