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Large deviations and support results fornonlinear Schrödinger equations with additive noise andapplications

Published online by Cambridge University Press:  15 November 2005

Éric Gautier
Éric Gautier*
Affiliation:
CREST-INSEE, URA D2200, 3 avenue Pierre Larousse, 92240 Malakoff, France. IRMAR, UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France; [email protected]
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Abstract

Sample path large deviationsfor the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero arepresented. The noise is a complex additive Gaussian noise. It iswhite in time and colored in space. The solutions may be global orblow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologiesanalogue to projective limit topologies. In this setting, thesupport of the law of the solution is also characterized. As aconsequence, results on the law of the blow-up time andasymptotics when the noise converges to zero are obtained. Anapplication to the transmission of solitary waves in fiber opticsis also given.

Keywords

Large deviationsstochastic partial differential equationsnonlinearSchrödinger equationswhite noiseprojective limitsupport theoremblow-upsolitary waves.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 74 - 97
Copyright
© EDP Sciences, SMAI, 2005

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