We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

中文翻译：

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用白噪声初始数据求解 4NLS

我们为圆上的（重新归一化的）三次四阶非线性薛定谔方程构建了全局时间奇异动力学，将白噪声测量作为不变测量。为此，我们引入了“随机共振/非线性分解”，它允许我们挑选出解的奇异分量。与经典的 McKean、Bourgain、Da Prato-Debussche 类型的论证不同，这个奇异分量是非线性的，由随机初始数据的任意高次幂组成。我们还采用了随机规范变换，导致随机傅里叶限制范数空间。对于这个问题，收缩论证不起作用，而是通过研究随机规范变换下的部分迭代 Duhamel 公式来建立平滑近似解的收敛性。