4 research outputs found
Realisability conditions for second order marginals of biphased media
16 pagesInternational audienceThis paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper
Random Measurable Sets and Covariogram Realisability Problems
We provide a characterization of the realisable set covariograms, bringing a
rigorous yet abstract solution to the problem in materials science. Our
method is based on the covariogram functional for random mesurable sets (RAMS)
and on a result about the representation of positive operators in a locally
compact space. RAMS are an alternative to the classical random closed sets in
stochastic geometry and geostatistics, they provide a weaker framework allowing
to manipulate more irregular functionals, such as the perimeter. We therefore
use the illustration provided by the problem to advocate the use of
RAMS for solving theoretical problems of 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.Comment: 35p