Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.
Mots clés : large deviations, stochastic partial differential equations, nonlinear Schrödinger equations, white noise, projective limit, support theorem, blow-up, solitary waves
@article{PS_2005__9__74_0,
author = {Gautier, \'Eric},
title = {Large deviations and support results for nonlinear {Schr\"odinger} equations with additive noise and applications},
journal = {ESAIM: Probability and Statistics},
pages = {74--97},
publisher = {EDP-Sciences},
volume = {9},
year = {2005},
doi = {10.1051/ps:2005005},
mrnumber = {2148961},
zbl = {1136.60346},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2005005/}
}
TY - JOUR AU - Gautier, Éric TI - Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications JO - ESAIM: Probability and Statistics PY - 2005 SP - 74 EP - 97 VL - 9 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2005005/ DO - 10.1051/ps:2005005 LA - en ID - PS_2005__9__74_0 ER -
%0 Journal Article %A Gautier, Éric %T Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications %J ESAIM: Probability and Statistics %D 2005 %P 74-97 %V 9 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2005005/ %R 10.1051/ps:2005005 %G en %F PS_2005__9__74_0
Gautier, Éric. Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 74-97. doi : 10.1051/ps:2005005. http://www.numdam.org/articles/10.1051/ps:2005005/
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