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Convergence rates for the $p$-Wasserstein distance of the empirical measures of an ergodic Markov process

René L. Schilling, Jian Wang, Bingyao Wu, Jie-Xiang Zhu

Abstract

Let $X:=(X_t)_{t\geq 0}$ be an ergodic Markov process on $\real^d$, and $p>0$. We derive upper bounds of the $p$-Wasserstein distance between the invariant measure and the empirical measures of the Markov process $X$. For this we assume, e.g.\ that the transition semigroup of $X$ is exponentially contractive in terms of the $1$-Wasserstein distance, or that the iterated Poincaré inequality holds together with certain moment conditions on the invariant measure. Typical examples include diffusions and underdamped Langevin dynamics.

Convergence rates for the $p$-Wasserstein distance of the empirical measures of an ergodic Markov process

Abstract

Let be an ergodic Markov process on , and . We derive upper bounds of the -Wasserstein distance between the invariant measure and the empirical measures of the Markov process . For this we assume, e.g.\ that the transition semigroup of is exponentially contractive in terms of the -Wasserstein distance, or that the iterated Poincaré inequality holds together with certain moment conditions on the invariant measure. Typical examples include diffusions and underdamped Langevin dynamics.
Paper Structure (11 sections, 9 theorems, 156 equations)

This paper contains 11 sections, 9 theorems, 156 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. More about the assumptions (H1)--(H3)
  4. Upper bounds for $\mathcal{T}_p$
  5. Proofs of Theorems \ref{['D']}, \ref{['IP']} and \ref{['S']}
  6. Proof of Theorem \ref{['D']}
  7. Proofs of Theorems \ref{['IP']} and \ref{['S']}
  8. Proof of Theorem \ref{['thm3']}
  9. Examples
  10. Diffusions
  11. Underdamped Langevin Dynamics

Key Result

Theorem 1.1

Let $(X_t)_{t\geq 0}$ be an ergodic Markov process on ${\mathds R}^d$ with invariant measure $\mu$. Assume that (H1) holds, let $p>0$, and assume that $\mu$ has for some $q> \max\{p, 1\}$ a $q$-th moment $\mu(|\cdot|^q) <\infty.$ Define Then there exists a constant $C>0$ such that for all $T \ge 2$,

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Example 5.1
  • proof
  • ...and 3 more