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Asymptotic Properties of Random Homology Induced by Diffusion Processes

Artem Galkin, Mauro Mariani

Abstract

We investigate the asymptotic behavior, in the long time limit, of the random homology associated to realizations of stochastic diffusion processes on a compact Riemannian manifold. In particular a rigidity result is established: if the rate is quadratic, then the manifold is a locally trivial fiber bundle over a flat torus, with fibers being minimal in a weighted sense (that is, regarding the manifold as a metric measured space, with the invariant probability being the weight measure). Surprisingly, this entails that at least for some classes of manifolds, the homology of non-reversible processes relaxes to equilibrium slower than its reversible counterpart (as opposed to the respective empirical measure, which relaxes faster).

Asymptotic Properties of Random Homology Induced by Diffusion Processes

Abstract

We investigate the asymptotic behavior, in the long time limit, of the random homology associated to realizations of stochastic diffusion processes on a compact Riemannian manifold. In particular a rigidity result is established: if the rate is quadratic, then the manifold is a locally trivial fiber bundle over a flat torus, with fibers being minimal in a weighted sense (that is, regarding the manifold as a metric measured space, with the invariant probability being the weight measure). Surprisingly, this entails that at least for some classes of manifolds, the homology of non-reversible processes relaxes to equilibrium slower than its reversible counterpart (as opposed to the respective empirical measure, which relaxes faster).

Paper Structure

This paper contains 17 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Rate function for random homologies
  5. Main result
  6. Quadratic bounds and symmetries
  7. Asymptotically Gaussian homologies
  8. Open problems
  9. Technical preliminaries
  10. Lifted generator
  11. Weighted harmonic forms
  12. Weighted minimality and harmonicity of submersions
  13. Weighted tension field
  14. Weighted minimality of submersions
  15. Weighted Albanese map
  16. ...and 2 more sections

Key Result

Proposition \oldthetheorem

There exists $\bar{h} \in H_1(M,R)$, that only depends on $g$ and $b$, such that and Moreover in the quasi-reversible case (see Definition d:quasireversible-(b)), for $\bar{c} \in H^1(M;{\mathbb R})$ the cohomology class of $gb$, $G$ satisfies

Theorems & Definitions (30)

  • Definition \oldthetheorem: Large Deviations Rates
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 20 more