紧致流形上的 Wasserstein 不等式与极小 Green 能量
经典分析与常微分方程
2019-07-23 v1 数学物理
math.MP
摘要
设 M M M d d d d ≥ 3 d \geq 3 d ≥ 3 G : M × M → R ∪ { ∞ } G: M \times M \rightarrow \mathbb{R} \cup \left\{\infty\right\} G : M × M → R ∪ { ∞ } − Δ -\Delta − Δ { x 1 , … , x n } ⊂ M \left\{x_1, \dots, x_n\right\} \subset M { x 1 , … , x n } ⊂ M d x dx d x W 2 ( 1 n ∑ k = 1 n δ x k , d x ) ≲ M 1 n 1 / d + 1 n ∣ ∑ k , ℓ = 1 k ≠ ℓ n G ( x k , x ℓ ) ∣ 1 / 2 . W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim_M \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}G(x_k, x_{\ell})\right|^{1/2}. W 2 ( n 1 k = 1 ∑ n δ x k , d x ) ≲ M n 1/ d 1 + n 1 k = ℓ k , ℓ = 1 ∑ n G ( x k , x ℓ ) 1/2 . S d \mathbb{S}^d S d d ≥ 3 d \geq 3 d ≥ 3 G ( x , y ) = 1 / ∥ x − y ∥ d − 2 G(x,y) = 1/\|x-y\|^{d-2} G ( x , y ) = 1/∥ x − y ∥ d − 2 W 2 ( 1 n ∑ k = 1 n δ x k , d x ) ≲ 1 n 1 / d + 1 n ∣ ∑ k , ℓ = 1 k ≠ ℓ n ( 1 ∥ x k − x ℓ ∥ d − 2 − c d ) ∣ 1 2 , W_2\left(\frac{1}{n} \sum_{k=1}^{n}{ \delta_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}{\left(\frac{1}{\|x_k - x_{\ell}\|^{d-2}} - c_d \right)} \right|^{\frac{1}{2}}, W 2 ( n 1 k = 1 ∑ n δ x k , d x ) ≲ n 1/ d 1 + n 1 k = ℓ k , ℓ = 1 ∑ n ( ∥ x k − x ℓ ∥ d − 2 1 − c d ) 2 1 , c d c_d c d W 2 ( 1 n ∑ k = 1 n δ x k , d x ) ≲ n − 1 / d W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim n^{-1/d} W 2 ( n 1 ∑ k = 1 n δ x k , d x ) ≲ n − 1/ d S d \mathbb{S}^d S d
引用
@article{arxiv.1907.09023,
title = {A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds},
author = {Stefan Steinerberger},
journal= {arXiv preprint arXiv:1907.09023},
year = {2019}
}