The Annals of Probability

Optimal transport from Lebesgue to Poisson

Martin Huesmann and Karl-Theodor Sturm

Full-text: Open access

Abstract

This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at $d=2$. The cost will be defined in terms of an arbitrary increasing function of the distance.

The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.

Article information

Source
Ann. Probab., Volume 41, Number 4 (2013), 2426-2478.

Dates
First available in Project Euclid: 3 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1372859757

Digital Object Identifier
doi:10.1214/12-AOP814

Mathematical Reviews number (MathSciNet)
MR3112922

Zentralblatt MATH identifier
1279.60024

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 49Q20: Variational problems in a geometric measure-theoretic setting

Keywords
Optimal transportation fair allocation Laguerre tessellation Poisson point process

Citation

Huesmann, Martin; Sturm, Karl-Theodor. Optimal transport from Lebesgue to Poisson. Ann. Probab. 41 (2013), no. 4, 2426--2478. doi:10.1214/12-AOP814. https://projecteuclid.org/euclid.aop/1372859757


Export citation

References

  • [1] Ajtai, M., Komlós, J. and Tusnády, G. (1984). On optimal matchings. Combinatorica 4 259–264.
  • [2] Ambrosio, L. (2003). Lecture notes on optimal transport problems. In Mathematical Aspects of Evolving Interfaces (Funchal, 2000). Lecture Notes in Math. 1812 1–52. Springer, Berlin.
  • [3] Ambrosio, L., Gigli, N. and Savaré, G. (2008). Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Birkhäuser, Basel.
  • [4] Ambrosio, L., Savaré, G. and Zambotti, L. (2009). Existence and stability for Fokker–Planck equations with log-concave reference measure. Probab. Theory Related Fields 145 517–564.
  • [5] Aurenhammer, F. (1991). Voronoi diagrams—A survey of a fundamental geometric data structure. ACM Computing Surveys (CSUR) 23 345–405.
  • [6] Beiglböck, M., Goldstern, M., Maresch, G. and Schachermayer, W. (2009). Optimal and better transport plans. J. Funct. Anal. 256 1907–1927.
  • [7] Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 375–417.
  • [8] Chatterjee, S., Peled, R., Peres, Y. and Romik, D. (2010). Phase transitions in gravitational allocation. Geom. Funct. Anal. 20 870–917.
  • [9] Chatterjee, S., Peled, R., Peres, Y. and Romik, D. (2010). Gravitational allocation to Poisson points. Ann. of Math. (2) 172 617–671.
  • [10] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge.
  • [11] Figalli, A. (2010). The optimal partial transport problem. Arch. Ration. Mech. Anal. 195 533–560.
  • [12] Gangbo, W. and McCann, R. J. (1996). The geometry of optimal transportation. Acta Math. 177 113–161.
  • [13] Hoffman, C., Holroyd, A. E. and Peres, Y. (2006). A stable marriage of Poisson and Lebesgue. Ann. Probab. 34 1241–1272.
  • [14] Holroyd, A. E. and Liggett, T. M. (2001). How to find an extra head: Optimal random shifts of Bernoulli and Poisson random fields. Ann. Probab. 29 1405–1425.
  • [15] Holroyd, A. E., Pemantle, R., Peres, Y. and Schramm, O. (2009). Poisson matching. Ann. Inst. Henri Poincaré Probab. Stat. 45 266–287.
  • [16] Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Probab. 33 31–52.
  • [17] Last, G. and Thorisson, H. (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Probab. 37 790–813.
  • [18] Lautensack, C. and Zuyev, S. (2008). Random Laguerre tessellations. Adv. in Appl. Probab. 40 630–650.
  • [19] Lott, J. and Villani, C. (2009). Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 903–991.
  • [20] Markó, R. and Timár, Á. (2011). A poisson allocation of optimal tail. Available at arXiv:1103.5259.
  • [21] Ohta, S.-I. (2009). Uniform convexity and smoothness, and their applications in Finsler geometry. Math. Ann. 343 669–699.
  • [22] Ohta, S.-I. and Sturm, K.-T. (2009). Heat flow on Finsler manifolds. Comm. Pure Appl. Math. 62 1386–1433.
  • [23] Otto, F. (2001). The geometry of dissipative evolution equations: The porous medium equation. Comm. Partial Differential Equations 26 101–174.
  • [24] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361–400.
  • [25] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems. Vol. I. Springer, New York.
  • [26] Sturm, K.-T. (2006). On the geometry of metric measure spaces. I. Acta Math. 196 65–131.
  • [27] Sturm, K.-T. (2006). On the geometry of metric measure spaces. II. Acta Math. 196 133–177.
  • [28] Talagrand, M. (1994). The transportation cost from the uniform measure to the empirical measure in dimension $\ge3$. Ann. Probab. 22 919–959.
  • [29] Timar, A. (2009). Invariant matchings of exponential tail on coin flips in $Z^{d}$. Available at arXiv:0909.1090.
  • [30] Touchard, J. (1956). Nombres exponentiels et nombres de Bernoulli. Canad. J. Math. 8 305–320.
  • [31] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
  • [32] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.