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

Published online by Cambridge University Press:  15 May 2003

Pierre Del Moral
Affiliation:
LSP, UMR 5583 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France; e-mail:
L. Miclo
Affiliation:
LSP, UMR 5583 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France; e-mail:
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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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2003

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References

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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. CrossRef
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).
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