Joint continuity of the local times of fractional Brownian sheets

Abstract

Let BH = {BH(t), t ∈ ℝ+N} be an (N, d)-fractional Brownian sheet with index H = (H 1 ,⋯,HN) ∈ (0,1)N defined by B H(t) = (B1H(t),⋯,(t), ⋯,B dH(t)) (t ∈ ℝ+N), where B1H,⋯,BdH are independent copies of a real-valued fractional Brownian sheet BH. We prove that if d< ∑l=1NHl-1, then the local times of BH are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global Hölder conditions for the local times of B H. These results are applied to study analytic and geometric properties of the sample paths of BH. © Association des Publications de l'Institut Henri Poincaré, 2008.