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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- 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.
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- P. Del Moral and L. Miclo, A Moran particle approximation of Feynman-Kac formulae. Stochastic Process. Appl.86 (2000) 193-216.
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- 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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