The stochastic Gross-Pitaevskii equation is used as a model of Bose-Einstein condensation (BEC) at positive temperature. The equation is a complex Ginzburg- Landau equation with a trapping potential and an additive space-time white noise. A positive temperature effect, for example, the spontaneous vortex formation by a sudden quench in BEC (seen as a phase transition) is of great interest in physics, and the Gibbs equilibrium is the key ingredient in the analysis from the point of view in statistical physics. In this talk we will present some mathematical results on the 2D stochastic Gross-Pitaevskii equation, where an `inhomogeneous' renormalization is required to give a sense to the nonlinearity.
This talk will be given in Japanese, but the slides are written in English.