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Optimal transport from Lebesgue to Poisson. (English) Zbl 1279.60024
The authors introduce a concept of optimal (semi-)couplings between the Lebesgue measure and a point process in $$\mathbb{R}^d$$. The optimal coupling minimizes an asymptotic mean cost functional $\mathfrak{E}_{\infty } (q) = \liminf_{n \to \infty} \frac{1}{ \lambda ^d (B_n)} \;\operatorname{E} \int_{\mathbb{R}^d \times B_n} \vartheta (|x-y|) \, dq (w,x,y),$ over all couplings $$q$$ of $$\lambda^d$$ and the point process; here $$B_n = [0,2^n)^d$$. The authors prove existence and uniqueness of an optimal semi-coupling whenever there exists one with finite asymptotic mean transportation cost. They prove the convergence of optimal couplings on finite doubling sequences of boxes $$(B_n(z,\gamma))$$ towards an optimal coupling between $$\lambda^d$$ and the point process. For $$d \leq 2$$, the asymptotic mean transportation cost is finite for the Poisson point process for $$L^p$$-costs with $$p < \frac{d}{2}$$ while for $$d \geq 3$$ or intensity $$\beta < 1$$ finiteness holds for any $$p < \infty$$. In the case $$\beta=1$$ for $$d > 2 (p \wedge 1)$$, the optimal asymptotic costs are of order $$d\,^{^{p\!_{/\!_2}}}$$.

##### MSC:
 60D05 Geometric probability and stochastic geometry 52A22 Random convex sets and integral geometry (aspects of convex geometry) 49Q20 Variational problems in a geometric measure-theoretic setting
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