Champs aléatoires: autosimilarité, anisotropie et étude directionnelle

Abstract : We study random fields that can modelize some porous media. We are mainly interested in second order statistics and focus on self-similarity properties. Under stationary assumptions, the field is characterized by a spectral measure. Asymptotic self-similarity properties are then given by the directional asymptotic homogeneity of the measure. Its parameter is given by the smallest coefficient of homogeneity in a logarithmic scale. To recover anisotropy one can perform a Radon transform of the field for which the self-similarity parameter depends on the direction. These second order results are well adapted to Gaussian models. In this case, the self-similarity parameter can be estimated with the quadratic variations. We consider injectivity problems for the Radon transform. In the last part, we study a Poissonian model obtained by small aggregated balls. Self-similarity properties are similar for second order statistics but unusual for convergence in law.
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Submitted on : Wednesday, September 21, 2005 - 1:28:37 AM
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Hermine Biermé. Champs aléatoires: autosimilarité, anisotropie et étude directionnelle. Mathématiques [math]. Université d'Orléans, 2005. Français. ⟨tel-00010227⟩

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