## Electronic Journal of Probability

### Stable transports between stationary random measures

#### 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

#### 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

#### References

• [1] Chatterjee, S., Peled, R., Peres, Y., and Romik, D. (2010). Gravitational allocation to Poisson points. Ann. Math., 172(1), 617–671.
• [2] Gale, D., and Shapley, L. S. (1962). College admissions and the stability of marriage. Amer. Math. Monthly, 9–15.
• [3] Hoffman, C., Holroyd, A. E., and Peres, Y. (2006). A stable marriage of Poisson and Lebesgue. Ann. Prob., 1241–1272.
• [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.
• [5] Holroyd, A. E., and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Prob., 31–52.
• [6] Huesmann, M., and Sturm, K. T. (2013). Optimal transport from Lebesgue to Poisson. Ann. Prob., 41(4), 2426–2478.
• [7] Korman, J., and McCann, R. (2015). Optimal transportation with capacity constraints. Trans. Amer. Math. Soc., 367(3), 1501–1521.
• [8] Last, G., Mörters, P., and Thorisson, H. (2014). Unbiased shifts of Brownian motion. Ann. Prob., 42(2), 431–463.
• [9] Last, G., and Thorisson, H. (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Prob., 790–813.
• [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.
• [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.
• [13] Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Prob., 24(4), 2057–2064.