Stochastic Nonlinear Schrödinger Equation. 1. A priori Estimates

We consider a nonlinear Schrödinger equation with a small real coefficient $\delta$ in front of the Laplacian. The equation is forced by a random forcing that is a white noise in time and is smooth in the space-variable $x$ from a unit cube; Dirichlet boundary conditions are assumed on the cube's boundary. We prove that the equation has a unique solution that vanishes at $t=0$. This solution is almost certainly smooth in $x$, and the $k$th moment of its $m$th Sobolev norm in $x$ is bounded by $C_{m,k}\delta^{-km-k/2}$. The proof is based on a lemma that can be treated as a stochastic maximum principle.