Wasserstein Convergence for Empirical Measures of Subordinated Diffusions on Riemannian Manifolds

Abstract

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 \(\mathbb {W}_{2}\) 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 \((\mu _{t}^{\alpha})_{t>0}\) (α ∈ (0,1)) be the empirical measures of the Markov process generated by Lα := −(−L)α, when ∂M is 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,$ where \(\mathbb {E}^{x}\) is the expectation for the process starting at point x, {λi}i≥ 1 are non-trivial (Neumann) eigenvalues of − L. In general, $\mathbb {E}^{x} [\mathbb {W}_{2}(\mu _{t}^{\alpha},\mu )^{2}] \begin {cases} \asymp t^{-1}, &\text {if}\ d<2(1+\alpha ),\\ \asymp t^{-\frac {2}{d-2\alpha }}, &\text {if} \ d>2(1+\alpha ),\\ \preceq t^{-1}\log (1+t), &\text {if} \ d=2(1+\alpha ), \text {i.e.}\ \alpha =\frac {1}{2}, d=3\end {cases}$ holds uniformly in x ∈ M, where in the last case \(\mathbb {E}^{x} [\mathbb {W}_{1}(\mu _{t}^{\alpha},\mu )^{2}]\succeq t^{-1}\log (1+t)\) holds for \(M=\mathbb {T}^{3}\) and V = 0.