Infinite Interacting Brownian Motions and EVI Gradient Flows (2509.06869v1)

Abstract: 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.