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.
Highlights
For a random measure Ψ on Rd, there are a number of equivalent definitions for the Palm distribution of Ψ
To obtain a shift-coupling, one can use a balancing transport kernel T that transports a multiple of the Lebesgue measure to Ψ, where by a transport kernel we mean a function that assigns to each point s ∈ Rd and ω in the probability space, a probability measure Tω(s, ·) on Rd
Given that T depends on Ψ in a translation-invariant manner, choosing Y with distribution Tω(0, ·) gives a shift-coupling of Ψ and its Palm version
Summary
For a random measure Ψ on Rd, there are a number of equivalent definitions for the Palm distribution of Ψ (see Section 2). Construct a flow-adapted transport kernel balancing two arbitrary jointly stationary and ergodic random measures on Rd with equal intensities (Theorem 4.8). Constructs a shift-coupling for an arbitrary stationary ergodic random measure on Rd and its Palm version (Theorem 4.9). The first, and deterministic, result (Theorem 4.13) is that stable constrained densities exist and one can be given by our algorithm (Algorithm 4.4) which is inspired by the continuum version of the Gale-Shapley algorithm in [3] Another important result is considering the algorithm in the random case described above. The construction and results in this paper can be generalized to random measures on a locally compact Abelian group and to non-ergodic cases (only equality of sample intensities is important) in the setting of [8].
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