Skip to main content Accessibility help
×
×
Home

Random measurable sets and covariogram realizability problems

  • Bruno Galerne (a1) and Raphael Lachièze-Rey (a1)
Abstract

We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S 2 problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S 2 problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.

Copyright
Corresponding author
Postal address: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 Rue des Saints-pères, 75006 Paris, France.
∗∗ Email address: raphael.lachieze-rey@parisdescartes.fr
References
Hide All
[1] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edn. Springer, Berlin.
[2] Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press.
[3] Bogachev, V. I. (2007). Measure Theory, Vol. II. Springer, Berlin.
[4] Caselles, V., Chambolle, A., Moll, S. and Novaga, M. (2008). A characterization of convex calibrable sets in R N with respect to anisotropic norms. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 803-832.
[5] Chilès, J.-P. and Delfiner, P. (1999). Geostatistics: Modeling Spatial Uncertainty. John Wiley, New York.
[6] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II: General Theory and Structure, 2nd edn. Springer, New York.
[7] Deza, M. M. and Laurent, M. (1997). Geometry of Cuts and Metrics. Springer, Berlin.
[8] Emery, X. (2010). On the existence of mosaic and indicator random fields with spherical, circular, and triangular variograms. Math. Geosci. 42, 969984.
[9] Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC, Boca Raton, FL.
[10] Fritz, T. and Chaves, R. (2013). Entropic inequalities and marginal problems. IEEE Trans. Inf. Theory 59, 803817.
[11] Galerne, B. (2011). Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal. Stereol. 30, 3951.
[12] Galerne, B. (2014). Random fields of bounded variation and computation of their variation intensity. Tech. Rep. 2014-25, Laboratoire MAP5, Université Paris Descartes.
[13] Himmelberg, C. J. (1975). Measurable relations. Fund. Math 87, 5372.
[14] Hirsch, F. and Lacombe, G. (1999). Elements of Functional Analysis (Graduate Texts Math. 192), Springer, New York.
[15] Jiao, Y., Stillinger, F. H. and Torquato, S. (2007). Modeling heterogeneous materials via two-point correlation functions: basic principles. Phys. Rev. E 76, 031110.
[16] Kallenberg, O. (1986). Random Measures, 4th edn. Akademie-Verlag, Berlin.
[17] Kuna, T., Lebowitz, J. L. and Speer, E. R. (2011). Necessary and sufficient conditions for realizability of point processes. Ann. Appl. Prob. 21, 12531281.
[18] Lachièze-Rey, R. (2013). Realisability conditions for second-order marginals of biphased media. Random Structures Algorithms 10.1002/rsa.20546.
[19] Lachiéze-Rey, R. and Molchanov, I. (2015). Regularity conditions in the realisability problem with applications to point processes and random closed sets. Ann. Appl. Prob. 25, 116149.
[20] Lantuéjoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin.
[21] Masry, E. (1972). On covariance functions of unit processes. SIAM J. Appl. Math. 23, 2833.
[22] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.
[23] Matheron, G. (1993). Une conjecture sur la covariance d'un ensemble aléatoire. In Cahiers de Géostatistique, Fascicule 3, Compte-Rendu des Journées de Géostatistique (Fontainebleau, 1993), pp. 107113.
[24] McMillan, B. (1955). History of a problem. J. Soc. Ind. Appl. Math. 3, 119128.
[25] Molchanov, I. (2005). Theory of Random Sets. Springer, London.
[26] Quintanilla, J. A. (2008). Necessary and sufficient conditions for the two-point phase probability function of two-phase random media. Proc. R. Soc. London A 464, 17611779.
[27] Rataj, J. (2014). Random sets of finite perimeter. Math. Nachr. 288, 10471056.
[28] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
[29] Shepp, L. A. (1963). On positive-definite functions associated with certain stochastic processes. Tech. Rep. 63-1213-11, Bell Laboratories.
[30] Straka, F. and Štěpán, J. (1988). Random sets in [0,1]. In Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Vol. B, Reidel, Dordrecht, pp. 349356.
[31] Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed