On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

Abstract

We study the wellposedness of the periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given byWe develop Strichartz estimates for this equation, which we then use to prove almost sure global wellposedness of this equation with L 2 initial data for nonlinearities with exponent 1 < p ≤ 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 (Commun. Part. Diff. Eq. 40(11):2047-2081 using rough path techniques. We also consider the quintic equation (p = 5), and show that it is analytically illposed in L 1 ω CtL 2 x .