Let M be a connected compact Riemannian manifold possibly with a boundary ∂M, let V ∈ C2(M) such that μ(dx) := eV (x)dx is a probability measure, where dx is the volume measure, and let L =Δ +∇V. As a continuation to Wang and Zhu (2019) where convergence in the quadratic Wasserstein distance W2\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {W}_{2}$\end{document} is studied for the empirical measures of the L-diffusion process (with reflecting boundary if ∂M≠∅), this paper presents the exact convergence rate for the subordinated process. In particular, letting (μtα)t>0\documentclass[12pt]{minimal}
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\begin{document}$(\mu _{t}^{\alpha})_{t>0}$\end{document} (α ∈ (0,1)) be the empirical measures of the Markov process generated by Lα := −(−L)α, when ∂M is empty or convex we have
limt→∞tEx[W2(μtα,μ)2]=∑i=1∞2λi1+αuniformly inx∈M,\documentclass[12pt]{minimal}
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\begin{document}$ \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,$\end{document} where Ex\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {E}^{x}$\end{document} is the expectation for the process starting at point x, {λi}i≥ 1 are non-trivial (Neumann) eigenvalues of − L. In general,
Ex[W2(μtα,μ)2]≍t−1,ifd<2(1+α),≍t−2d−2α,ifd>2(1+α),≼t−1log(1+t),ifd=2(1+α),i.e.α=12,d=3\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {E}^{x} [\mathbb {W}_{2}(\mu _{t}^{\alpha},\mu )^{2}] \begin {cases} \asymp t^{-1}, &\text {if}\ d<2(1+\alpha ),\\ \asymp t^{-\frac {2}{d-2\alpha }}, &\text {if} \ d>2(1+\alpha ),\\ \preceq t^{-1}\log (1+t), &\text {if} \ d=2(1+\alpha ), \text {i.e.}\ \alpha =\frac {1}{2}, d=3\end {cases}$\end{document} holds uniformly in x ∈ M, where in the last case Ex[W1(μtα,μ)2]≽t−1log(1+t)\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {E}^{x} [\mathbb {W}_{1}(\mu _{t}^{\alpha},\mu )^{2}]\succeq t^{-1}\log (1+t)$\end{document} holds for M=T3\documentclass[12pt]{minimal}
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\begin{document}$M=\mathbb {T}^{3}$\end{document} and V = 0.
