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Grandes déviations pour des équations de Schrödinger non linéaires stochastiques et applications

Abstract : This thesis is dedicated to the study of the small noise asymptotic in random perturbations of nonlinear Schrödinger equations. The noises are Gaussian, mostly white in time and always colored in space, of additive and multiplicative types. Large deviations are such that the behavior of the stochastic system differs significantly from the deterministic one. As the noise goes to zero the probability of such rare events goes to zero on a logarithmic scale with speed given by the noise amplitude. We prove large deviation principles at the level of paths. The rate of convergence to zero of the logarithm of the probabilities is related to an optimal control problem. Our first application is to the blow-up times. We then apply our results to the study of the small noise asymptotic of the tails of the mass and position of the soliton-like pulse in a "white noise limit". The fluctuations of these quantities are the main causes of error in optical soliton transmission. We also consider the problem of the mean exit times and the exit points from a neighborhood of zero for weakly damped equations. Finally we present large deviations and support theorem for fractional additive Gaussian noises.
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Contributor : Eric Gautier <>
Submitted on : Sunday, January 1, 2006 - 7:57:42 PM
Last modification on : Friday, July 10, 2020 - 4:04:14 PM
Long-term archiving on: : Saturday, April 3, 2010 - 7:56:27 PM

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  • HAL Id : tel-00011274, version 1

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Eric Gautier. Grandes déviations pour des équations de Schrödinger non linéaires stochastiques et applications. Mathématiques [math]. ENSAE ParisTech, 2005. Français. ⟨tel-00011274⟩

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