用白噪声初值求解4-NLS
Solving the 4NLS with white noise initial data
摘要：
以白噪声测度为不变测度，构造了圆上三次四阶非线性薛定谔方程的时间域整体奇异动力学.为此，我们引入了“随机共振/非线性分解”，这使我们能够挑选出的解决方案的奇异分量。与经典的McKean，Bourgain，Da Prato-Debussche型参数不同，这个奇异分量是非线性的，由随机初始数据的任意高的幂组成。我们还采用了随机规范变换，导致随机傅立叶限制范数空间。对于这个问题，压缩参数不起作用，而是通过研究随机规范变换下的部分迭代Duhamel公式来建立光滑逼近解的收敛性。我们减少了关键的非线性估计的白噪声的某些随机多重线性泛函的有界性。
英文摘要：
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 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.
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