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A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations. (English) Zbl 1026.60110

The authors study McKean-Vlasov equations with random initial conditions. They give a probabilistic representation of the moments of the solutions in order to approximate them by a stochastic particle method. This method is based on random weights which are defined through nonparametric estimators of a regression function. It is both original and efficient, and the convergence rate is estimated in terms of the number of simulated particles and the time discretization step. Singular interactions as in Burgers equation or Navier-Stokes equations are unfortunately not in the scope of this work.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
65C35 Stochastic particle methods
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[1] BENVENISTE, A., JUDITSKY, A., DELy ON, B., ZHANG, Q. and GLORENNEC, P.-Y. (1996). Wavelets in identification. In Indentification, Adaptation, Learning. The Science of Learning Models from Data (S. Bittanti and G. Picci, eds.) 435-478. Springer, Berlin. · Zbl 0856.93101
[2] BOULEAU, N. and LEPINGLE, D. (1994). Numerical Methods for Stochastic Processes. Wiley, New York. · Zbl 0822.60003
[3] CANNARSA, P. and VESPRI, V. (1987). Generation of analytic semigroups by elliptic operators with unbounded coefficients. SIAM J. Math. Anal. 18 857-872. · Zbl 0623.47039
[4] CARRARO, L. and DUCHON, J. (1994). Solutions statistiques intrinsèques de l’équation de Burgers et processus de Lévy. Note aux Comptes-Rendus de l’Académie des Sciences Sér. I 319 855-858. · Zbl 0822.58008
[5] CESSENAT, M., LEDANOIS, G., LIONS, P. L., PARDOUX, E. and SENTIS, R. (1989). Méthodes Probabilistes pour les Équations de la physique. Sy ntheses. Collection du Commissariat à l’Energie Atomique.
[6] CHORIN, A. J., KAST, A. P. and KUPFERMAN, R. (2000). Optimal prediction and the Mori-Zwanzig representation of irreversible processes. Proc. Natl. Acad. Sci. USA 97 2968-2973. · Zbl 0968.60036
[7] CHU, C. K. and MARRON, J. S. (1991). Choosing a kernel regression estimator. Statist. Sci. 6 404-436. · Zbl 0955.62561
[8] COLLOMB, G. (1976). Estimation non paramétrique de la régression par la méthode du noy au. Ph.D. thesis, Univ. P. Sabatier, Toulouse, France.
[9] CONSTANTIN, P. and WU, J. (1997). Statistical solutions of the Navier-Stokes equations on the phase space of vorticity and the inviscid limit. J. Math. Phy s 38 3031-3045. · Zbl 0893.76013
[10] FOIAS, C. (1973). Statistical study of the Navier-Stokes equations i and ii. Rend. Sem. Mat. Univ. Padova 48 219-348; 49 9-123.
[11] FOIAS, C. and TEMAM, R. (1980). Homogeneous statistical solutions of Navier-Stokes equations. Indiana Univ. Math. J. 29 913-957. · Zbl 0463.76056
[12] FOIAS, C. and TEMAM, R. (1983). Self-similar universal homogeneous statistical solutions of the Navier-Stokes equations. Comm. Math. Phy s. 90 187-206. · Zbl 0532.60058
[13] FOX, R. O. (1996). Computational methods for turbulent reacting flows in the chemical process industry. Revue de l’Institut Français du Pétrole 51 215-243.
[14] FRIEDMAN, A. (1975). Stochastic Differential Equations and Applications 1. Academic Press, New York. · Zbl 0323.60056
[15] HARDLE, W. (1990). Applied Nonparametric Regression. Cambridge Univ. Press.
[16] HARDLE, W., KERKy ACHARIAN, G., PICARD, D. and TSy BAKOV, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statist. 129. Springer, New York.
[17] HOPF, E. (1952). Statistical hy drody namics and functional calculus. J. Rational Mechanical Analy sis 1 98-123.
[18] LUNARDI, A. (1995). Analy tic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Boston. · Zbl 0816.35001
[19] MÉLÉARD, S. (1996). Asy mptotic behaviour of some interacting particle sy stems; McKean- Vlasov and Boltzmann models. Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math. 1627 42-95. Springer, Berlin. · Zbl 0864.60077
[20] MOHAMMADI, B. and PIRONNEAU, O. (1994). Analy sis of the K-Epsilon Turbulence Model. Masson, Paris.
[21] MONIN, A. S. and YAGLOM, A. M. (1975). Statistical Fluid Mechanics 2. MIT Press.
[22] SZNITMAN, A. S. (1991). Topics in propagation of chaos. Ecole d’Eté de Probabilités de SaintFlour XIX. Lecture Notes in Math. 1464 165-251. Springer, Berlin. · Zbl 0732.60114
[23] TALAY, D. (1986). Discrétisation d’une E.D.S. et calcul approché d’espérances de fonctionnelles de la solution. Math. Modelling Numer. Anal. 20 141-179. · Zbl 0662.65129
[24] TALAY, D. (1996). Probabilistic Numerical methods for partial differential equations: Elements of analysis. Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math. 1627 148-196. Springer, Berlin. · Zbl 0854.65116
[25] TALAY, D. and VAILLANT, O. (2001). A stochastic particle method with random weights for statistical solutions of McKean-Vlasov equations. INRIA report.
[26] TALAY, D. and VAILLANT, O. (2000). Vitesse de convergence d’une methode particulaire stochastique avec poids d’interaction aléatoires. Note aux Comptes-Rendus de l’Académie des Sciences Sér. I 330 821-824. · Zbl 0960.65013
[27] VAILLANT, O. (2000). Une méthode particulaire stochastique à poids aléatoires pour l’approximation de solutions statistiques d’équations de McKean-Vlasov-Fokker- Planck. Ph.D. thesis, Univ. Provence, Marseille.
[28] VISHIK, M. J. and FURSIKOV, A. V. (1988). Mathematical Problems of Statistical Hy dromechanics. Kluwer, Dordrecht.
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