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

TL;DR

This paper studies the problem of constructing translation-invariant allocations that transport one diffuse random measure to another on Euclidean space, providing conditions for existence and counterexamples when conditions fail.

Contribution

It introduces a method to construct allocations between diffuse random measures on , and identifies necessary conditions for their existence, including counterexamples.

Findings

Allocations exist under mild conditions with an auxiliary point process.

Counterexamples show allocations may not exist without these conditions.

The approach extends the theory of measure transport in stochastic geometry.

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.