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).
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