On the wellposedness for periodic nonlinear Schr\"odinger equations with white noise dispersion

TL;DR

This paper establishes the almost sure global well-posedness of a periodic nonlinear Schrödinger equation with white noise dispersion for certain nonlinearities, extending previous solutions and analyzing ill-posedness at higher nonlinearities.

Contribution

It proves stochastic Strichartz estimates for the equation, extends Fourier restriction spaces to stochastic settings, and compares solutions with rough path techniques, addressing well-posedness and ill-posedness.

Findings

Global well-posedness for 1 < p ≤ 3 in L^2 initial data

Solutions coincide with rough path solutions by Chouk and Gubinelli

Ill-posedness of the quintic case (p=5) in L^1_ω C_t L^2_x

Abstract

We consider a periodic nonlinear Schr\"odinger equation with white noise dispersion and a power nonlinearity given by \begin{equation*} idu = \Delta u \circ dW_t + |u|^{p-1}u\;dt \end{equation*} By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with $L^2$ initial data for nonlinearities with exponent $1 < p \leq 3$. By generalizing the Fourier restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation ($p=5$), and show that it is analytically illposed in $L^1_\omega C_t L^2_x$.