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Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups

  • Pierre Del Moral (a1) and L. Miclo (a1)
Abstract

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue of Schrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine a class of models extending the hard obstacle model of K. Burdzy, R. Holyst and P. March and including the Moran type scheme presented by the authors in a previous work. We provide precise uniform estimates with respect to the time parameter and we analyze the fluctuations of continuous time particle models.

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References
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[1] Burdzy, K., Holyst, R., Ingerman, D. and March, P., Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A 29 (1996) 2633-2642.
[2] Burdzy, K., Holyst, R. and March, P., Fleming-Viot, A particle representation of Dirichlet Laplacian. Comm. Math. Phys. 214 (2000) 679-703.
[3] Del Moral, P. and Guionnet, A., On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré 37 (2001) 155-194.
[4] P. Del Moral and L. Miclo, Branching and interacting particle system approximations of Feynman-Kac formulae with applications to nonlinear filtering, in Séminaire de Probabilités XXXIV, edited by J. Azéma, M. Emery, M. Ledoux and M. Yor. Springer, Lecture Notes in Math. 1729 (2000) 1-145. Asymptotic stability of non linear semigroups of Feynman-Kac type. Ann. Fac. Sci. Toulouse (to be published).
[5] P. Del Moral and L. Miclo, Asymptotic stability of nonlinear semigroup of Feynman-Kac type. Publications du Laboratoire de Statistique et Probabilités, No. 04-99 (1999).
[6] Del Moral, P. and Miclo, L., Moran, A particle approximation of Feynman-Kac formulae. Stochastic Process. Appl. 86 (2000) 193-216.
[7] P. Del Moral and L. Miclo, About the strong propagation of chaos for interacting particle approximations of Feynman-Kac formulae. Publications du Laboratoire de Statistiques et Probabilités, Toulouse III, No 08-00 (2000).
[8] Del Moral, P. and Miclo, L., Genealogies and increasing propagation of chaos for Feynman-Kac and genetic models. Ann. Appl. Probab. 11 (2001) 1166-1198.
[9] M.D. Donsker and R.S. Varadhan, Asymptotic evaluation of certain Wiener integrals for large time in Functional Integration and its Applications, edited by A.M. Arthur. Oxford Universtity Press (1975) 15-33.
[10] J. Feng and T. Kurtz, Large deviations for stochastic processes. http://www.math.umass.edu/ feng/Research.html
[11] J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes. Springer-Verlag, A Series of Comprehensive Studies in Math. 288 (1987).
[12] T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York (1980).
[13] M. Reed and B. Simon, Methods of modern mathematical physics, II, Fourier analysis, self adjointness. Academic Press, New York (1975).
[14] A.S. Sznitman, Brownian motion, obstacles and random media. Springer, Springer Monogr. in Math. (1998).
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ESAIM: Probability and Statistics
  • ISSN: 1292-8100
  • EISSN: 1262-3318
  • URL: /core/journals/esaim-probability-and-statistics
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