Joint continuity of the local times of fractional Brownian sheets

TL;DR

This paper proves the joint continuity of local times for fractional Brownian sheets under certain conditions, confirming a conjecture and providing sharp regularity results that enhance understanding of their sample path properties.

Contribution

It establishes the joint continuity of local times for fractional Brownian sheets when the dimension condition is met, confirming a prior conjecture and deriving sharp Hölder regularity conditions.

Findings

Local times are jointly continuous if d<∑H_ℓ^{-1}.

Sharp Hölder conditions for local times are established.

Results apply to the analysis of sample path properties.

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 $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ 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 $B^H$.