JournalsrmiVol. 20, No. 2pp. 395–412

Hausdorff dimension of the graph of the Fractional Brownian Sheet

  • Antoine Ayache

    Université Lille 1, Villeneuve d'Asq, France
Hausdorff dimension of the graph of the Fractional Brownian Sheet cover
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Abstract

Let {B(α)(t)}tRd\{B^{(\alpha)}(t)\}_{t\in\mathbb{R}^{d}} be the Fractional Brownian Sheet with multi-index α=(α1,,αd)\alpha=(\alpha_1,\ldots, \alpha_d), 0<αi<10< \alpha_i< 1. In \cite{Kamont1996}, Kamont has shown that, with probability 11, the box dimension of the graph of a trajectory of this Gaussian field, over a non-degenerate cube QRdQ\subset\mathbb{R}^{d} is equal to d+1min(α1,,αd)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.

Cite this article

Antoine Ayache, Hausdorff dimension of the graph of the Fractional Brownian Sheet. Rev. Mat. Iberoam. 20 (2004), no. 2, pp. 395–412

DOI 10.4171/RMI/394