Wasserstein Convergence Rate for Empirical Measures on Noncompact Manifolds

2020/07/29 by Wang, Feng-Yu · 2 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2007.14667

Abstract

Let X_t be the (reflecting) diffusion process generated by L:=Δ+∇ V on a complete connected Riemannian manifold M possibly with a boundary ∂ M, where V∈ C¹(M) such that μ(d x):= e^V(x)d x is a probability measure. We estimate the convergence rate for the empirical measure μ_t:=\frac 1 t ∫₀^t δ_X_s\d s under the Wasserstein distance. As a typical example, when M=\mathbb R^d and V(x)= c₁- c₂ |x|^p for some constants c₁∈ \mathbb R, c₂>0 and p>1, the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when either d

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• Convergence rates for the p-Wasserstein distance of the empirical measures of an ergodic Markov process
• Exact rate of convergence for the empirical measure of a subordinated process in p-Wasserstein distance

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