Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise

Anne de Bouard (1) , Arnaud Debussche (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 IRMAR - Institut de Recherche Mathématique de Rennes

We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrödinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability

Mots clés: Nonlinear Schrödinger equations; stochastic partial differential equations; white noise; blow-up; variance identity; support theorem

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Anne de Bouard, Arnaud Debussche. Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise. Annals of Probability, 2005, 33 (3), pp.1078-1110. ⟨10.1214/009117904000000964⟩. ⟨hal-00383396⟩