Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schr\"odinger Equation with Noise

We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear Schr\"odinger equation with a focusing nonlinearity of order $p<6$. Key properties of these Gibbs measures, in particular absence of "phase transitions" and regularity properties of field samples, are established. We then study a time evolution of this system given by the Hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature. The noise is of Ornstein-Uhlenbeck type for the Fourier modes of the field, with the strength of the noise decaying to zero, as the frequency of the mode tends to $\infty$. We prove exponential approach of the state of the system to a grand-canonical Gibbs measure at a temperature and "chemical potential" determined by the stochastic noise term.