Local times of multifractional Brownian sheets

TL;DR

This paper investigates the local times of multifractional Brownian sheets, establishing their existence, regularity, and the Hausdorff dimensions of their level sets under certain conditions, extending previous results for related processes.

Contribution

It extends the theory of local times and Hausdorff dimensions from fractional and multifractional Brownian motions to the more complex multifractional Brownian sheets.

Findings

Existence and joint continuity of local times.

Hölder regularity of local times.

Hausdorff dimensions of level sets determined.

Abstract

Denote by $H(t)=(H_1(t),...,H_N(t))$ a function in $t\in{\mathbb{R}}_+^N$ with values in $(0,1)^N$. Let $\{B^{H(t)}(t)\}=\{B^{H(t)}(t),t\in{\mathbb{R}}^N_+\}$ be an $(N,d)$-multifractional Brownian sheet (mfBs) with Hurst functional $H(t)$. Under some regularity conditions on the function $H(t)$, we prove the existence, joint continuity and the H\"{o}lder regularity of the local times of $\{B^{H(t)}(t)\}$. We also determine the Hausdorff dimensions of the level sets of $\{B^{H(t)}(t)\}$. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets.