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Fractal random series generated by Poisson-Voronoi tessellations

Authors: Pierre Calka and Yann Demichel
Journal: Trans. Amer. Math. Soc. 367 (2015), 4157-4182
MSC (2010): Primary 28A80, 60D05; Secondary 26B35, 28A78, 60G55
Published electronically: September 24, 2014
MathSciNet review: 3324923
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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.

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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

Yann Demichel
Affiliation: Laboratoire MODAL’X, EA 3454, Université Paris Ouest Nanterre La Défense, 200 avenue de la République, 92001 Nanterre, France

Keywords: Poisson-Voronoi tessellation, Poisson point process, random functions, Takagi series, fractal dimension, Hausdorff dimension
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).
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