Invariant Transports of Stationary Random Measures and Mass - Stationarity

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. 1. Introduction. We consider (jointly) stationary random measures on a locally compact Abelian group G, for instance, G = R d. A transport-kernel is a Markovian kernel T that distributes mass over G and depends on both ω in the underlying sample space Ω and a location s ∈ G. The number T (ω, s, B) is the proportion of mass transported from location s to the set B. More generally, a weighted transport-kernel is a kernel T which need not be Markovian. If T is finite, then the mass at s is weighted by T (ω, s, G) before being transported by the normalized T. In general, we assume that T (ω, s, B) is finite for compact B but allow that T (ω, s, G) = ∞. A kernel T is invariant if it is invariant under joint shifts of all three arguments. If ξ and η are random measures on G such that ξT = η, then T is (ξ, η)-balancing and, in particular, if ξ = η, then T is ξ-preserving. Sometimes an invariant T can be reduced to an allocation rule τ (depending on ω ∈ Ω) that maps each location s to a new location τ (s) in an invariant way. In fact, we might think of an invariant transport-kernel T as the conditional distribution of a randomized allocation rule.