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.
Narn-Rueih Shieh
Department of Mathematics,
National Taiwan University
Taipei 10617, Taiwan
Yimin Xiao
Department of Statistics and Probability,
A-413 Wells Hall
Michigan State University
East Lansing, MI 48824, USA

Last Update: September 25, 2008
© 2006 - Davar Khoshnevisan, Narn-Rueih Shieh, and Yimin Xiao