Wasserstein Convergence Rate for Empirical Measures on Noncompact Manifolds

Abstract

Let $X_t$ be the (reflecting) diffusion process generated by $L:=\Delta +\nabla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $\partial M$, where $V\in C^1(M)$ such that $\mu (dx):=e^{V(x)}dx$ is a probability measure. We estimate the convergence rate for the empirical measure $μ_t:=\frac 1 t \int_0^t δ_{X_s\d s$ under the Wasserstein distance. As a typical example, when $M={\mathbb{R}}^d$ and $V(x)=c_1-c_2|x{|}^p$ for some constants $c_1\in \mathbb{R},c_2>0$ and $p>1$, the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when either $d<\frac{4(p-1)}{p}$ or $d\ge 4$ and $p\to \infty$.

19 pages

References in corpus (1)

• Convergence in Wasserstein Distance for Empirical Measures of Dirichlet Diffusion Processes on Manifolds