Gaussian fluctuations from random Schrödinger equation

The Schr{ö}dinger equation with spatial white noise potential

We consider the linear and nonlinear Schr{ö}dinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities.

A nonlinear Schr{ö}dinger equation with fractional noise

We study a stochastic Schr{ö}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schr{ö}dinger group, which allows us to co…

全空间上带空间白噪声的对数薛定谔方程

The logarithmic Schr{ö}dinger equation with spatial white noise on the full space

本文研究全空间上带乘性空间白噪声的对数薛定谔方程，因非线性无法用正则结构等方法，借助指数变换求解，适用于d≤2。

双线性 Strichartz 估计与非线性 Schrödinger 方程的几乎必然全局解

Bilinear Strichartz estimates and almost sure global solutions for the nonlinear Schr{ö}dinger equation

本文通过谐振子的 Bourgain 型双线性估计、lens 变换及 Kato 型光滑估计，构造超临界非线性 Schrödinger 方程在随机初值下的几乎必然全局解。

从随机 Zakharov 系统到乘性随机非线性 Schr{ö}dinger 方程

From Stochastic Zakharov System to Multiplicative Stochastic Nonlinear Schr{ö}dinger Equation

研究时间白噪声驱动、空间有色噪声的 Zakharov 系统在离子声速趋于无穷时向乘性随机非线性 Schr{ö}dinger 方程的收敛性，通过扩散近似框架与扰动测试函数方法克服噪声导致的奇异性，并证明一维情形下解的存在唯一性与依概率收敛。

Brenier-Schr{ö}dinger problem on compact manifold with boundary

We consider the Brenier-Schr{ö}dinger problem on compact manifolds with boundary. In the spirit of a work by Arnaudon, Cruzeiro, L{é}onard and Zambrini, we study the kinetic property of regular solutions and obtain a link to the Navier-Stokes equations with an impermeability condition. We also enhance the class of models for which the problem admits a unique solution. This involves a method of taking quotients by reflection groups for which we give several examples.