标题： 无限相互作用的布朗运动与EVI梯度流 显示英文标题

标题： Infinite Interacting Brownian Motions and EVI Gradient Flows

摘要： 我们提供了一个充分条件，使得无限维配置空间$\mathbf \Upsilon$中$\mu$对称扩散过程$\mathsf{X}$的时间边缘分布是相对熵（又称）的唯一 Wasserstein$\mathsf{W}_{2, \mathsf{d}_\mathbf\Upsilon}$ $\mathsf{EVI}$ 梯度流 Kullback-Leibler散度) $\mathrm{Ent}_{\mu}$ 在概率测度空间 $P(\mathbf \Upsilon)$ 上的 $\mathbf \Upsilon$。 这里，$\mathbf \Upsilon$配备了$\ell^2$匹配的扩展距离$\mathsf d_\mathbf \Upsilon$和一个 Borel 概率$\mu$，而$P(\mathbf \Upsilon)$配备了运输扩展距离$\mathsf W_{2, \mathsf d_\mathbf \Upsilon}$，成本为$\mathsf d_\mathbf \Upsilon^2$。 我们的结果包括 $\mu=\mathsf{sine}_2$ 和 $\mu= \mathsf{Airy}_2$ 点过程，其中相关的扩散过程分别是体相和软边极限下无限维Dyson型随机微分方程的无标签解。 作为应用，我们证明扩展度量测度空间 $(\mathbf \Upsilon , \mathsf{d}_\mathbf \Upsilon, \mu)$ 满足黎曼曲率-维数（RCD）条件以及扭曲的Brunn-Minkowski不等式、HWI不等式和其他几个函数不等式。 最后我们证明 $\mathsf{X}$ 的定律的时间边缘传播了数刚性与尾部平凡性。 显示英文摘要

摘要： We provide a sufficient condition under which the time marginal of the law of $\mu$-symmetric diffusion process $\mathsf{X}$ in the infinite dimensional configuration space $\mathbf \Upsilon$ is the unique Wasserstein $\mathsf{W}_{2, \mathsf{d}_\mathbf\Upsilon}$ $\mathsf{EVI}$-gradient flow of the relative entropy (a.k.a. Kullback-Leibler divergence) $\mathrm{Ent}_{\mu}$ on the space $P(\mathbf \Upsilon)$ of probability measures on $\mathbf \Upsilon$. Here, $\mathbf \Upsilon$ is equipped with the $\ell^2$-matching extended distance $\mathsf d_\mathbf \Upsilon$ and a Borel probability $\mu$ while $P(\mathbf \Upsilon)$ is endowed with the transportation extended distance $\mathsf W_{2, \mathsf d_\mathbf \Upsilon}$ with cost $\mathsf d_\mathbf \Upsilon^2$. Our results include the cases $\mu=\mathsf{sine}_2$ and $\mu= \mathsf{Airy}_2$ point processes, where the associated diffusion processes are unlabelled solutions to the infinite-dimensional Dyson-type stochastic differential equations in the bulk and soft-edge limit respectively. As an application, we show that the extended metric measure space $(\mathbf \Upsilon , \mathsf{d}_\mathbf \Upsilon, \mu)$ satisfies the Riemannian curvature-dimension (RCD) condition as well as the distorted Brunn-Minkowski inequality, the HWI inequality and several other functional inequalities. Finally we prove that the time marginal of the law of $\mathsf{X}$ propagates number rigidity and tail triviality.