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Wasserstein asymptotics for empirical measures of diffusions on four dimensional closed manifolds

Dario Trevisan, Feng-Yu Wang, Jie-Xiang Zhu

Abstract

We identify the leading term in the asymptotics of the quadratic Wasserstein distance between the invariant measure and empirical measures for diffusion processes on closed weighted four-dimensional Riemannian manifolds. Unlike results in lower dimensions, our analysis shows that this term depends solely on the Riemannian volume of the manifold, remaining unaffected by the potential and vector field in the diffusion generator.

Wasserstein asymptotics for empirical measures of diffusions on four dimensional closed manifolds

Abstract

We identify the leading term in the asymptotics of the quadratic Wasserstein distance between the invariant measure and empirical measures for diffusion processes on closed weighted four-dimensional Riemannian manifolds. Unlike results in lower dimensions, our analysis shows that this term depends solely on the Riemannian volume of the manifold, remaining unaffected by the potential and vector field in the diffusion generator.

Paper Structure

This paper contains 6 sections, 8 theorems, 93 equations.

Table of Contents

  1. Introduction and main result
  2. Asymptotics for the smoothed empirical measures
  3. Estimate of $\mathbb E \left[ \mu( |\nabla f_{T,\varepsilon}|^2) \right]$
  4. Estimate on $\mathbb E[|\mathbb W_2^2(\mu_{T,\varepsilon},\mu)-\mu(|\nabla f_{T,\varepsilon}|^2)|]$
  5. Improved contractivity estimate
  6. Proof of theorem \ref{['T1']}

Key Result

Theorem 1.1

With the notation introduced above, for a closed Riemannian manifold $M$ with dimension $d=4$, weighted volume measure $\mu$, and the occupation measure $\mu_T$ of the diffusion process with generator $L = \Delta + \nabla V \nabla + Z$, it holds

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['lemma:ito-tanaka-eigen']}
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 5 more