Quantitative Bounds on the Rate of Approach to Equilibrium for some One-Dimensional Stochastic Non-Linear Schr\'odinger Equations

We establish quantitative bounds on the rate of approach to equilibrium for a system with infinitely many degrees of freedom evolving according to a one-dimensional focusing nonlinear Schr\"odinger equation with diffusive forcing. Equilibrium is described by a generalized grand canonical ensemble. Our analysis also applies to the easier case of defocusing nonlinearities