Document ESAIM: P&S, April 2005, Vol. 9, pp. 74-97
DOI: 10.1051/ps:2005005
Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications
Éric Gautier1, 21 CREST-INSEE, URA D2200, 3 avenue Pierre Larousse, 92240 Malakoff, France.
2 IRMAR, UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France; eric.gautier@bretagne.ens-cachan.fr
(Received May 25, 2004. Revised December 15, 2004.)
Abstract
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.
Mathematics Subject Classification. 35Q51, 35Q55, 60F10, 60H15.
Key words: Large deviations, stochastic partial differential equations, nonlinear Schrödinger equations, white noise, projective limit, support theorem, blow-up, solitary waves.
© EDP Sciences, SMAI 2005