通过 Föllmer 流构造 Lipschitz 传输映射

Exponential and Gaussian concentration of 1-Lipschitz maps

In this paper, we prove an exponential and Ganssian concentration inequality for 1-Lipschitz maps from mm-spaces to Hadamard manifolds. In particular, we give a complete answer to a question by M. Gromov.

1-Lipschitz的中心和$L^p$ -浓度映射到$\mathbb{R}$ -树中

Central and $L^p$-concentration of 1-Lipschitz maps into $\mathbb{R}$-trees

本文研究了从mm-空间到$\mathbb{R}$ -树的1-Lipschitz映射的Lévy-Milman集中现象.我们的主要定理断言，集中到$\mathbb{R}$ -树是等价的集中到真正的线。

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the stochastic recursion $X_{n}^{x}=\p_{θ_{n}}(X_{n-1}^{x})$, starting at $x\in\R^d$, where $θ_{1}, θ_{2},...$ are i.i.d. random variables taking their values in a metric space $(Θ, \mathfrak{r}),$ and $\p_{θ_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $ν$. Under appropriate assumptions on $\p_{θ_n}$, we will show that the measure $ν$ has a heavy tail with the exponent $α>0$ i.e. $ν(\{x\in\R^d: |x|>t\})\asymp t^{-α}$. Using this result we show that properly norm…

A Measure on the space of Lipschitz isometric maps of a compact 1-manifold into $\mathbb R^2$

Let $M$ be a compact 1-manifold. Given a continuous function $g:M\to \mathbb R_+$ we consider the following ordinary differential equation: $\|\dot{f}(t)\|=g(t)$, where $f:M\to \mathbb R^2$. We construct a probability measure on the space of almost everywhere differentiable solutions of this differential equation and study this measure. A solution of this equation can be viewed as an isometric immersion of a compact 1-manifold into $\mathbb R^2$. Nash's convergence technique in the proof of isometric $C^1$-immersion theorem plays an important role in the construction.

热流、对数凹性与Lipschitz传输映射

Heat flow, log-concavity, and Lipschitz transport maps

本文研究热方程解的log-Hessian估计，对强对数凹测度的对数Lipschitz扰动给出显式一致下界，并据此估计从标准高斯到该类测度的Lipschitz传输映射常数，同时讨论与分数扩散模型及对数Sobolev不等式的联系。

Lipschitz传输映射的耦合方法

A coupling approach to Lipschitz transport maps

本文提出一种基于耦合技术的概率方法，用于界定Langevin流映射的Lipschitz常数，并成功将其应用于构造对数凹测度到对数Lipschitz扰动的传输映射，同时放宽了凸性条件并移除了势能三阶导数界限。