Optimal transport between random measures

Abstract

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