Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups
ESAIM: Probability and Statistics, Volume 7 (2003), pp. 171-208.

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.

DOI: 10.1051/ps:2003001
Classification: 62L20,  65C05,  81Q05,  82C22
Keywords: Feynman-Kac formula, Schrödinger operator, spectral radius, Lyapunov exponent, spectral decomposition, semigroups on a Banach space, interacting particle systems, genetic algorithms, asymptotic stability, central limit theorems
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     author = {Del Moral, Pierre and Miclo, L.},
     title = {Particle approximations of {Lyapunov} exponents connected to {Schr\"odinger} operators and {Feynman-Kac} semigroups},
     journal = {ESAIM: Probability and Statistics},
     pages = {171--208},
     publisher = {EDP-Sciences},
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     year = {2003},
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     url = {http://www.numdam.org/articles/10.1051/ps:2003001/}
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Del Moral, Pierre; Miclo, L. Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: Probability and Statistics, Volume 7 (2003), pp. 171-208. doi : 10.1051/ps:2003001. http://www.numdam.org/articles/10.1051/ps:2003001/

[1] K. Burdzy, R. Holyst, D. Ingerman and P. March, Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A 29 (1996) 2633-2642. | Zbl

[2] K. Burdzy, R. Holyst and P. March, A Fleming-Viot particle representation of Dirichlet Laplacian. Comm. Math. Phys. 214 (2000) 679-703. | Zbl

[3] P. Del Moral and A. Guionnet, On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré 37 (2001) 155-194. | Numdam | MR | Zbl

[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). | Numdam | Zbl

[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] P. Del Moral and L. Miclo, A Moran particle approximation of Feynman-Kac formulae. Stochastic Process. Appl. 86 (2000) 193-216. | Zbl

[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] P. Del Moral and L. Miclo, Genealogies and increasing propagation of chaos for Feynman-Kac and genetic models. Ann. Appl. Probab. 11 (2001) 1166-1198. | Zbl

[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. | MR | Zbl

[10] J. Feng and T. Kurtz, Large deviations for stochastic processes. http://www.math.umass.edu/ feng/Research.html | MR

[11] J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes. Springer-Verlag, A Series of Comprehensive Studies in Math. 288 (1987). | MR | Zbl

[12] T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York (1980). | MR | Zbl

[13] M. Reed and B. Simon, Methods of modern mathematical physics, II, Fourier analysis, self adjointness. Academic Press, New York (1975). | MR | Zbl

[14] A.S. Sznitman, Brownian motion, obstacles and random media. Springer, Springer Monogr. in Math. (1998). | MR | Zbl

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