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

Pierre Del Moral; L. Miclo

ESAIM: Probability and Statistics (2003)

  • Volume: 7, page 171-208
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Del Moral, Pierre, and Miclo, L.. "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups." ESAIM: Probability and Statistics 7 (2003): 171-208. <http://eudml.org/doc/244833>.

@article{DelMoral2003,
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.},
author = {Del Moral, Pierre, Miclo, L.},
journal = {ESAIM: Probability and Statistics},
keywords = {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; Feynman-Kac formula; semigroups on Banach spaces},
language = {eng},
pages = {171-208},
publisher = {EDP-Sciences},
title = {Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups},
url = {http://eudml.org/doc/244833},
volume = {7},
year = {2003},
}

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
PB - EDP-Sciences
VL - 7
SP - 171
EP - 208
AB - 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.
LA - eng
KW - 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; Feynman-Kac formula; semigroups on Banach spaces
UR - http://eudml.org/doc/244833
ER -

References

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  1. [1] K. Burdzy, R. Holyst, D. Ingerman and P. March, Configurational transition in a Fleming–Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A 29 (1996) 2633-2642. Zbl0901.60054
  2. [2] K. Burdzy, R. Holyst and P. March, A Fleming–Viot particle representation of Dirichlet Laplacian. Comm. Math. Phys. 214 (2000) 679-703. Zbl0982.60078
  3. [3] P. Del Moral and A. Guionnet, On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré 37 (2001) 155-194. Zbl0990.60005MR1819122
  4. [4] 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). Zbl0963.60040
  5. [5] 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). 
  6. [6] P. Del Moral and L. Miclo, A Moran particle approximation of Feynman–Kac formulae. Stochastic Process. Appl. 86 (2000) 193-216. Zbl1030.65004
  7. [7] 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). 
  8. [8] P. Del Moral and L. Miclo, Genealogies and increasing propagation of chaos for Feynman–Kac and genetic models. Ann. Appl. Probab. 11 (2001) 1166-1198. Zbl1040.60031
  9. [9] 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. Zbl0333.60078MR486395
  10. [10] J. Feng and T. Kurtz, Large deviations for stochastic processes. http://www.math.umass.edu/ feng/Research.html Zbl1113.60002MR2260560
  11. [11] J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes. Springer-Verlag, A Series of Comprehensive Studies in Math. 288 (1987). Zbl0635.60021MR959133
  12. [12] T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York (1980). Zbl0435.47001MR1335452
  13. [13] M. Reed and B. Simon, Methods of modern mathematical physics, II, Fourier analysis, self adjointness. Academic Press, New York (1975). Zbl0308.47002MR493420
  14. [14] A.S. Sznitman, Brownian motion, obstacles and random media. Springer, Springer Monogr. in Math. (1998). Zbl0973.60003MR1717054

Citations in EuDML Documents

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  1. Gideon Simpson, Mitchell Luskin, Numerical analysis of parallel replica dynamics
  2. Mathias Rousset, On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes
  3. Mohamed El Makrini, Benjamin Jourdain, Tony Lelièvre, Diffusion Monte Carlo method: Numerical Analysis in a Simple Case
  4. Pierre Del Moral, Arnaud Doucet, Sumeetpal S. Singh, A backward particle interpretation of Feynman-Kac formulae
  5. Denis Villemonais, General approximation method for the distribution of Markov processes conditioned not to be killed
  6. Pierre Del Moral, Nicolas G. Hadjiconstantinou, An introduction to probabilistic methods with applications

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