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 interacting particles evolving in an environment with soft obstacles related to a potential function . 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.
Mots-clés : 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
@article{PS_2003__7__171_0,
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},
volume = {7},
year = {2003},
doi = {10.1051/ps:2003001},
zbl = {1040.81009},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps:2003001/}
}
TY - JOUR AU - Del Moral, Pierre AU - Miclo, L. TI - Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups JO - ESAIM: Probability and Statistics PY - 2003 SP - 171 EP - 208 VL - 7 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2003001/ DO - 10.1051/ps:2003001 LA - en ID - PS_2003__7__171_0 ER -
%0 Journal Article %A Del Moral, Pierre %A Miclo, L. %T Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups %J ESAIM: Probability and Statistics %D 2003 %P 171-208 %V 7 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps:2003001/ %R 10.1051/ps:2003001 %G en %F PS_2003__7__171_0
Del Moral, Pierre; Miclo, L. Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 171-208. doi : 10.1051/ps:2003001. http://archive.numdam.org/articles/10.1051/ps:2003001/
[1] ,, and, Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A 29 (1996) 2633-2642. | Zbl
[2] , and, A Fleming-Viot particle representation of Dirichlet Laplacian. Comm. Math. Phys. 214 (2000) 679-703. | Zbl
[3] and, 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] and, 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] and, Asymptotic stability of nonlinear semigroup of Feynman-Kac type. Publications du Laboratoire de Statistique et Probabilités, No. 04-99 (1999).
[6] and, A Moran particle approximation of Feynman-Kac formulae. Stochastic Process. Appl. 86 (2000) 193-216. | Zbl
[7] and, 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] and, Genealogies and increasing propagation of chaos for Feynman-Kac and genetic models. Ann. Appl. Probab. 11 (2001) 1166-1198. | Zbl
[9] and, 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] and, Large deviations for stochastic processes. http://www.math.umass.edu/ feng/Research.html | MR
[11] and, Limit theorems for stochastic processes. Springer-Verlag, A Series of Comprehensive Studies in Math. 288 (1987). | MR | Zbl
[12] , Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York (1980). | MR | Zbl
[13] and, Methods of modern mathematical physics, II, Fourier analysis, self adjointness. Academic Press, New York (1975). | MR | Zbl
[14] , Brownian motion, obstacles and random media. Springer, Springer Monogr. in Math. (1998). | MR | Zbl
Cité par Sources :
