Joint continuity of the local times of fractional Brownian sheets

Abstract

Let $B^H=\{B^H(t),t\in {\mathbb{R}}_{+}^N\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in (0,1{)}^N$ defined by $B^H(t)=(B_1^H(t),...,B_d^H(t))(t\in {\mathbb{R}}_{+}^N),$ where $B_1^H,...,B_d^H$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<{\sum }_{\ell =1}^N{H}_{\ell }^{-1}$, then the local times of $B^H$ 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 $B^H$.

Published in at http://dx.doi.org/10.1214/07-AIHP131 the Annales de l'Institut Henri Poincaré - Probabilités et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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