Hausdorff dimension of the graph of the Fractional Brownian Sheet

Abstract

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.