Representation formula for stochastic Schrödinger evolution equations and applications

Anne de Bouard (1) , Reika Fukuizumi (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Division of Mathematics, Graduate School of Information Sciences

We prove a representation formula for solutions of Schrödinger equations with potentials multiplied by a temporal real-valued white noise in the Stratonovich sense. Using this formula, we obtain a dispersive estimate which allows us to study the Cauchy problem in L2 or in the energy space of model equations arising in Bose-Einstein condensation or in fiber optics. Our results also give a justification of diffusion-approximation for stochastic nonlinear Schrödinger equations.

Mots clés: Nonlinear Schrödinger equation; stochastic partial differential equations; white noise; Feynman path integral; stationary phase method; diffusion-approximation

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

Anne de Bouard, Reika Fukuizumi. Representation formula for stochastic Schrödinger evolution equations and applications. Nonlinearity, 2012, 25 (11), pp.2993-3022. ⟨10.1088/0951-7715/25/11/2993⟩. ⟨hal-00804577⟩