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Non-equilibrium functional inequalities for finite Markov chains

Bastian Hilder, Patrick van Meurs, Upanshu Sharma

Abstract

Functional inequalities such as the Poincaré and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications, including multiscale and non-reversible dynamics, require analysing probability measures that are not at equilibrium, where the classical theory tied to steady states no longer applies. We introduce generalised versions of these inequalities for arbitrary positive measures on a finite state space, retaining key structural properties of their classical counterparts. In particular, we prove continuity of the associated constants with respect to the reference measure and establish explicit positive lower bounds. As an application, we derive quantitative coarse-graining error estimates for non-reversible Markov chains, both with and without explicit scale separation, and propose a quantitative criterion for assessing the quality of coarse-graining maps.

Non-equilibrium functional inequalities for finite Markov chains

Abstract

Functional inequalities such as the Poincaré and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications, including multiscale and non-reversible dynamics, require analysing probability measures that are not at equilibrium, where the classical theory tied to steady states no longer applies. We introduce generalised versions of these inequalities for arbitrary positive measures on a finite state space, retaining key structural properties of their classical counterparts. In particular, we prove continuity of the associated constants with respect to the reference measure and establish explicit positive lower bounds. As an application, we derive quantitative coarse-graining error estimates for non-reversible Markov chains, both with and without explicit scale separation, and propose a quantitative criterion for assessing the quality of coarse-graining maps.
Paper Structure (22 sections, 13 theorems, 168 equations, 2 figures)

This paper contains 22 sections, 13 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the article and summary of notation.
  3. The classical functional inequalities
  4. Generalised functional inequalities
  5. Notation
  6. Generalised Dirichlet form and the Fisher information
  7. Generalised PI and LSI inequalities
  8. Properties of the generalised PI and LSI constants
  9. gLSI implies gPI and tensorisation
  10. Continuity of the generalised LSI constant
  11. Lower bounds on $\alpha_{\mathop{\mathrm{gPI}}\nolimits}(\zeta)$ and $\alpha_{\mathop{\mathrm{gLSI}}\nolimits}(\zeta)$
  12. Applications
  13. Comparing two solutions to the forward Kolmogorov equation
  14. Clustering in Markov chains
  15. Clustering in Markov chains without explicit scale separation
  16. ...and 7 more sections

Key Result

Proposition 3.2

For any $\zeta\in \mathcal{P}_{+}(\mathcal{Z})$, the generalised Dirichlet form def:genDirichlet and the generalised Fisher information def:genFisher with respect to $\zeta$ satisfy the following.

Figures (2)

  • Figure 1: Energy landscape with two macro-states and six micro-states.
  • Figure 2: Depiction of the transitions of the Markov chain generated by \ref{['eq:generator-example']}.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 32 more