On global existence for the stochastic nonlinear Schrödinger equation with time-dependent linear loss/gain

This paper was focused on global existence for the stochastic nonlinear Schrödinger equation with time-dependent loss/gain, which read $ \mathrm{i}du+(\Delta u+\lambda|u|^\alpha u+ia(t)u)dt = dW $. We proved the global existence and uniqueness of the solution in $ H^1(\mathbb{R}^N) $ through the uniform boundedness of the momentum and energy functionals. The global existence result of the solution for this type of equation depended on the ranges of time-dependent loss/gain coefficient.