Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles

TL;DR

This paper establishes the hydrodynamic limit of the symmetric exclusion process on complex geometric structures, deriving associated Fokker-Planck equations on Riemannian manifolds and principal bundles.

Contribution

It extends the hydrodynamic limit results to weighted Riemannian manifolds and principal bundles, incorporating geometric and probabilistic techniques.

Findings

Hydrodynamic limit described by Fokker-Planck equations on manifolds and bundles

Extension of previous results to new geometric settings

Convergence of single particle random walk to diffusion process

Abstract

We prove that the hydrodynamic limit of the symmetric exclusion process (SEP) is a Fokker-Planck equation in the setting of Poisson random neighborhood graphs approximating a weighted Riemannian manifold with Ricci curvature bounded from below. We also consider the lift of the SEP to a principal bundle, and obtain a Fokker-Planck equation with a weighted horizontal Laplacian as its hydrodynamic limit. Both results significantly extend the geometric settings in which one can prove the hydrodynamic limit from duality combined with convergence of the single particle random walk towards a diffusion process.