Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Eric Gautier (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CREST - Centre de Recherche en Économie et Statistique

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 space wise. 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 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: projective limit; support theorem; blow-up; solitary waves; white noise; nonlinear Schrödinger equations; large deviations; stochastic partial differential equations

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Eric Gautier. Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications. ESAIM: Probability and Statistics, 2005, 9, pp.74-97. ⟨10.1051/ps:2005005⟩. ⟨hal-00002106⟩