Fractal random series generated by Poisson-Voronoi tessellations
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- by Pierre Calka and Yann Demichel PDF
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Abstract:
In this paper, we construct a new family of random series defined on $\mathbb {R}^D$, indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of $\mathbb {R}^D$ are not the usual dyadic meshes but random Voronoi tessellations generated by Poisson point processes. This approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and Hausdorff dimensions. The proof of this main result is based on several new distributional properties of the Poisson-Voronoi tessellation on the one hand, and an estimate of the oscillations of the function coupled with an application of a Frostman-type lemma on the other hand. Finally, we introduce two related models and provide in particular a box-dimension calculation for a derived deterministic Takagi-Knopp series with hexagonal bases.References
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Additional Information
- Pierre Calka
- Affiliation: Laboratoire de Mathématiques Raphaël Salem, UMR 6085, Université de Rouen, avenue de l’Université, Technopôle du Madrillet, 76801 Saint-Etienne-du-Rouvray, France
- Email: pierre.calka@univ-rouen.fr
- Yann Demichel
- Affiliation: Laboratoire MODAL’X, EA 3454, Université Paris Ouest Nanterre La Défense, 200 avenue de la République, 92001 Nanterre, France
- Email: yann.demichel@u-paris10.fr
- Received by editor(s): September 19, 2012
- Received by editor(s) in revised form: May 14, 2013
- Published electronically: September 24, 2014
- Additional Notes: This work was partially supported by the French ANR grant PRESAGE (ANR-11-BS02-003), the French ANR grant MATAIM (ANR-09-BLAN-0029-01) and the French research group GeoSto (CNRS-GDR3477).
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 4157-4182
- MSC (2010): Primary 28A80, 60D05; Secondary 26B35, 28A78, 60G55
- DOI: https://doi.org/10.1090/S0002-9947-2014-06267-3
- MathSciNet review: 3324923