Intersections and polar functions of fractional brownian sheets

Abstract

Let B H = B H (t), t∈ R N + be a real-valued ( N, d) fractional Brownian sheet with Hurst index H=(H 1, …, 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.