Exponential rate of convergence independent of the dimension in a mean-field system of particles

• Autorzy: Dyda B. , Tugaut J.

Identyfikatory

DOI

10.19195/0208-4147.37.1.6

• Języki publikacji: EN

Abstrakty

EN

This article deals with a mean-field model. We consider a large number of particles interacting through their empirical law. We know that there is a unique invariant probability for this diffusion.We look at functional inequalities. In particular, we briefly show that the diffusion satisfies a Poincaré inequality. Then, we establish a so-called WJ-inequality, which is independent of the number of particles.

Słowa kluczowe

EN

mean-field model; Poincaré inequality; transportation inequality; high dimension

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2017
• Tom: Vol. 37, Fasc. 1
• Strony: 145--161
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Dyda B.

• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

autor

Tugaut J.

• Université Jean Monnet, Saint-Étienne and Institut Camille Jordan, Lyon, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France

Bibliografia

• [1] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel 2008.
• [2] D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Electron. Commun. Probab. 13 (2008), pp. 60-66.
• [3] S. Benachour, B. Roynette, D. Talay, and P. Vallois, Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos, Stochastic Process. Appl. 75 (2) (1998), pp. 173-201.
• [4] G. Ben Arous and O. Zeitouni, Increasing propagation of chaos for mean field models, Ann. Inst. Henri Poincaré Probab. Stat. 35 (1) (1999), pp. 85-102.
• [5] F. Bolley, I. Gentil, and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal. 263 (8) (2012), pp. 2430-2457.
• [6] F. Bolley, I. Gentil, and A. Guillin, Uniform convergence to equilibrium for granular media, Arch. Ration. Mech. Anal. 208 (2) (2013), pp. 429-445.
• [7] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), pp. 375-417.
• [8] F. Collet, P. Dai Pra, and E. Sartori, A simple mean field model for social interactions: dynamics, fluctuations, criticality, J. Stat. Phys. 139 (5) (2010), pp. 820-858.
• [9] D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics 82 (1-3) (2010), pp. 53-68.
• [10] H. P. McKean, Jr., A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA 56 (1966), pp. 1907-1911.
• [11] 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), Air Force Office Sci. Res., Arlington, VA, 1967, pp. 41-57.
• [12] S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, in: Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995), Lecture Notes in Math., Vol. 1627, Springer, Berlin 1996, pp. 42-95.
• [13] K.-T. Sturm and M.-K. von Renesse, Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math. 68 (2005), pp. 923-940.
• [14] A-S. Sznitman, Topics in propagation of chaos, in: École d’Été de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Math., Vol. 1464, Springer, Berlin 1991, pp. 165-251.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).