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Let $X_t$ be the (reflecting) diffusion process generated by $L:=Delta+ abla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $partial M$, where $Vin C^1(M)$ such that $mu(d x):= e^{V(x)}d x$ is a probability measure. We estimate the convergence rate for the empirical measure $mu_t:=frac 1 t int_0^t delta_{X_sd s$ 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 $c_1in mathbb R, c_2>0$ 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 $dge 4$ and $ptoinfty$.
Let $M$ be a connected compact Riemannian manifold possibly with a boundary, let $Vin C^2(M)$ such that $mu(d x):=e^{V(x)}d x$ is a probability measure, where $d x$ is the volume measure, and let $L=Delta+ abla V$. The exact convergence rate in Wasse
The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is
We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $infty-$Wasserstein distance betwe
This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimens
We prove that moderate deviations for empirical measures for countable nonhomogeneous Markov chains hold under the assumption of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chains in Ces`aro sense.