Cauchy Problem for Stochastic Nonlinear Schrödinger Equation with Nonlinear Energy-Critical Damping

We consider the Cauchy problem for the stochastic nonlinear Schr & ouml;dinger equation augmented by nonlinear energy-critical damping term arising in nonlinear optics and quantum field theory. Through examining the behavior of the momentum and energy functionals, we almost surely prove the existence and uniqueness of global solutions with continuous H1(Rd) valued paths. The results cover either defocusing nonlinearity in the full energy critical and subcritical range of exponents or focusing nonlinearity in the full subcritical range, as in the deterministic case.