论文标题
瓦斯汀的收敛速率，用于非紧密歧管的经验措施
Wasserstein Convergence Rate for Empirical Measures on Noncompact Manifolds
论文作者
论文摘要
令$ x_t $为$ l：=δ+\ nabla v $生成的（反射）扩散过程，在完整连接的riemannian歧管$ m $上可能使用边界$ \ partial m $，其中$ v \ in c^1（m）$ in c^1（m）$ in c^1（m）$ y y $μ（d x）我们估计经验度量的收敛速率$μ_t：= \ frac 1 t \ int_0^tΔ__{x_s \ d s $在Wasserstein距离下。作为一个典型的例子，当$ m = \ mathbb r^d $和$ v（x）= c_1- c_1- c_1- c_2 | x |^p $对于某些常数$ c_1 \ in \ mathbb r，c_2> 0 $和$ p> 1 $，明确的上限和下限是$ drac $ n时，即$ d <d < $ d \ ge 4 $和$ p \ to \ infty $。
Let $X_t$ be the (reflecting) diffusion process generated by $L:=Δ+\nabla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $\partial M$, where $V\in C^1(M)$ such that $μ(d x):= e^{V(x)}d x$ is a probability measure. We estimate the convergence rate for the empirical measure $μ_t:=\frac 1 t \int_0^t δ_{X_s\d s$ 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 $p\to\infty$.
