Wasserstein Convergence for Empirical Measures of Subordinated Diffusions on Riemannian Manifolds

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