Gautier, Eric (2005) Large deviations for stochastic nonlinear Schrödinger equations and applications. PhD thesis Mathématiques et Applications, ENSAE.

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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 a support theorem for fractional additive Gaussian noises.

Table of content

1 Introduction
1.1 Eléments sur les équations déterministes
1.1.1 Propriétés du groupe linéaire
1.1.2 L'équation non linéaire, caractère localement bien posé du problème de Cauchy
1.1.3 Invariants de l'équation, existence globale et explosion en temps fini
1.1.4 Les ondes solitaires
1.2 Les perturbations aléatoires
1.2.1 Des motivations
1.2.2 Le bruit, le processus de Wiener et les mesures Gaussiennes
1.2.3 Les équations de Schrödinger non linéaires stochastiques
1.3 Les grandes déviations
1.3.1 Présentation
1.3.2 Des résultats généraux
1.3.3 Transport par image directe de PGDs
2 Présentation des résultats
2.1 Grandes déviations et support pour un bruit additif
2.2 Grandes déviations uniformes pour un bruit multiplicatif
2.3 Application à la transmission par solitons
2.4 Application `a la sortie d'un domaine d'attraction
2.5 Le cas d'un bruit additif fractionnaire
3 Conclusion et perspectives
A Large deviations in the additive case and applications
A.1 Introduction
A.2 Notations and preliminary results
A.2.1 Properties of the group
A.2.2 Topology and trajectory spaces
A.2.3 Statistical properties of the noise
A.2.4 The random perturbation
A.2.5 Continuity with respect to the perturbation
A.3 Sample path large deviations
A.4 The support of the law of the solution
A.5 Applications to the blow-up times
A.5.1 Probability of blow-up after time T
A.5.2 Probability of blow-up before time T
A.5.3 Bounds for the approximate blow-up time
A.6 Applications to nonlinear optics
A.6.1 Upper bounds
A.6.2 Lower bounds
B Large deviations in the multiplicative case
B.1 Introduction
B.2 Preliminaries and statement of the result
B.3 Continuity of the skeleton with respect to the control on the sets Ca and exponential tail estimates
B.4 Almost continuity of the Iô map
B.5 End of the proof of the uniform LDP
B.6 Applications to the blow-up times
C Application to the timing jitter 119
C.1 Introduction
C.2 Notations and preliminaries
C.3 Tails of the the mass and center with additive noise
C.4 Tails of the center in the multiplicative case
C.5 Annex - proof of Theorem C.2 - 1
D Exit from a domain for weakly damped equations
D.1 Introduction
D.2 Preliminaries
D.3 Exit from a domain of attraction in L2
D.4 Exit from a domain of attraction in H1
D.5 Annex - proof of Theorem D.2 - 1
E The case of a fractional additive noise
E.1 Introduction
E.2 Preliminaries
E.3 Local well posedness and necessary results
E.4 The main results

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