Electronic Journal of Probability

Stable transports between stationary random measures

Mir-Omid Haji-Mirsadeghi and Ali Khezeli

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We give an algorithm to construct a translation-invariant transport kernel between two arbitrary ergodic stationary random measures on $\mathbb R^d$, given that they have equal intensities. As a result, this yields a construction of a shift-coupling of an arbitrary ergodic stationary random measure and its Palm version. This algorithm constructs the transport kernel in a deterministic manner given a pair of realizations of the two measures. The (non-constructive) existence of such a transport kernel was proved in [9]. 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 transport densities and transport kernels. We give a definition of stability of constrained transport densities and introduce our construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable constrained transport densities, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 51, 25 pp.

Received: 14 April 2015
Accepted: 3 April 2016
First available in Project Euclid: 5 August 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G57: Random measures 60G55: Point processes 60G10: Stationary processes

stationary random measure palm distribution mass transport allocation stable matching capacity constrained transport kernel Voronoi transport kernel shift-coupling

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Haji-Mirsadeghi, Mir-Omid; Khezeli, Ali. Stable transports between stationary random measures. Electron. J. Probab. 21 (2016), paper no. 51, 25 pp. doi:10.1214/16-EJP4237. https://projecteuclid.org/euclid.ejp/1470414022

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