Wasserstein Convergence for Empirical Measures of Subordinated Diffusions on Riemannian Manifolds

Let M be a connected compact Riemannian manifold possibly with boundary ∂M, let V ∈ C2(M) such that μ(dx) := eV (x)dx is probability measure, where dx the volume and L =Δ +∇V. As continuation to Wang Zhu (2019) convergence in quadratic Wasserstein distance \(\mathbb {W}_{2}\) studied for empirical measures of L-diffusion process (with reflecting if ∂M≠∅), this paper presents exact rate subordinated process. In particular, letting \((\mu _{t}^{\alpha})_{t>0}\) (α (0,1)) Markov generated by Lα −(−L)α, when ∂M empty or convex we have $ \lim _{t\to \infty } \big \{t \mathbb {E}^{x} [\mathbb {W}_{2}(\mu _{t}^{\alpha},\mu )^{2}]\big \}= \sum \limits_{i=1}^{\infty }\frac {2}{\lambda _{i}^{1+\alpha }}\ \text { uniformly\ in\ x\in M,$ {E}^{x}\) expectation starting at point x, {λi}i≥ 1 are non-trivial (Neumann) eigenvalues − L. general, $\mathbb )^{2}] \begin {cases} \asymp t^{-1}, &\text {if}\ d<2(1+\alpha ),\\ t^{-\frac {2}{d-2\alpha }}, {if} \ d>2(1+\alpha \preceq t^{-1}\log (1+t), d=2(1+\alpha ), {i.e.}\ \alpha =\frac {1}{2}, d=3\end {cases}$ holds uniformly x M, last case {W}_{1}(\mu )^{2}]\succeq (1+t)\) \(M=\mathbb {T}^{3}\) = 0.