论文标题
分数非线性schrödinger方程的高定期措施高斯度量的准不转变
Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations
论文作者
论文摘要
我们考虑了一维非线性的分数非线性schrödinger方程（FNLS）的库奇问题，具有立方非线性和高分散参数$α> 1 $，遵守负sobolev juromity $σ<s- \ s- \ frac \ frac {1} $ s的高斯随机初始数据，我们表明，对于所有$ s _ {*}（α）<s \ leq \ frac {1} {2} $，该方程几乎肯定是全球范围的。此外，在$ h^{s}（\ mathbb t）上支持的关联的高斯度量$在等式的流程下是准不变。对于$α<\ frac {1} {20}（17 + 3 \ sqrt {21}）\大约1.537 $，初始数据的规律性低于确定性良好的度量理论提供的。 我们通过遵循Diperna-Lions的方法（1989）获得了这一结果；首先显示无限二维liouville方程解决方程的全局界限，用于运输高斯度量，然后通过将波尔加因的不变性测量参数转移到准抗变异的设置中，将这些边界传输到方程的解。这使我们能够从其概率的局部良好性理论中引导（FNL）的解决方案的全局界限。
We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter $α> 1$, subject to a Gaussian random initial data of negative Sobolev regularity $σ<s-\frac{1}{2}$, for $s \le \frac 12$. We show that for all $s_{*}(α) <s\leq \frac{1}{2}$, the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on $H^{s}(\mathbb T)$ is quasi-invariant under the flow of the equation. For $α< \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537$, the regularity of the initial data is lower than the one provided by the deterministic well-posedness theory. We obtain this result by following the approach of DiPerna-Lions (1989); first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting Bourgain's invariant measure argument to the quasi-invariance setting. This allows us to bootstrap almost sure global bounds for the solution of (FNLS) from its probabilistic local well-posedness theory.
