Infinite Interacting Brownian Motions and EVI Gradient Flows

We provide a sufficient condition under which the time marginal of the law of μ-symmetric diffusion process X in the infinite dimensional configuration space Υ is the unique Wasserstein W₂, ₃_⏤ EVI-gradient flow of the relative entropy (a. k. a. Kullback-Leibler divergence) Ent_μ on the space P (Υ) of probability measures on Υ. Here, Υ is equipped with the ²-matching extended distance d_ Υ and a Borel probability μ while P (Υ) is endowed with the transportation extended distance W₂, ₃_ ⏤ with cost d_ Υ². Our results include the cases μ=sine₂ and μ= Airy₂ 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 (Υ, d_ Υ, μ) 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 X propagates number rigidity and tail triviality.