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

TL;DR

This paper establishes the existence of classical solutions in the Sobolev space $H^2(R^d)$ for stochastic nonlinear Schrödinger equations with multiplicative noise, extending classical deterministic methods to stochastic settings.

Contribution

It demonstrates the well-posedness of stochastic nonlinear Schrödinger equations in $H^2(R^d)$ using Kato's techniques, a novel extension to stochastic PDEs.

Findings

Existence of classical solutions in $H^2(R^d)$

Extension of Kato's methods to stochastic equations

Well-posedness results for stochastic NLS with imaginary noise coefficients

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.