Electronic Journal of Probability

Stable transports between stationary random measures

Mir-Omid Haji-Mirsadeghi and Ali Khezeli

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1470414022

Digital Object Identifier
doi:10.1214/16-EJP4237

Mathematical Reviews number (MathSciNet)
MR3539645

Zentralblatt MATH identifier
1346.60068

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

  • [1] Chatterjee, S., Peled, R., Peres, Y., and Romik, D. (2010). Gravitational allocation to Poisson points. Ann. Math., 172(1), 617–671.
    Mathematical Reviews (MathSciNet): MR2680428
    Zentralblatt MATH: 1206.60013
    Digital Object Identifier: doi:10.4007/annals.2010.172.617
  • [2] Gale, D., and Shapley, L. S. (1962). College admissions and the stability of marriage. Amer. Math. Monthly, 9–15.
    Zentralblatt MATH: 0109.24403
    Digital Object Identifier: doi:10.1080/00029890.1962.11989827
  • [3] Hoffman, C., Holroyd, A. E., and Peres, Y. (2006). A stable marriage of Poisson and Lebesgue. Ann. Prob., 1241–1272.
    Zentralblatt MATH: 1111.60008
    Digital Object Identifier: doi:10.1214/009117906000000098
    Project Euclid: euclid.aop/1158673318
  • [4] Hoffman, C., Holroyd, A. E., and Peres, Y. (2009). Tail bounds for the stable marriage of Poisson and Lebesgue. Canad. J. Math., 61(6), 1279.
    Zentralblatt MATH: 1191.60015
    Digital Object Identifier: doi:10.4153/CJM-2009-060-7
  • [5] Holroyd, A. E., and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Prob., 31–52.
    Zentralblatt MATH: 1097.60032
    Digital Object Identifier: doi:10.1214/009117904000000603
    Project Euclid: euclid.aop/1108141719
  • [6] Huesmann, M., and Sturm, K. T. (2013). Optimal transport from Lebesgue to Poisson. Ann. Prob., 41(4), 2426–2478.
    Zentralblatt MATH: 1279.60024
    Digital Object Identifier: doi:10.1214/12-AOP814
    Project Euclid: euclid.aop/1372859757
  • [7] Korman, J., and McCann, R. (2015). Optimal transportation with capacity constraints. Trans. Amer. Math. Soc., 367(3), 1501–1521.
    Zentralblatt MATH: 1305.90065
    Digital Object Identifier: doi:10.1090/S0002-9947-2014-06032-7
  • [8] Last, G., Mörters, P., and Thorisson, H. (2014). Unbiased shifts of Brownian motion. Ann. Prob., 42(2), 431–463.
    Zentralblatt MATH: 1295.60093
    Digital Object Identifier: doi:10.1214/13-AOP832
    Project Euclid: euclid.aop/1393251292
  • [9] Last, G., and Thorisson, H. (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Prob., 790–813.
    Zentralblatt MATH: 1176.60036
    Digital Object Identifier: doi:10.1214/08-AOP420
    Project Euclid: euclid.aop/1241099929
  • [10] Liggett, T. M. (2002). Tagged particle distributions or how to choose a head at random. In In and out of equilibrium, 133–162, Birkhäuser Boston.
    Zentralblatt MATH: 1108.60319
  • [11] Mecke, J. (1967). Stationäre zufällige Maße auf lokalkompakten abelschen Gruppen. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 9(1), 36–58.
  • [12] Schneider, R., and Weil, W. (2008). Stochastic And Integral Geometry, Springer.
    Zentralblatt MATH: 1175.60003
  • [13] Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Prob., 24(4), 2057–2064.
    Zentralblatt MATH: 0879.60051
    Digital Object Identifier: doi:10.1214/aop/1041903217
    Project Euclid: euclid.aop/1041903217

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