Geometric Properties of Fractional Brownian Sheets

Abstract

Let $B^H=\{B^H(t), t\in {\Bbb R}_+^N\}$ be an (N, d)-fractional Brownian sheet with Hurst index H = (H1,...,HN) ∈ (0, 1)N. Our objective of the present article is to characterize the anisotropic nature of BH in terms of H. We prove the following results: (1) BH is sectorially locally nondeterministic. (2) By introducing a notion of "dimension" for Borel measures and sets, which is suitable for describing the anisotropic nature of BH, we determine ${\rm dim}_{\cal H}B^H(E)$ for an arbitrary Borel set $E \subset (0, \infty)^N.$ Moreover, when Bα is an (N, d)-fractional Brownian sheet with index 〈α〉 = (α,..., α) (0 < α < 1), we prove the following uniform Hausdorff dimension result for its image sets: If N ≤ αd, then with probability one, ${\rm dim}_{\cal H}B^{\langle\alpha\rangle}(E)=\frac{1}{\alpha}{\rm dim}_{\cal H}E {\rm for\ all\ Borel\ sets}\ E \subset (0, \infty)^N.$ (3) We provide sufficient conditions for the image BH(E) to be a Salem set or to have interior points. The results in (2) and (3) describe the geometric and Fourier analytic properties of BH. They extend and improve the previous theorems of Mountford [35], Khoshnevisan and Xiao [29] and Khoshnevisan, Wu, and Xiao [28] for the Brownian sheet, and Ayache and Xiao [5] for fractional Brownian sheets.