Stable transports between stationary random measures

TL;DR

This paper presents a deterministic algorithm for constructing translation-invariant transport kernels between ergodic stationary random measures with equal intensities, generalizing previous work and inspired by stable marriage algorithms.

Contribution

It introduces a novel algorithm for stable transport kernels between random measures, extending prior results to more general cases with constrained densities.

Findings

The algorithm guarantees existence of stable transport kernels.

It ensures uniqueness and monotonicity of solutions.

The method generalizes previous constructions for Lebesgue measure and point processes.

Abstract

We give an algorithm to construct a translation-invariant transport kernel between ergodic stationary random measures $\Phi$ and $\Psi$ on $\mathbb R^d$, given that they have equal intensities. As a result, this yields a construction of a shift-coupling of an ergodic stationary random measure and its Palm version. This algorithm constructs the transport kernel in a deterministic manner given realizations $\varphi$ and $\psi$ of the measures. The (non-constructive) existence of such a transport kernel was proved in [8]. Our algorithm is a generalization of the work of [3], in which a construction is provided for the Lebesgue measure and an ergodic simple point process. In the general case, we limit ourselves to what we call constrained densities and transport kernels. We give a definition of stability of constrained densities and introduce our construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable constrained densities, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness.