The well-posedness of the stochastic nonlinear Schr\"odinger equations in $H^2(\mathbb{R}^d)$

Abstract

The Cauchy problem for the stochastic nonlinear Schr\"odinger equation with multiplicative noise is considered where the nonlinear term is of power type and the noise coefficients are purely imaginary numbers. The main purpose of this paper is to construct classical solutions in $H^2(\mathbb{R}^d)$ for the problem. The techniques of Kato \cite{K87,K89} work well in overcoming this difficulty even for the stochastic equations.