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,...,H_N) ∈ (0, 1)_N. Our objective of the present article is to characterize the anisotropic nature of B_H in terms of H. We prove the following results: (1) B_H 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 B_H, we determine ${\rm dim}_{\cal H}B^H(E)$ for an arbitrary Borel set $E \subset (0, \infty)^N.$ Moreover, when B_α 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 B_H(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 B_H. 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.