Multifractal analysis and self-similarity / Analyse multifractale et auto-similarité

Collection Multifractal analysis and self-similarity / Analyse multifractale et auto-similarité

Organisateur(s) Barral, Julien ; Batakis, Athanasios ; Berthé, Valérie ; Seuret, Stéphane ; Thuswaldner, Jörg
Date(s) 26/06/2023 - 30/06/2023
URL associée https://conferences.cirm-math.fr/2751.html
00:00:00 / 00:00:00
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Fractional Gaussian and Stable randoms fields on fractals

De Céline Lacaux

In this talk, we adopt the viewpoint about fractional fields which is given in Lodhia and al. Fractional Gaussian fields: a survey, Probab. Surv. 13 (2016), 1-56. As example, we focus on random fields defined on the Sierpiński gasket but random fields defined on fractional metric spaces can also be considered. Hence, for $s \geq 0$, we consider the random measure $X=(-\Delta)^{-s} W$ where $\Delta$ is a Laplacian on the Sierpiński gasket $K$ equipped with its Hausdorff measure $\mu$ and where $W$ is a Gaussian random measure with intensity $\mu$. For a range of values of the parameter $s$, the random measure $X$ admits a Gaussian random field $(X(x))_{x \in K}$ as density with respect to $\mu$. Moreover, using entropy method, an upper bound of the modulus of continuity of $(X(x))_{x \in K}$ is obtained, which leads to the existence of a modification with Hölder sample paths. Along the way we prove sharp global Hölder regularity estimates for the fractional Riesz kernels on the gasket. In addition, the fractional Gaussian random field $X$ is invariant by the symmetries of the gasket. If time allows, some extension to $\alpha$-stable random fields will also be presented. Especially, for $s \geq s_0$ there still exists a modification of the $\alpha$-stable field $\mathrm{X}$ with Hölder sample paths whereas for $s< s_{0}$, such modification does not exist. This is a joint work with Fabrice Baudoin (University of Connecticut).

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Données de citation

  • DOI 10.24350/CIRM.V.20063803
  • Citer cette vidéo Lacaux, Céline (27/06/2023). Fractional Gaussian and Stable randoms fields on fractals. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20063803
  • URL https://dx.doi.org/10.24350/CIRM.V.20063803

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