Infinite Interacting Brownian Motions and EVI Gradient Flows

Abstract

We give a sufficient condition under which the time-marginal law of $\mu$-reversible infinite interacting Brownian motions is characterised as the steepest gradient descent of the relative entropy in the Wasserstein space in the sense of evolution variational inequality (EVI). This is an infinite-dimensional generalisation of Jordan-Kinderlehrer-Otto/Ambrosio-Gigli-Savar\'e theory. Consequently, the configuration space (the space of locally finite point measures) endowed with the reversible measure $\mu$ is an RCD space and the time-marginal law is identified to the heat flow on this space. Our result covers the infinite-dimensional Dyson Brownian motion with bulk and soft-edge limits; the latter yields Airy line ensemble as its stationary process, a central object in KPZ universality. Our result therefore provides an optimal transport characterisation of these models as Wasserstein gradient flows, and establishes a range of new functional inequalities (HWI, distorted Brunn-Minkowski, dimension-free Harnack and many others) as a corollary. As an application, we discover the new phenomena, dynamical number rigidity and dynamical tail triviality, that the time-marginal law possesses number rigidity and tail triviality for every time $t>0$, revealing a propagation of random crystal and extremal structures by the Dyson Brownian motions.