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

Jonathan Junné, Frank Redig, Rik Versendaal

Abstract

We define the symmetric exclusion process (SEP) on random neighbourhood graphs drawn from Poisson point processes on complete connected Riemannian manifolds equipped with a Gibbs reference measure. We obtain diffusions with drift induced by the potential of the Gibbs measure as hydrodynamic limits. This is a consequence of the duality between the (SEP) and random walks approximating weighted Laplacians. We also lift the (SEP) to principal bundles, and we obtain horizontal diffusions with drift as hydrodynamic limits. Examples include the orthonormal frame bundle with the horizontal Laplacian which plays a central role in the Eells-Elworthy-Émery construction of Riemannian Brownian motion.

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

Abstract

We define the symmetric exclusion process (SEP) on random neighbourhood graphs drawn from Poisson point processes on complete connected Riemannian manifolds equipped with a Gibbs reference measure. We obtain diffusions with drift induced by the potential of the Gibbs measure as hydrodynamic limits. This is a consequence of the duality between the (SEP) and random walks approximating weighted Laplacians. We also lift the (SEP) to principal bundles, and we obtain horizontal diffusions with drift as hydrodynamic limits. Examples include the orthonormal frame bundle with the horizontal Laplacian which plays a central role in the Eells-Elworthy-Émery construction of Riemannian Brownian motion.

Paper Structure

This paper contains 6 sections, 14 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Abstract hydrodynamic limit via duality
  4. Convergence of discrete random walks
  5. Lifting the (SEP) to principal bundles
  6. Computational lemmas

Key Result

Theorem 2.3

(Hydrodynamic limit of the (SEP)) Let $(G_N=(V_N,E_N))_{N \ge 1}$ be a sequence of random neighbourhood graphs on $\mathcal{M}$ in the sense of def:Random neighbourhood graph with symmetric weights $(W_N)_{N\ge1}$ given by eq:symmetric weights for thm. Let $\eta_0^N$ be an initial configuration of t Suppose that the bandwidth parameters $(h_N)_{N\ge1}$ are chosen in such a way that both $h_N \to 0

Theorems & Definitions (36)

  • Definition 2.1: Random neighbourhood graph
  • Definition 2.2: Symmetric exclusion process (SEP)
  • Theorem 2.3
  • Remark 2.4
  • Definition 3.1: Random walk (RW)
  • Theorem 3.2
  • proof
  • Definition 4.1: Weighted Laplacian
  • Remark 4.2
  • Theorem 4.3
  • ...and 26 more