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 $\mathrm{i}du+(\mathrm{\Delta }u+\lambda |u{|}^{\alpha }u+ia(t)u)dt=dW$. 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.