Propagation of chaos, Wasserstein gradient flows and toric Kahler-Einstein metrics

TL;DR

This paper develops a probabilistic framework linking particle systems, Wasserstein gradient flows, and Kahler-Einstein metrics, revealing new evolution equations and a real analog of the Yau-Tian-Donaldson conjecture in complex geometry.

Contribution

It introduces a novel large N-limit evolution equation derived from stochastic particle systems, connecting Kahler geometry with Wasserstein gradient flows and extending the Yau-Tian-Donaldson conjecture.

Findings

Derivation of a drift-diffusion PDE coupled with Monge-Ampere operator

Identification of the PDE as a Wasserstein gradient flow of the K-energy functional

Establishment of a probabilistic analog of the Yau-Tian-Donaldson conjecture

Abstract

Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general non-equilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasi-convex N-particle interaction energy. We show that a deterministic "macroscopic" evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at "negative temperature" the corresponding limiting evolution equation yields a drift-diffusion equation, coupled to the Monge-Ampere operator, whose static solutions correspond to toric Kahler-Einstein metrics. This drift-diffusion equation is the gradient flow on the Wasserstein space of probability measures of the K-energy functional in Kahler geometry and it can be seen as a fully non-linear version of various extensively studied dissipative evolution equations and conservations laws, including the Keller-Segel equation and Burger's equation. We also obtain a real probabilistic (and tropical) analog of the complex geometric Yau-Tian-Donaldson conjecture in this setting. In a companion paper applications to singular pair interactions are given.