Image sets of fractional Brownian sheets

TL;DR

This paper investigates the Hausdorff dimension of image sets of fractional Brownian sheets, providing uniform dimensional properties and introducing a refined local nondeterminism property relevant for fractal analysis.

Contribution

It establishes uniform Hausdorff dimension results for fractional Brownian sheets and introduces a refined sectorial local nondeterminism property.

Findings

Determined Hausdorff dimension of image sets in the specified dimension range.

Answered open questions on the dimensional properties of fractional Brownian sheets.

Introduced a new refinement of local nondeterminism property for fractional Brownian sheets.

Abstract

Let $B^H = \{ B^H(t), t\in\mathbb{R}^N \}$ be an $(N,d)$-fractional Brownian sheet with Hurst index $H=(H_1,\dotsc,H_N)\in (0,1)^N$. The main objective of the present paper is to study the Hausdorff dimension of the image sets $B^H(F+t)$, $F\subset\mathbb{R}^N$ and $t\in\mathbb{R}^N$, in the dimension case $d<\tfrac{1}{H_1}+\cdots+\tfrac{1}{H_N}$. Following the seminal work of Kaufman (1989), we establish uniform dimensional properties on $B^H$, answering questions raised by Khoshnevisan et al (2006) and Wu and Xiao (2009). For the purpose of this work, we introduce a refinement of the sectorial local-nondeterminism property which can be of independent interest to the study of other fine properties of fractional Brownian sheets.