Transportation of diffuse random measures on ℝ^d

2021/12/24 by Günter Last, Last, Günter, Hermann Þórisson +1
Mathematics · #60G55 #60G57 #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2112.13053

openalex publication_date 2021/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider two jointly stationary and ergodic random measures ξ and η on ℝ^d with equal finite intensities, assuming ξ to be diffuse. An allocation is a random mapping taking ℝ^d to ℝ^d∪\∞\ in a translation invariant way. We construct allocations transporting the diffuse ξ to arbitrary η, under the mild condition of existence of an `auxiliary' point process which is needed only in the case when η is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting ξ to η need not exist.

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