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Equivariant discretizations of diffusions, random walks, and harmonic functions

Werner Ballmann, Panagiotis Polymerakis

TL;DR

The paper develops Lyons-Sullivan discretizations for diffusion processes on manifolds with covering or properly discontinuous group actions, connecting spaces of L-harmonic functions to μ-harmonic functions on a discrete fiber X. It proves that, in cocompact or recurrent base scenarios, harmonic-function spaces and both Martin and Poisson boundaries correspond between the manifold and the discrete model, enabling transfer of Liouville-type results. The work emphasizes amenability and FC-hypercentrality as structural constraints, and extends the discretization framework to random walks on countable sets, expanding the analytic toolkit for harmonic analysis on spaces with symmetry. Collectively, it unifies continuous and discrete potential theory in geometric settings and provides new boundary-identification results across coverings and group actions.

Abstract

For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory.

Equivariant discretizations of diffusions, random walks, and harmonic functions

TL;DR

The paper develops Lyons-Sullivan discretizations for diffusion processes on manifolds with covering or properly discontinuous group actions, connecting spaces of L-harmonic functions to μ-harmonic functions on a discrete fiber X. It proves that, in cocompact or recurrent base scenarios, harmonic-function spaces and both Martin and Poisson boundaries correspond between the manifold and the discrete model, enabling transfer of Liouville-type results. The work emphasizes amenability and FC-hypercentrality as structural constraints, and extends the discretization framework to random walks on countable sets, expanding the analytic toolkit for harmonic analysis on spaces with symmetry. Collectively, it unifies continuous and discrete potential theory in geometric settings and provides new boundary-identification results across coverings and group actions.

Abstract

For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory.

Paper Structure

This paper contains 21 sections, 47 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Main results in the cocompact case
  3. Main results in the recurrent case
  4. Main results for normal coverings
  5. The motivating example (after Sullivan Su)
  6. Discussion
  7. Structure of the article
  8. Preliminaries
  9. Diffusion operators and processes
  10. Martin boundary
  11. Poisson boundary
  12. $FC$-hypercenters and $FC$-hypercentral groups
  13. Lyons-Sullivan discretization
  14. Balayage and $L$-harmonic functions
  15. $LS$-discretization and $L$-harmonic functions
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose that $N$ is compact and that $X$ is endowed with the family $\mu$ of $LS$-measures associated to appropriate $LS$-data. Then

Theorems & Definitions (87)

  • Example 1.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Remark 1.4
  • Example 1.6
  • ...and 77 more