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On the Kantorovich contraction of Markov semigroups

Pierre Del Moral, Mathieu Gerber

TL;DR

The paper develops an operator-theoretic framework to study contraction of Markov semigroups with respect to Kantorovich semi-distances, including Wasserstein distances, by combining a simple Lyapunov drift with a local contraction property. Under standard drift (H1) and local contraction (H2) conditions, it proves exponential decay of a broad class of Dobrushin-type contraction coefficients, extending existing $V$-norm results to Kantorovich distances. It provides explicit contraction bounds for both strict and exponential regimes and shows how these translate into Wasserstein contraction for finite-dimensional settings and continuous-time semigroups, including diffusion models. The framework yields new contraction results for models with boundaries, two-block Gibbs samplers, iterated random functions, and overdamped Langevin diffusions, all with couplings avoided in favor of Lyapunov-based proofs and operator theory. This offers a versatile, principled toolkit for stability analysis of Markov processes across statistics and applied probability, with clear criteria for establishing unique invariant measures and quantitative convergence rates.

Abstract

This paper develops a novel operator theoretic framework to study the contraction properties of Markov semigroups with respect to a general class of Kantorovich semi-distances, which notably includes Wasserstein distances. The rather simple contraction cost framework developed in this article, which combines standard Lyapunov techniques with local contraction conditions, helps to unifying and simplifying many arguments in the stability of Markov semigroups, as well as to improve upon some existing results. Our results can be applied to both discrete time and continuous time Markov semigroups, and we illustrate their wide applicability in the context of (i) Markov transitions on models with boundary states, including bounded domains with entrance boundaries, (ii) operator products of a Markov kernel and its adjoint, including two-block-type Gibbs samplers, (iii) iterated random functions and (iv) diffusion models, including overdampted Langevin diffusion with convex at infinity potentials.

On the Kantorovich contraction of Markov semigroups

TL;DR

The paper develops an operator-theoretic framework to study contraction of Markov semigroups with respect to Kantorovich semi-distances, including Wasserstein distances, by combining a simple Lyapunov drift with a local contraction property. Under standard drift (H1) and local contraction (H2) conditions, it proves exponential decay of a broad class of Dobrushin-type contraction coefficients, extending existing -norm results to Kantorovich distances. It provides explicit contraction bounds for both strict and exponential regimes and shows how these translate into Wasserstein contraction for finite-dimensional settings and continuous-time semigroups, including diffusion models. The framework yields new contraction results for models with boundaries, two-block Gibbs samplers, iterated random functions, and overdamped Langevin diffusions, all with couplings avoided in favor of Lyapunov-based proofs and operator theory. This offers a versatile, principled toolkit for stability analysis of Markov processes across statistics and applied probability, with clear criteria for establishing unique invariant measures and quantitative convergence rates.

Abstract

This paper develops a novel operator theoretic framework to study the contraction properties of Markov semigroups with respect to a general class of Kantorovich semi-distances, which notably includes Wasserstein distances. The rather simple contraction cost framework developed in this article, which combines standard Lyapunov techniques with local contraction conditions, helps to unifying and simplifying many arguments in the stability of Markov semigroups, as well as to improve upon some existing results. Our results can be applied to both discrete time and continuous time Markov semigroups, and we illustrate their wide applicability in the context of (i) Markov transitions on models with boundary states, including bounded domains with entrance boundaries, (ii) operator products of a Markov kernel and its adjoint, including two-block-type Gibbs samplers, (iii) iterated random functions and (iv) diffusion models, including overdampted Langevin diffusion with convex at infinity potentials.

Paper Structure

This paper contains 44 sections, 9 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Set-up
  3. Kantorovich semi-distances
  4. Contraction of Markov transport maps
  5. Contribution of the paper
  6. Contraction theorems
  7. Assumptions
  8. Additional notation
  9. Some comparison principles
  10. A strict contraction theorem
  11. Exponential decay of $\phi$-Dobrushin coefficients
  12. Exponential decay of the $(\varphi_V,\phi)$-Dobrushin contraction coefficient
  13. Application 1: Wasserstein exponential contraction estimates
  14. Application 2: Continuous time Markov semigroups
  15. Discussion
  16. ...and 29 more sections

Key Result

Lemma 1.1

Let $\phi$ and $\psi$ be two $V$-semi-distances. Then,

Theorems & Definitions (22)

  • Lemma 1.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 12 more