Preprint:
Hausdorff Dimension of the Contours
of Symmetric Additive Lévy Processes
Davar Khoshnevisan, Narn-Rueih Shieh, and Yimin Xiao
Abstract.
Let \(X_1\,,\ldots,X_N\) denote N independent,
symmetric Lévy processes on \({\bf R}^d.\)The
corresponding additive Lévy process is
defined as the following
\(N\)-parameter random field on \({\bf R}^d\):
$$
X({\bf t}) := X_1(t_1)+\cdots+X_N(t_N)\hskip1in
({\bf t}\in{\bf R}^N_+).
$$
Khoshnevisan and Xiao (2002) have found a necessary and sufficient
condition for the zero-set \(X^{-1}\{0\}\) of
\(X\) to be non-trivial with positive probability.
They also provide bounds for the Hausdorff dimension
of \(X^{-1}\{0\}\) which hold with positive probability
in the case that \(^{-1}\{0\}\) can be non void.
Here, we prove that the Hausdorff dimension of \(X^{-1}\{0\}\) is
a constant almost surely on the event \(\{X^{-1}\{0\}\cap F\neq\emptyset\}\).
Moreover, we derive a formula for the said constant. This portion of our work
extends the one-parameter formulas of Horowitz (1968)
and Hawkes (1974).
More generally, we prove that for every non-random
Borel set \(F\) in \((0\,,\infty)^N\),
the Hausdorff dimension of
\(X^{-1}\{0\}\cap F\) is a constant almost surely
on the event \(\{X^{-1}\{0\}\cap F\neq\emptyset\}\).
This constant is computed explicitly in many cases.
Keywords.
Additive Lévy processes, level sets, Hausdorff dimension
AMS Classification (2000)
Primary. 60G70
Secondary. 60F15
Support. Research supported in part by a grant from
the National Science Foundation grant DMS-0404729.
Pre/E-Prints. This paper is available in
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
davar@math.utah.edu
|
Narn-Rueih Shieh
Department of Mathematics,
National Taiwan University
Taipei 10617, Taiwan
shiehnr@math.ntu.edu.tw
|
Yimin Xiao
Department of Statistics and Probability,
A-413 Wells Hall
Michigan State University
East Lansing, MI 48824, USA
xiao@stt.msu.edu |
Last Update: September 25, 2008
© 2006 - Davar Khoshnevisan, Narn-Rueih Shieh, and Yimin Xiao