Abstract:

Let B^H={B^H(t), t ∈ R₊^N} be a real-valued (N, d) fractional Brownian sheet with Hurst index H=(H₁, …, H_N). The characteristics of the polar functions for B^H are discussed. The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of B^H is obtained. The Hausdorff dimension about the fixed points and the inequality about the Kolmogorov’s entropy index for B^H are presented. Furthermore, it is proved that any two independent fractional Brownian sheets are nonintersecting in some conditions. A problem proposed by LeGall about the existence of no-polar continuous functions satisfying the Hölder condition is also solved.

Key words: Fractional Brownian sheet, polar function, Hausdorff dimension, intersection