Invariant transports of stationary random measures and mass-stationarity

TL;DR

This paper develops a theoretical framework for invariant transport-kernels balancing stationary random measures, introduces the concept of mass-stationarity, and establishes a key equivalence with Palm measures.

Contribution

It provides a new invariance property for Palm measures, a criterion for the existence of invariant transport-kernels, and links mass-stationarity with Palm measures.

Findings

Invariance property of Palm measures derived from Neveu's exchange formula.

A necessary and sufficient condition for the existence of invariant transport-kernels.

Equivalence between mass-stationarity and Palm measures.

Abstract

We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu's exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary.