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

Del Moral, Pierre; Miclo, L.

ESAIM: Probability and Statistics, Tome 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.

Classification : 62L20, 65C05, 81Q05, 82C22
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

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