Abstract
We investigate here the optimal transportation problem on configuration space for the quadratic cost. It is shown that, as usual, provided that the corresponding Wasserstein is finite, there exists one unique optimal measure and that this measure is supported by the graph of the derivative (in the sense of the Malliavin calculus) of a “concave” (in a sense to be defined below) function. For finite point processes, we give a necessary and sufficient condition for the Wasserstein distance to be finite.
Similar content being viewed by others
References
Albeverio, S., Yu. Kondratiev, G., Rockner, M.: Analysis and geometry on configuration spaces. J. Funct. Anal. 154(2), 444–500 (1998)
Barbour, A.D., Brown, T.C.: Stein’s method and point process approximation. Stochastic Process. Appl. 43(1), 9–31 (1992)
Barbour, A.D., Chryssaphinou, O.: Compound Poisson approximation: a user’s guide. Ann. Appl. Probab. 11(3), 964–1002 (2001)
Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation, Oxford Studies in Probability, vol. 2. The Clarendon Press Oxford University Press, Oxford Science Publications (1992)
Barbour, A.D., Maansson, M.: Compound Poisson process approximation. Ann. Probab. 30(3), 1492–1537 (2002)
Barbour, A.D., Xia, A.: Estimating Stein’s constants for compound Poisson approximation. Bernoulli 6(4), 581–590 (2000)
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. vol. I, 2nd edn. Probability and its Applications (New York), Elementary theory and methods. Springer-Verlag, New York (2003)
Feyel, D., Üstünel, A.S.: Monge–Kantorovitch measure transportation and Monge–Ampère equation on Wiener space. Probab. Theory Related Fields 128(3), 347–385 (2004)
Kallenberg, O.: Random Measures, 3rd edn. Academic Press (1983)
Levin, V.: Abstract cyclical monotonicity and Monge solutions for the general Monge–Kantorovich problem. Set-Valued Anal. 7(1), 7–32 (1999)
Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. I. Probability and its Applications (New York), Theory Springer-Verlag, New York (1998)
Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. II. Probability and its Applications (New York), Applications Springer-Verlag, New York (1998)
Röckner, M., Schied, A.: Rademacher’s theorem on configuration spaces and applications. J. Funct. Anal. 169(20), 325–356 (1999)
Rüschendorf, L.: On c-optimal random variables. Stat. Probab. Lett. 27(3), 267–270 (1996)
Thorisson, H.: Coupling, Stationarity, and Regeneration. Probability and its Applications (New York), pp. xiv+517. Springer-Verlag, New York (2000)
Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003)
Xia, A.: Poisson approximation, compensators and coupling. Stochastic Anal. Appl. 18(1), 159–177 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Decreusefond, L. Wasserstein Distance on Configuration Space. Potential Anal 28, 283–300 (2008). https://doi.org/10.1007/s11118-008-9077-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-008-9077-5