The nonlinear stochastic Schr\"odinger equation via stochastic Strichartz estimates

TL;DR

This paper establishes local existence and uniqueness of solutions for the stochastic nonlinear Schrödinger equation with nonlinear Stratonovich noise, and proves global solutions under certain conditions, using stochastic Strichartz estimates.

Contribution

It introduces a novel approach using stochastic Strichartz estimates to analyze the stochastic NLS with nonlinear noise, covering both subcritical and critical cases.

Findings

Proved local existence and uniqueness of solutions.

Established global existence under additional noise assumptions.

Developed new stochastic Strichartz estimates for the analysis.

Abstract

We consider the stochastic NLS with nonlinear Stratonovic noise for initial values in $L^2(R^d)$ and prove local existence and uniqueness of a mild solution for subcritical and critical nonlinearities. The proof is based on deterministic and stochastic Strichartz estimates. In the subcritical case, we prove that the solution is global, if we impose an additional assumption on the nonlinear noise.