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Quantitative concentration inequalities for empirical measures on non-compact spaces. (English) Zbl 1113.60093

Quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances, are established. Typical estimates are of the form \[ P\left[\sup_{\| \varphi\| _{\text{Lip}}\leq 1} \left( N^{-1} \sum_{i=1}^N \varphi(X_t^i)- \int \varphi \,d\mu_t \right) > \varepsilon \right] \leq C e^{-\lambda N \varepsilon^2}, \] where \(\| \varphi\| _{\text{Lip}}=\sup_{x \neq y} | \varphi(x)-\varphi(y)| d^{-1}(x,y)\) and \(d(\cdot)\) stands for the distance in phase space (say Euclidean norm in \(\mathbb{R}^d\)). The core of estimates is based on variants of Sanov’s theorem.
These results are applied to provide some error bounds for particle simulations in a simple mean-field kinetic model for granular media. A system of interacting particles is governed by the system of coupled stochastic differential equations. The tools include coupling arguments, as well as regularity and moment estimates for solutions of differential equations.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] Araujo, A.; Giné, E., The central limit theorem for real and Banach valued random variables (1980), New York: Wiley, New York · Zbl 0457.60001
[2] Benachour, S.; Roynette, B.; Talay, D.; Vallois, P., Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos, Stoch Process Appl, 75, 2, 173-201 (1998) · Zbl 0932.60063
[3] Benachour, S.; Roynette, B.; Vallois, P., Nonlinear self-stabilizing processes. II. Convergence to invariant probability, Stoch Process Appl, 75, 2, 203-224 (1998) · Zbl 0932.60064
[4] Benedetto, D.; Caglioti, E.; Carrillo, J. A.; Pulvirenti, M., A non-Maxwellian steady distribution for one-dimensional granular media, J Stat Phys, 91, 5-6, 979-990 (1998) · Zbl 0921.60057
[5] Bobkov, S.; Gentil, I.; Ledoux, M., Hypercontractivity of Hamilton-Jacobi equations, J Math Pures Appl, 80, 7, 669-696 (2001) · Zbl 1038.35020
[6] Bobkov, S.; Götze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J Funct Anal, 163, 1-28 (1999) · Zbl 0924.46027
[7] Bolley, F.: Quantitative concentration inequalities on sample path space for mean field interaction. Preprint (2005) available online via http://www.lsp.ups-tlse. fr/Fp/Bolley · Zbl 1208.82038
[8] Bolley, F.; Villani, C., Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities, Ann Fac Sci Toulouse, 14, 3, 331-352 (2005) · Zbl 1087.60008
[9] Carrillo, J. A.; McCann, R. J.; Villani, C., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev Mat Iberoamericana, 19, 3, 971-1018 (2003) · Zbl 1073.35127
[10] Carrillo, J. A.; McCann, R. J.; Villani, C., Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch Rational Mech. Anal, 179, 217-263 (2006) · Zbl 1082.76105
[11] Cattiaux, P., Guillin, A.: Talagrand’s like quadratic transportation cost inequalities.Preprint (2004), available online via http://www.ceremade.dauphine.fr/  guillin/index3.html · Zbl 1118.58017
[12] Dembo, A.; Zeitouni, O., Large deviations techniques and applications (1998), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0896.60013
[13] Desvillettes, L.; Villani, C., On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm Partial Differential Equations, 25, 1-2, 179-259 (2000) · Zbl 0946.35109
[14] Djellout, H.; Guillin, A.; Wu, L., Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann Probab, 32, 3, 2702-2732 (2004) · Zbl 1061.60011
[15] Gao, F., Moderate deviations and large deviations for kernel density estimators, J Theor Prob, 16, 401-418 (2003) · Zbl 1041.62025
[16] Giné, E.; Zinn, J., Empirical processes indexed by Lipschitz functions, Ann Probab, 14, 4, 1329-1338 (1986) · Zbl 0611.60029
[17] Ledoux, M.: The concentration of measure phenomenon. In: Mathematical surveys and monographs, vol 89. Providence: American Mathematical Society, 2001 · Zbl 0995.60002
[18] Ledoux, M.; Talagrand, M., Probability in Banach spaces (1991), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0748.60004
[19] Malrieu, F., Logarithmic Sobolev inequalities for some nonlinear PDE’s, Stoch Process Appl, 95, 1, 109-132 (2001) · Zbl 1059.60084
[20] Malrieu, F., Convergence to equilibrium for granular media equations and their Euler schemes, Ann Appl Probab, 13, 2, 540-560 (2003) · Zbl 1031.60085
[21] Marchioro, C.; Pulvirenti, M., Mathematical theory of incompressible nonviscuous fluids (1994), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[22] Massart, P.: Saint-Flour Lecture Notes. Available at http://www.math.u-psud. fr/  massart 2003
[23] Otto, F.; Villani, C., Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J Funct Anal, 173, 361-400 (2000) · Zbl 0985.58019
[24] Petrov, V. V., Limit theorems of probability theory (1995), New York: The Clarendon Press, Oxford University Press, New York · Zbl 0826.60001
[25] Schochet, S., The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm Pure Appl Math, 49, 9, 911-965 (1996) · Zbl 0862.35092
[26] Sion, M., On general minimax theorems, Pac J Math, 8, 171-176 (1958) · Zbl 0081.11502
[27] Sznitman, A.-S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX—1989 Lecture Notes in Mathematics.vol. 1464. Berlin Heidelberg New York: Springer, 1991
[28] Villani, C.: Topics in optimal transportation. In: Graduate studies in mathematics (58). Providence: American Mathematical Society, 2003 · Zbl 1106.90001
[29] Wang, F.-Y., Probability distance inequalities on Riemannian manifolds and path spaces, J Funct Anal, 206, 1, 167-190 (2004) · Zbl 1048.58013
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