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 Electronic Journal of Probability > Vol. 11 (2006) > Paper 32 open journal systems 


Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet

Yimin Xiao, Michigan State University
Davar Khoshnevisan, The University of Utah
Dongsheng Wu, Michigan State University


Abstract
We prove the following results about the images and multiple points of an N-parameter, d-dimensional Brownian sheet B ={B(t)}t in R+N:
(1)
If dim F ≤ d/2, then B(F) is almost surely a Salem set.
(2)
If N ≤ d/2, then with probability one dim B(F) = 2 dim F for all Borel sets of R+N, where ``dim'' could be everywhere replaced by the ``Hausdorff,'' ``packing,'' ``upper Minkowski,'' or ``lower Minkowski dimension.''
(3)
Let Mk be the set of k-multiple points of B. If N ≤ d/2 and Nk > (k-1)d/2, then dimh Mk = dimp Mk = 2 Nk - (k-1)d a.s.
The Hausdorff dimension aspect of (2) was proved earlier; see Mountford (1989) and Lin (1999). The latter references use two different methods; ours of (2) are more elementary, and reminiscent of the earlier arguments of Monrad and Pitt (1987) that were designed for studying fractional Brownian motion. If N>d/2 then(2) fails to hold. In that case, we establish uniform-dimensional properties for the (N,1)-Brownian sheet that extend the results of Kaufman (1989) for 1-dimensional Brownian motion. Our innovation is in our use of the sectorial local nondeterminism of the Brownian sheet (Khoshnevisan and Xiao, 2004).


Full text: PDF

Pages: 817-843

Published on: September 19, 2006
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Electronic Journal of Probability. ISSN: 1083-6489