布朗运动驱动的非线性随机薛定谔方程的基本性质

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.

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.

Dynamical aspects of generalized Schr{ö}dinger problem via Otto calculus -- A heuristic point of view

The defining equation $(\ast):\ \dot ω\_t=-F'(ω\_t),$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the family of slowed down gradient flow equations: $\dot ω^{ \varepsilon}\_t=- \varepsilon F'( ω^{ \varepsilon}\_t),$ where $\varepsilon>0$, and (ii) by considering the \emph{accelerations} $\ddot ω^{ \varepsilon}\_t$. We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus.A special formulation of the Schr{ö}dinger problem c…

killed Feynman-Kac 半群的长期行为与奇异 Schr{ö}dinger 势

Long time behavior of killed Feynman-Kac semigroups with singular Schr{ö}dinger potentials

本文研究了来自统计物理的各种过程的 killed Feynman-Kac 半群的紧性和长期行为，考虑了包括分数拉普拉斯算子在内的广泛过程，通过 Perron-Frobenius 定理和谱半径工具分析了奇异 Schr{ö}dinger 势的影响。

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。