Two dimensional Gross-Pitaevskii equation with space-time white noise

TL;DR

This paper investigates the two-dimensional stochastic Gross-Pitaevskii equation, a model for Bose-Einstein condensation at positive temperature, focusing on well-posedness, invariant measures, and stationary solutions with space-time white noise.

Contribution

It introduces an inhomogeneous Wick renormalization to establish well-posedness and proves the existence of invariant measures and stationary solutions for the stochastic model.

Findings

Proved well-posedness of the stochastic Gross-Pitaevskii equation.

Established existence of invariant measures.

Constructed stationary martingale solutions.

Abstract

In this paper we consider the two-dimensional stochastic Gross-Pitaevskii equation, which is a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg-Landau equation with a harmonic potential and an additive space-time white noise. We study the well-posedness of the model using an inhomogeneous Wick renormalization due to the potential, and prove the existence of an invariant measure and of stationary martingale solutions.