Geometric properties of fractional Brownian sheets
The aim of the present paper is to prove special properties of \((N,d)\)-fractional Brownian sheets \(B^H=\{B^H(t) : t\in\mathbb R_+^N\}\) with Hurst index \(H=(H_1,\ldots,H_N)\in (0,1)^N\). First it is shown that those processes are sectorial local non-deterministic in the case \(d=1\), i.e., given \(\varepsilon>0\) there is a constant \(c>0\) only depending on \(\varepsilon\), on \(H\) and on \(N\) such that \[ \text{ Var}(B^H(u)\, | \, B^H(t^1),\dots,B^H(t^n))\geq c\,\sum_{j=1}^N\min_{0\leq k\leq n} | u_j-t_j^k| ^{2 H_j} \] for all \(u,t^1,\dots,t^n\in[\varepsilon,\infty)^N\). Here one takes \(t^0=0\). The authors introduce a notion of dimension for Borel measures and sets, which is suitable to describe the anisotropic nature of \(B_H\), and they determine this dimension for \(B_H(E)\) where \(E\) is some Borel set in \((0,\infty)^N\). Furthermore, sufficient conditions are given in order that \(B_H(E)\) is a Salem set or that it contains interior points.
scientific article; zbMATH DE number 5189562