Well-posedness of a parametrically forced nonlinear Schr\"odinger equation driven by translation-invariant noise

TL;DR

This paper establishes well-posedness for a parametrically forced nonlinear Schrödinger equation with multiplicative noise, demonstrating existence, uniqueness, and regularity of solutions in Sobolev spaces, and extending results to the standard cubic NLS.

Contribution

It proves well-posedness of a stochastic NLS with translation-invariant noise in all Sobolev spaces, using fixed-point and Itô calculus methods, and provides a blow-up criterion.

Findings

Solutions exist and are unique in $H^{\sigma}(\mathbb{R})$ for all $\sigma \geq 0$.

A blow-up criterion based on the $L^2$-norm is established.

Results extend to the standard cubic NLS with multiplicative noise.

Abstract

We prove well-posedness in $H^{\sigma}(\mathbb{R})$ for any $\sigma \in [0,\infty)$ of a parametrically forced nonlinear Schr\"odinger equation (PFNLS) in one dimension driven by multiplicative Stratonovich noise which has spatially homogeneous statistics. The noise is white in time and correlated in space. We first construct local mild solutions via a fixed-point argument. We then formulate a blow-up criterion by showing that the equation has persistence of integrability and regularity as long as the $L^2(\mathbb{R})$-norm of the solution remains finite. Afterwards we derive a pathwise estimate on the $L^2(\mathbb{R})$-norm using a mild It\^o formula. Our results also apply to the standard cubic NLS equation driven by multiplicative translation-invariant Stratonovich noise.