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

ESAIM: Probability and Statistics (2003)

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

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topDel 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

top- [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] 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] 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] 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] 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] P. Del Moral and L. Miclo, A Moran particle approximation of Feynman–Kac formulae. Stochastic Process. Appl. 86 (2000) 193-216. Zbl1030.65004
- [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] 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] 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] J. Feng and T. Kurtz, Large deviations for stochastic processes. http://www.math.umass.edu/ feng/Research.html Zbl1113.60002MR2260560
- [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] T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York (1980). Zbl0435.47001MR1335452
- [13] M. Reed and B. Simon, Methods of modern mathematical physics, II, Fourier analysis, self adjointness. Academic Press, New York (1975). Zbl0308.47002MR493420
- [14] A.S. Sznitman, Brownian motion, obstacles and random media. Springer, Springer Monogr. in Math. (1998). Zbl0973.60003MR1717054

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