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

ESAIM: Probability and Statistics (2010)

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

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topMoral, Pierre Del, and Miclo, L.. "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups." ESAIM: Probability and Statistics 7 (2010): 171-208. <http://eudml.org/doc/104302>.

@article{Moral2010,

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 = {Moral, Pierre Del, 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; Schrödinger operator; spectral decomposition; semigroups on Banach spaces},

language = {eng},

month = {3},

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/104302},

volume = {7},

year = {2010},

}

TY - JOUR

AU - Moral, Pierre Del

AU - Miclo, L.

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

JO - ESAIM: Probability and Statistics

DA - 2010/3//

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; Schrödinger operator; spectral decomposition; semigroups on Banach spaces

UR - http://eudml.org/doc/104302

ER -

## References

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- K. Burdzy, R. Holyst and P. March, A Fleming-Viot particle representation of Dirichlet Laplacian. Comm. Math. Phys.214 (2000) 679-703.
- P. Del Moral and A. Guionnet, On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré37 (2001) 155-194.
- 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).
- P. Del Moral and L. Miclo, A Moran particle approximation of Feynman-Kac formulae. Stochastic Process. Appl.86 (2000) 193-216.
- 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).
- 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.
- 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).

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