caa a22 4500 46324673X CHVBK 20180405153328.0 cr unu---uuuuu 170326e20070201xx s 000 0 eng 10.1007/s00041-005-5078-y doi (NATIONALLICENCE)springer-10.1007/s00041-005-5078-y Geometric Properties of Fractional Brownian Sheets [Elektronische Daten] [Dongsheng Wu, Yimin Xiao] Let $B^H=\{B^H(t), t\in {\Bbb R}_+^N\}$ be an (N, d)-fractional Brownian sheet with Hurst index H = (H1,...,HN) ∈ (0, 1)N. Our objective of the present article is to characterize the anisotropic nature of BH in terms of H. We prove the following results: (1) BH is sectorially locally nondeterministic. (2) By introducing a notion of "dimension" for Borel measures and sets, which is suitable for describing the anisotropic nature of BH, we determine ${\rm dim}_{\cal H}B^H(E)$ for an arbitrary Borel set $E \subset (0, \infty)^N.$ Moreover, when Bα is an (N, d)-fractional Brownian sheet with index 〈α〉 = (α,..., α) (0 < α < 1), we prove the following uniform Hausdorff dimension result for its image sets: If N ≤ αd, then with probability one, ${\rm dim}_{\cal H}B^{\langle\alpha\rangle}(E)=\frac{1}{\alpha}{\rm dim}_{\cal H}E {\rm for\ all\ Borel\ sets}\ E \subset (0, \infty)^N.$ (3) We provide sufficient conditions for the image BH(E) to be a Salem set or to have interior points. The results in (2) and (3) describe the geometric and Fourier analytic properties of BH. They extend and improve the previous theorems of Mountford [35], Khoshnevisan and Xiao [29] and Khoshnevisan, Wu, and Xiao [28] for the Brownian sheet, and Ayache and Xiao [5] for fractional Brownian sheets. Birkhauser Boston, 2007 Wu Dongsheng Department of Statistics and Probability, A-413, Wells Hall, Michigan State University, East Lansing, MI 48824, USA aut Xiao Yimin Department of Statistics and Probability, A-413, Wells Hall, Michigan State University, East Lansing, MI 48824, USA aut Journal of Fourier Analysis and Applications Birkhäuser-Verlag; www.birkhauser.com 13/1(2007-02-01), 1-37 1069-5869 13:1<1 2007 13 41 https://doi.org/10.1007/s00041-005-5078-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00041-005-5078-y text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Wu Dongsheng Department of Statistics and Probability, A-413, Wells Hall, Michigan State University, East Lansing, MI 48824, USA aut NATIONALLICENCE 700 1- Xiao Yimin Department of Statistics and Probability, A-413, Wells Hall, Michigan State University, East Lansing, MI 48824, USA aut NATIONALLICENCE 773 0- Journal of Fourier Analysis and Applications Birkhäuser-Verlag; www.birkhauser.com 13/1(2007-02-01), 1-37 1069-5869 13:1<1 2007 13 41 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer