Stochastic nonlinear Schr\"odinger equation with almost space-time white noise

TL;DR

This paper establishes local well-posedness for a stochastic nonlinear Schrödinger equation driven by almost space-time white noise on a one-dimensional torus, advancing understanding of stochastic PDEs with highly irregular noise.

Contribution

It proves local well-posedness of the renormalized SNLS with almost space-time white noise, and discusses criticality in this stochastic setting.

Findings

Proved local well-posedness for the SNLS with almost space-time white noise.

Compared criticality in SNLS with stochastic heat equation.

Analyzed the effects of highly irregular noise on solution behavior.

Abstract

We study the stochastic cubic nonlinear Schr\"odinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space-time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).