Wasserstein Convergence for Empirical Measures of Subordinated Diffusions on Riemannian Manifolds

TL;DR

This paper derives the precise rate at which empirical measures of subordinated diffusion processes on compact Riemannian manifolds converge in Wasserstein distance, extending understanding of convergence behavior in geometric stochastic analysis.

Contribution

It provides the first explicit Wasserstein convergence rate for empirical measures of subordinated diffusions on Riemannian manifolds.

Findings

Explicit Wasserstein convergence rate derived

Applicable to reflected diffusions on manifolds

Enhances understanding of empirical measure convergence

Abstract

Let $M$ be a connected compact Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $\mu(\d x):=\e^{V(x)}\d x$ is a probability measure, where $\d x$ is the volume measure, and let $L=\Delta+\nabla V$. The exact convergence rate in Wasserstein distance is derived for empirical measures of subordinations for the (reflecting) diffusion process generated by $L$.