Geometric characteristics of smooth anisotropic random fields.

Abstract : This thesis deals with anisotropic regular random fields, defined on the Euclidian space and studied from a geometric perspective. Some of our framework is Gaussian. We focus on three geometric characteristics: the number of critical points, the level sets measure and the Euler characteristic of excursion sets. Our main tools are Rice formulas for the expectation and the variance. We first address the question of the finiteness of the variance of the number of critical points of a stationary and Gaussian random field. The so-called Geman condition, which is known as a sufficient condition in dimension one, is extended to higher dimensions and to an anisotropic setting. Then two different anisotropic models are studied. On the one hand, the anisotropy of the deformed random field model (studied in dimension two) is due to a deterministic deformation of the parameter space. We give an explicit characterization of the deformations that preserve the isotropy of deformed random field. The cases of isotropy are proved to match a certain invariance property of the expected Euler characteristic of some excursion sets. This geometric characteristic also allows to identify the deformation of the model, when the latter is unknown. On the other hand, the anisotropy of the random wave model stems from the spectral domain. Our anisotropic random wave model allows to generalize existing models, for instance Berry’s planar waves and a spatiotemporal sea wave model. Our purpose is to link geometric characteristics of a random wave, such as the expected measure of its level sets, with the distribution of its random wavevector, in particular its moments of finite order and its directional statistics. Considering Berry’s anisotropic planar waves, we prove that the expected length of its nodal lines is a decreasing function of the anisotropy of the random wavevector.
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Julie Fournier. Geometric characteristics of smooth anisotropic random fields.. Probability [math.PR]. Sorbonne Université; Université Paris 5 René Descartes, 2018. English. ⟨tel-02155827⟩

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