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 $\alpha =({\alpha }_1,\dots ,{\alpha }_d)$, $0<{\alpha }_i<1$. In [14], 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 $d+1-\min ({\alpha }_1,\dots ,{\alpha }_d)$. In this paper, we prove that this result remains true when the box dimension is replaced by the Hausdorff dimension or the packing dimension.