Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T10:45:08.157Z Has data issue: false hasContentIssue false
  • English
  • Français

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  and
L. Miclo
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:
Get access

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 ofSchrödinger 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 modelsextending the hard obstacle modelof 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.

Keywords

Feynman–Kac formulaSchrödinger operatorspectral radiusLyapunov exponentspectral decompositionsemigroups on a Banach spaceinteracting particle systemsgeneticalgorithmsasymptotic stabilitycentral limit theorems.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 171 - 208
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burdzy, K., Holyst, R., Ingerman, D. and March, P., Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A 29 (1996) 2633-2642. CrossRef
Burdzy, K., Holyst, R. and March, P., Fleming-Viot, A particle representation of Dirichlet Laplacian. Comm. Math. Phys. 214 (2000) 679-703. CrossRef
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).
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).
Del Moral, P. and Miclo, L., Moran, A particle approximation of Feynman-Kac formulae. Stochastic Process. Appl. 86 (2000) 193-216. CrossRef
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).
Del Moral, P. and Miclo, L., Genealogies and increasing propagation of chaos for Feynman-Kac and genetic models. Ann. Appl. Probab. 11 (2001) 1166-1198.
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.
J. Feng and T. Kurtz, Large deviations for stochastic processes. http://www.math.umass.edu/ feng/Research.html
J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes. Springer-Verlag, A Series of Comprehensive Studies in Math. 288 (1987).
T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York (1980).
M. Reed and B. Simon, Methods of modern mathematical physics, II, Fourier analysis, self adjointness. Academic Press, New York (1975).
A.S. Sznitman, Brownian motion, obstacles and random media. Springer, Springer Monogr. in Math. (1998).