Parametrized Kantorovich-Rubinstein theorem and application to the coupling of random variables

On a Kantorovich-Rubinstein inequality

An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d \rightarrow \infty$ is Lipschitz and $\left\{x_1, \dots, x_N \right\} \subset [0,1]^d$, then $$ \left| \int_{[0,1]^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N}{f(x_k)} \right| \leq \left\| \nabla f \right\|_{L^{\infty}} \cdot W_1\left( \frac{1}{N} \sum_{k=1}^{N}{δ_{x_k}} , dx\right),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Mover's) Distance. We prove another such inequality with a smaller norm on $\nabla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e…

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Relative Optimal Transport

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$\mathrm{P}_2(\mathbb{R}^d)$ 中 Zolotarev 距离与 Wasserstein 距离之间的尖锐不等式

Sharp inequalities between Zolotarev and Wasserstein distances in $\mathrm{P}_2(\mathbb{R}^d)$

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Stability of measure solutions to a generalized Boltzmann equation with collisions of a random number of particles

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随机几何图投影中交叉数与应力的极限定理

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Disintegration theorem for multifunctions, with applications to empirical Wasserstein distances and average-case statistical bounds

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