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

Free access article

Issue		ESAIM: PS Volume 7, 2003


Page(s)		171 - 208
DOI		10.1051/ps:2003001

ESAIM: P&S, March 2003, Vol. 7, pp. 171-208
DOI: 10.1051/ps:2003001

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

Pierre Del Moral and L. Miclo

LSP, UMR 5583 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France;
e-mail: delmoral@cict.fr, miclo@cict.fr

(Received October 30, 2001. Revised September 11, 2002.)

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.

Mathematics Subject Classification. 62L20, 65C05, 81Q05, 82C22

Key words: 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.

© EDP Sciences, SMAI 2003

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