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On a strong form of propagation of chaos for McKean-Vlasov equations. (English) Zbl 1396.65013

Summary: This note shows how to considerably strengthen the usual mode of convergence of an \(n\)-particle system to its McKean-Vlasov limit, often known as propagation of chaos, when the volatility coefficient is nondegenerate and involves no interaction term. Notably, the empirical measure converges in a much stronger topology than weak convergence, and any fixed \(k\) particles converge in total variation to their limit law as \(n\rightarrow \infty\). This requires minimal continuity for the drift in both the space and measure variables. The proofs are purely probabilistic and rather short, relying on Girsanov’s and Sanov’s theorems. Along the way, some modest new existence and uniqueness results for McKean-Vlasov equations are derived.

MSC:

65C35 Stochastic particle methods
60K35 Interacting random processes; statistical mechanics type models; percolation theory
35K59 Quasilinear parabolic equations
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References:

[1] G.B. Arous and O. Zeitouni, Increasing propagation of chaos for mean field models, Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol. 35, Elsevier, 1999, pp. 85–102. · Zbl 0928.60092
[2] D. Bertsekas and S. Shreve, Stochastic optimal control: The discrete time case, Athena Scientific, 1996. · Zbl 0471.93002
[3] M. Bossy and D. Talay, Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation, The Annals of Applied Probability 6 (1996), no. 3, 818–861. · Zbl 0860.60038
[4] L. Campi and M. Fischer, \( n \)-player games and mean field games with absorption, arXiv preprint arXiv:1612.03816 (2016). · Zbl 1420.91020
[5] R. Carmona and D. Lacker, A probabilistic weak formulation of mean field games and applications, The Annals of Applied Probability 25 (2015), no. 3, 1189–1231. · Zbl 1332.60100
[6] P.-E. Chaudru de Raynal, Strong well-posedness of McKean-Vlasov stochastic differential equation with Hölder drift, arXiv preprint arXiv:1512.08096 (2015).
[7] T.-S. Chiang, McKean-Vlasov equations with discontinuous coefficients, Soochow J. Math 20 (1994), no. 4, 507–526. · Zbl 0817.60071
[8] D. Crisan, T.G. Kurtz, and Y. Lee, Conditional distributions, exchangeable particle systems, and stochastic partial differential equations, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 50, Institut Henri Poincaré, 2014, pp. 946–974. · Zbl 1306.60086
[9] D. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics: An International Journal of Probability and Stochastic Processes 20 (1987), no. 4, 247–308. · Zbl 0613.60021
[10] A. Dembo and O. Zeitouni, Large deviations techniques and applications, vol. 38, Springer Science & Business Media, 2009. · Zbl 0896.60013
[11] N. Dunford and J.T. Schwartz, Linear operators, part I: General theory, Interscience, 1957.
[12] J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Mathematische Nachrichten 137 (1988), no. 1, 197–248. · Zbl 0678.60100
[13] N. Gozlan and C. Léonard, Transport inequalities. A survey, arXiv preprint arXiv:1003.3852 (2010).
[14] B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers’ equations, ESAIM: Probability and Statistics 1 (1997), 339–355. · Zbl 0929.60062
[15] B. Jourdain and J. Reygner, Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation, Stochastic Partial Differential Equations: Analysis and Computations 1 (2013), no. 3, 455–506. · Zbl 1279.65012
[16] I. Karatzas and S. Shreve, Brownian motion and stochastic calculus, vol. 113, Springer Science & Business Media, 2012. · Zbl 0734.60060
[17] V.N. Kolokoltsov, Nonlinear diffusions and stable-like processes with coefficients depending on the median or var, Applied Mathematics & Optimization 68 (2013), no. 1, 85–98. · Zbl 1273.49031
[18] J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics 2 (2007), no. 1, 229–260. · Zbl 1156.91321
[19] H.P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences 56 (1966), no. 6, 1907–1911. · Zbl 0149.13501
[20] Y.S. Mishura and A.Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, arXiv preprint arXiv:1603.02212 (2016).
[21] M. Shkolnikov, Large systems of diffusions interacting through their ranks, Stochastic Processes and their Applications 122 (2012), no. 4, 1730–1747. · Zbl 1276.60087
[22] A.-S. Sznitman, Topics in propagation of chaos, Ecole d’Eté de Probabilités de Saint-Flour XIX-1989 (1991), 165–251.
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