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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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