# Hydrodynamic limit of the Symmetric Exclusion Process on a compact Riemannian manifold

## TL;DR

This paper proves that the symmetric exclusion process on random grids approximating a compact Riemannian manifold converges to the heat equation, establishing a hydrodynamic limit linking microscopic particle dynamics to macroscopic diffusion.

## Contribution

It demonstrates the convergence of random walks to Brownian motion and the empirical density field to the heat equation on the manifold, bridging stochastic processes and PDEs on curved spaces.

## Key findings

- Random walks on grids converge to Brownian motion on the manifold.
- Empirical density fields converge to the heat equation solution.
- Provides a rigorous hydrodynamic limit for exclusion processes on manifolds.

## Abstract

We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it converges to the solution of the heat equation on the manifold.