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October  2020, 40(10): 5729-5754. doi: 10.3934/dcds.2020243

A gradient flow approach of propagation of chaos

Samir Salem ,

Institut de Mathématiques de Toulouse, UMR 5219, 118 Route de Narbonne, 31400, Toulouse, France

* Corresponding author

Received  June 2019 Revised  February 2020 Published  June 2020

Fund Project: The author is supported by the Labex CIMI

We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting $ N $-particle system, and the $ N $ times tensorized product of the solution to the corresponding limit nonlinear conservation law. It then enables to recover classical propagation of chaos results [20] in the case of Lipschitz coefficients, uniform in time propagation of chaos in [17] in the case of strictly convex coefficients. And some recent results [7] as the case of particle in a double well potential.

Keywords: Interacting diffusions, Wasserstein distance, functional inequalities, propgation of chaos.
Mathematics Subject Classification: 35Q84, 37A50, 49Q25.
Citation: Samir Salem. A gradient flow approach of propagation of chaos. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5729-5754. doi: 10.3934/dcds.2020243
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

F. Bolley, I. Gentil and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.  doi: 10.1016/j.jfa.2012.07.007.  Google Scholar

[3]

F. Bolley, I. Gentil and A. Guillin, Uniform convergence to equilibrium for granular media, Arch. Rational Mech. Anal., 208 (2013), 429-445.  doi: 10.1007/s00205-012-0599-z.  Google Scholar

[4]

E. A. Carlen, Superadditivity of Fisher's Information and Logarithmic Sobolev Inequalities, J. Funct. Anal., 101 (1991), 194-211. doi: 10.1016/0022-1236(91)90155-X.  Google Scholar

[5]

D. Cordero-Erausquin and A. Figalli, Regularity of monotone maps between unbounded domains, Discrete & Contin. Dyn. Syst., 39 (2019), 7101-7112.  doi: 10.3934/dcds.2019297.  Google Scholar

[6]

R. L. Dobrušin, Vlasov equations, Anal. i Prilozhen., 13 (1979), 48-58.   Google Scholar

[7]

A. Durmus, A. Eberle, A. Guillin and R. Zimmer, An elementary approach to uniform in time propagation of chaos., preprint, arXiv: 1805.11387. Google Scholar

[8]

A. Eberle, Reflection couplings and contraction rates for diffusions, Probab. Theory Relat. Fields, 166 (2016), 851-886.  doi: 10.1007/s00440-015-0673-1.  Google Scholar

[9]

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

[10]

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

[11]

F. Golse, C. Mouhot and T. Paul, On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., 343 (2016), 165-205.  doi: 10.1007/s00220-015-2485-7.  Google Scholar

[12]

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

[13]

S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.  doi: 10.1016/j.spa.2010.03.009.  Google Scholar

[14]

T. Holding, Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli, Preprint, arXiv: 1608.02877, (2016). 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.  doi: 10.1016/j.jfa.2016.09.014.  Google Scholar

[16]

P.-E. Jabin and Z. Wang, Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1,\infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.  Google Scholar

[17]

F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.  doi: 10.1016/S0304-4149(01)00095-3.  Google Scholar

[18]

H.-P. McKean Jr., A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[19]

S. Salem, Propagation of chaos for some 2 dimensional fractional Keller Segel equations in diffusion dominated and fair competition cases, preprint, arXiv: 1712.06677. Google Scholar

[20]

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

[21]

C. Villani, Of Triangles, Gases, Prices and Men, Huawei-IHES Workshop on Mathematical Science, Tuesday May 5th 2015. Google Scholar

[22]

C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

F. Bolley, I. Gentil and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.  doi: 10.1016/j.jfa.2012.07.007.  Google Scholar

[3]

F. Bolley, I. Gentil and A. Guillin, Uniform convergence to equilibrium for granular media, Arch. Rational Mech. Anal., 208 (2013), 429-445.  doi: 10.1007/s00205-012-0599-z.  Google Scholar

[4]

E. A. Carlen, Superadditivity of Fisher's Information and Logarithmic Sobolev Inequalities, J. Funct. Anal., 101 (1991), 194-211. doi: 10.1016/0022-1236(91)90155-X.  Google Scholar

[5]

D. Cordero-Erausquin and A. Figalli, Regularity of monotone maps between unbounded domains, Discrete & Contin. Dyn. Syst., 39 (2019), 7101-7112.  doi: 10.3934/dcds.2019297.  Google Scholar

[6]

R. L. Dobrušin, Vlasov equations, Anal. i Prilozhen., 13 (1979), 48-58.   Google Scholar

[7]

A. Durmus, A. Eberle, A. Guillin and R. Zimmer, An elementary approach to uniform in time propagation of chaos., preprint, arXiv: 1805.11387. Google Scholar

[8]

A. Eberle, Reflection couplings and contraction rates for diffusions, Probab. Theory Relat. Fields, 166 (2016), 851-886.  doi: 10.1007/s00440-015-0673-1.  Google Scholar

[9]

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

[10]

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

[11]

F. Golse, C. Mouhot and T. Paul, On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., 343 (2016), 165-205.  doi: 10.1007/s00220-015-2485-7.  Google Scholar

[12]

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

[13]

S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.  doi: 10.1016/j.spa.2010.03.009.  Google Scholar

[14]

T. Holding, Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli, Preprint, arXiv: 1608.02877, (2016). 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.  doi: 10.1016/j.jfa.2016.09.014.  Google Scholar

[16]

P.-E. Jabin and Z. Wang, Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1,\infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.  Google Scholar

[17]

F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.  doi: 10.1016/S0304-4149(01)00095-3.  Google Scholar

[18]

H.-P. McKean Jr., A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[19]

S. Salem, Propagation of chaos for some 2 dimensional fractional Keller Segel equations in diffusion dominated and fair competition cases, preprint, arXiv: 1712.06677. Google Scholar

[20]

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

[21]

C. Villani, Of Triangles, Gases, Prices and Men, Huawei-IHES Workshop on Mathematical Science, Tuesday May 5th 2015. Google Scholar

[22]

C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

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