A gradient flow approach of propagation of chaos

Previous Article
A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $
DCDS Home
This Issue
Next Article
On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity

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

A gradient flow approach of propagation of chaos

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 and 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.
[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.
[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.
[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.
[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.
[6]	R. L. Dobrušin, Vlasov equations, Anal. i Prilozhen., 13 (1979), 48-58.
[7]	A. Durmus, A. Eberle, A. Guillin and R. Zimmer, An elementary approach to uniform in time propagation of chaos., preprint, arXiv: 1805.11387.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	T. Holding, Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli, Preprint, arXiv: 1608.02877, (2016).
[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.
[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.
[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.
[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.
[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.
[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.
[21]	C. Villani, Of Triangles, Gases, Prices and Men, Huawei-IHES Workshop on Mathematical Science, Tuesday May 5th 2015.
[22]	C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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.
[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.
[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.
[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.
[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.
[6]	R. L. Dobrušin, Vlasov equations, Anal. i Prilozhen., 13 (1979), 48-58.
[7]	A. Durmus, A. Eberle, A. Guillin and R. Zimmer, An elementary approach to uniform in time propagation of chaos., preprint, arXiv: 1805.11387.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	T. Holding, Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli, Preprint, arXiv: 1608.02877, (2016).
[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.
[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.
[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.
[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.
[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.
[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.
[21]	C. Villani, Of Triangles, Gases, Prices and Men, Huawei-IHES Workshop on Mathematical Science, Tuesday May 5th 2015.
[22]	C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[1]	Xing Huang, Feng-Yu Wang. Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1667-1679. doi: 10.3934/dcds.2020336
[2]	Lingyan Cheng, Ruinan Li, Liming Wu. Exponential convergence in the Wasserstein metric $ W_1 $ for one dimensional diffusions. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5131-5148. doi: 10.3934/dcds.2020222
[3]	Jonathan Zinsl. The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 919-933. doi: 10.3934/dcdss.2017047
[4]	Kimun Ryu, Inkyung Ahn. Positive steady--states for two interacting species models with linear self-cross diffusions. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1049-1061. doi: 10.3934/dcds.2003.9.1049
[5]	Qing Ma, Yanjun Wang. Distributionally robust chance constrained svm model with $\ell_2$-Wasserstein distance. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021212
[6]	Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855
[7]	Nikolaos Roidos, Yuanzhen Shao. Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation. Evolution Equations and Control Theory, 2022, 11 (3) : 793-825. doi: 10.3934/eect.2021026
[8]	Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300
[9]	Ruiqiang He, Xiangchu Feng, Xiaolong Zhu, Hua Huang, Bingzhe Wei. RWRM: Residual Wasserstein regularization model for image restoration. Inverse Problems and Imaging, 2021, 15 (6) : 1307-1332. doi: 10.3934/ipi.2020069
[10]	José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921
[11]	Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565
[12]	Hongyong Zhao, Qianjin Zhang, Linhe Zhu. The spatial dynamics of a zebrafish model with cross-diffusions. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1035-1054. doi: 10.3934/mbe.2017054
[13]	Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090
[14]	Amarjit Budhiraja, John Fricks. Molecular motors, Brownian ratchets, and reflected diffusions. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 711-734. doi: 10.3934/dcdsb.2006.6.711
[15]	S.-I. Ei, M. Mimura, M. Nagayama. Interacting spots in reaction diffusion systems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 31-62. doi: 10.3934/dcds.2006.14.31
[16]	Silvia Caprino, Carlo Marchioro. On a charge interacting with a plasma of unbounded mass. Kinetic and Related Models, 2011, 4 (1) : 215-226. doi: 10.3934/krm.2011.4.215
[17]	Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191
[18]	Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[19]	Arsen R. Dzhanoev, Alexander Loskutov, Hongjun Cao, Miguel A.F. Sanjuán. A new mechanism of the chaos suppression. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 275-284. doi: 10.3934/dcdsb.2007.7.275
[20]	Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Samir Salem
on Google Scholar
Samir Salem

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Samir Salem
on Google Scholar
Samir Salem

[Back to Top]