Infinite Interacting Brownian Motions and EVI Gradient Flows
Kohei Suzuki
TL;DR
This work extends the gradient-flow paradigm for diffusion to the infinite-dimensional configuration space of locally finite point measures, establishing a sufficient condition under which time-marginals of ${\mu}$-reversible infinite interacting Brownian motions are ${\EVI}_K$ gradient flows of the relative entropy ${\mathcal H}_{\mu}$ in the Wasserstein space with the ${\ell^2}$-matching distance ${\mssd_{\boldsymbol\Upsilon}}$. The authors prove that the resulting extended metric measure space ${\boldsymbol\Upsilon}$ endowed with ${\mu}$ is ${\RCD}(K,\infty)$ and identify the time-marginals with the corresponding heat flow, linking Dyson Brownian motions in both bulk and soft edge to optimal transport. The framework yields a suite of functional inequalities (HWI, distorted Brunn–Minkowski, log-Harnack, Bakry–Émery) and dynamical phenomena such as dynamical number rigidity and tail triviality for all positive times, revealing propagation of random crystal structures. As applications, the sine${}_2$ and Airy${}_2$ ensembles satisfy the framework, offering a unified, transport-theoretic perspective on KPZ-universal objects like the Airy line ensemble and providing a robust foundation for further analysis of infinite-particle diffusions in geometry and probability.
Abstract
We give a sufficient condition under which the time-marginal law of $μ$-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-Savaré theory. Consequently, the configuration space (the space of locally finite point measures) endowed with the reversible measure $μ$ 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.