Construction of fractal random series with point processes and Voronoi tessellations

Calka, P., Universite de Rouen, France

Weierstrass and Takagi-Knopp type series are infinite sums of sinusoidal or sawtooth signals with fractal graph. For more than a century, they have been intensively studied as models for irregular signals and rough surfaces. Their definition requires an underlying sequence of partitions of the space which is in general derived from the dyadic cubes. In this talk, we construct a new family of random series defined on $\R^d$. They are close to the Takagi-Knopp model with the extra ingredient of a sequence of random partitions of the space, distributed as Voronoi tessellations of an i.i.d. sequence of underlying point processes. These series are continuous functions, indexed by one scaling parameter and two Hurst-like exponents. We show that the graph of such function is fractal with explicit and equal box dimension and Hausdorff dimension. The proof of this result relies on the adaptation of general criteria from fractal geometry (oscillation estimates, Frostman-type lemma) on the one hand and new distributional properties of Voronoi tessellations on the other hand. Particular emphasis will be given to the particular ingredients from stochastic geometry. We will finally discuss some extensions of the model. This talk is based on joint works with Yann Demichel.