The Hausdorff Dimension of the Level Sets for a Fractional Brownian Sheet

Abstract

Let be a (N, d, α) fractional Brownian sheet, where Z ₁,…, Z _d are independent copies of a real valued fractional Brownian sheet taking value in R ¹. Let ℜ denote the class of all N-dimensional closed intervals I ⊂ (0, ∞) ^N with sides parallel to the axes and E(x, T) = {t ∈ T: Z(t) = x} be the level set of Z(t) at x. In this paper, the Hausdorff dimension of E(x, T) is established. We get that for any T ∈ ℜ, almost every x ∈ R ^d , dimE(x, T) = N − dα with probability one under the condition N > dα.

Key Words:

• Hausdorff dimension
• Level set
• Fractional Brownian sheet

2000MR Subject Classification:

• 60G17
• 60J65
• 60F15

Acknowledgment

Project supported by NSFC(10131040), SRFDP(2002335090), KRF(D00008).