Anne De Bouard

The stochastic Gross-Pitaevskii equation is a mean-field model designed to describe a Bose-Einstein condensate close to the critical condensation temperature. It is a complex Ginzburg-Landau equation, with a harmonic confining potential and additive space-time white noise. In this talk, we will discuss the space dimension-two case, for which renormalization is required. We will construct the Gibbs measure for this equation, which is also formally invariant for the nonlinear Schr ̈odinger equation with harmonic potential in dimension two. It will also be shown that this measure is singular with respect to the underlying Gaussian measure. This talk is based on joint works with A. Debussche (ENS Rennes) and R. Fukuizumi (Waseda University, Japan).