Exact rate of convergence for the empirical measure of a subordinated process in $p$-Wasserstein distance

TL;DR

This work derives exact $p$-Wasserstein convergence rates for the empirical measure of a class of non-symmetric subordinated diffusions on compact Riemannian manifolds, with generator $\mathcal{L}^\alpha=-(-L)^\alpha+Z$. The authors combine a Bernstein-type deviation bound for time averages with the Benamou–Brenier framework to translate Wasserstein distances into gradient resolvent terms, and they perform a detailed spectral analysis of the base diffusion. They establish sharp upper and matching lower bounds expressed through the rate function $\gamma_{\alpha,d}(T)$, and they obtain a precise renormalization limit for the quadratic case in the critical dimensions, namely $(\alpha,d)=(1/2,3)$ with $Z=0$, where $\lim_{T\to\infty} (T/\log T) \mathds{E}^\mu[W_2^2(\mu_T^\alpha,\mu)] = \mathrm{Vol}(M)/(2\pi^2)$. The results extend previous work to a broader class of subordinated processes and provide exact constants for both rates and renormalization, with implications for convergence analysis in stochastic and geometric settings.

Abstract

We establish exact rates of convergence in the $p$-Wasserstein distance for the empirical measure of a class of non-symmetric jump processes, which are subordinated to a diffusion process on a compact Riemannian manifold. For the quadratic Wasserstein distance, we determine the renormalization limit. We extend the main results of \cite{WW} and \cite{WWZ}. Our method uses two key elements: a Bernstein-type inequality for the subordinated process and the PDE approach established in \cite{AMB}

Theorems & Definitions (28)

• Theorem 1.1: Upper bound
• Theorem 1.2: Lower bound
• Theorem 1.3
• Lemma 2.1
• Proposition 2.2
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['Wpq']}