Transportation of diffuse random measures on ${\mathbb{R}}^d$

Abstract

We consider two jointly stationary and ergodic random measures $\xi$ and $\eta$ on ${\mathbb{R}}^d$ with equal finite intensities, assuming $\xi$ to be diffuse. An allocation is a random mapping taking ${\mathbb{R}}^d$ to ${\mathbb{R}}^d\cup \{\infty \}$ in a translation invariant way. We construct allocations transporting the diffuse $\xi$ to arbitrary $\eta$, under the mild condition of existence of an `auxiliary' point process which is needed only in the case when $\eta$ is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting $\xi$ to $\eta$ need not exist.