We study the wellposedness of the periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by \(idu = \Delta u \circ dW_t + |u |^{p-1}u\;dt\). We 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 \(1<p\le 3\). 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 Differ Equ 40(11):2047–2081, 2015) using rough path techniques. We also consider the quintic equation (\(p=5\)), and show that it is analytically illposed in \(L^1_\omega C_t L^2_x\).

我们研究了具有白噪声色散和由\(idu = \Delta u \circ dW_t + |u |^{p-1}u\;dt\)给出的功率非线性的周期性非线性薛定谔方程的适定性。我们对该方程进行了 Strichartz 估计，然后用该估计来证明该方程与指数为\(1<p\le 3\)的非线性的\(L^2\)初始数据几乎确定的全局适定性。通过将傅立叶限制空间\(X^{s,b}\)推广到随机设置，我们还证明我们的解决方案与 Chouk 和 Gubinelli 构造的解决方案一致（Commun Part Differ Equ 40(11):2047–2081 ，2015）使用粗糙路径技术。我们还考虑五次方程 ( \(p=5\)），并表明它在\(L^1_\omega C_t L^2_x\)中是分析不适定的。