Joint continuity of the local times of fractional Brownian sheets

Details Joint continuity of the local times of fractional Brownian sheets Joint continuity of the local times of fractional Brownian sheets

2008-08-22

arxiv.org/abs/0808.3054v1

Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques 2008, Vol. 44, No. 4, 727-748

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AIHP131 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiqu

Scientific paper

10.1214/07-AIHP131

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 $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, 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 H\"{o}lder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.

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