Optimal transport between random measures

Abstract

We study couplings q • of two equivariant random measures λ • and µ • on a Riemannian manifold (M, d, m). Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the minimal/optimal cost is finite and λ ω m we prove that there is a unique equivariant coupling minimizing the mean transportation cost per volume. Moreover, the optimal coupling is induced by a transportation map, q • = (id, T ) * λ • . We show that the optimal transportation map can be approximated by solutions to classical optimal transportation problems on bounded regions. In case of L p −cost the optimal transportation cost per volume defines a metric on the space of equivariant random measure with unit intensity.

Transport cost estimates for random measures in dimension one

“…transporting λ to µ, induces a shift-coupling of P with its Palm-measure P µ . By (1.2), c ∞ = inf T,T * λ=µ E[|T |] and, by the results of [4], the infimum is attained by a unique mapT which is measurably dependent only on the σ-algebra generated by µ. Hence, X :=T (0) shift-couples P and P µ and, by (1.2), we need to show that E[|X|] = ∞.…”

“…The main results of [5,4] show that there is a unique optimal coupling between λ and µ provided that c ∞ < ∞. In particular, eventhough there are arbitrarily many asymptotically optimal couplings there is a unique invariant one.…”

“…In [5,4] it was shown that there is a unique optimal coupling between the Lebesgue measure λ d on R d and an invariant random measure µ on R d of unit intensity provided that the asymptotic mean transportation cost…”

The One‐Dimensional Log‐Gas Free Energy Has a Unique Minimizer

“…We cannot argue via displacement convexity directly on the level of P 0 ; P 1 since they are probability measures on infinite point configurations. Optimal transport theory for random stationary measures as initiated in [7,10,11] is not yet developed well enough to be directly applicable. Instead, we use transport theory between finite measures together with a careful approximation argument relying on screening of electric fields.…”

There is no stationary cyclically monotone Poisson matching in 2d

“…Our interest in this problem is motivated on the one hand by work on geometric properties of matchings by Holroyd [13] and Holroyd et al [15,14], and on the other hand by work on optimally coupling random measures by the first author and Sturm [17] and the first author [16]. In [14], Holroyd, Janson, and Wästlund analyze (stationary) matchings satisfying the local optimality condition (1.2) with the exponent 2 replaced by γ ∈ [−∞, ∞].…”