Revista Matemática Iberoamericana


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Volume 20, Issue 2, 2004, pp. 395–412
DOI: 10.4171/RMI/394

Published online: 2004-08-31

Hausdorff dimension of the graph of the Fractional Brownian Sheet

Antoine Ayache[1]

(1) Université Lille 1, Villeneuve d'Asq, France

Let $\{B^{(\alpha)}(t)\}_{t\in\mathbb{R}^{d}}$ be the Fractional Brownian Sheet with multi-index $\alpha=(\alpha_1,\ldots, \alpha_d)$, $0< \alpha_i< 1$. In \cite{Kamont1996}, Kamont has shown that, with probability $1$, the box dimension of the graph of a trajectory of this Gaussian field, over a non-degenerate cube $Q\subset\mathbb{R}^{d}$ is equal to $d+1-\min(\alpha_1,\ldots,\alpha_d)$. In this paper, we prove that this result remains true when the box dimension is replaced by the Hausdorff dimension or the packing dimension.

Keywords: Gaussian fields, fractional brownian motion, random wavelet series, Hausdorff dimension, packing dimension

Ayache Antoine: Hausdorff dimension of the graph of the Fractional Brownian Sheet. Rev. Mat. Iberoamericana 20 (2004), 395-412. doi: 10.4171/RMI/394