Harmonic analysis of additive Lévy processes

D. Khoshnevisan and Yimin Xiao

Abstract. Let \(X_1\,,\ldots,X_N\) denote \(N\) independent \(d\)-dimensional Lévy processes, and consider the \(N\)-parameter random field

\(\displaystyle {\bf X}({\bf t}) := X_1(t_1) + \cdots + X_N(t_N). \)
First we demonstrate that for all nonrandom Borel sets \(F \subset{\bf R}^d\), the Minkowski sum \({\bf X}({\bf R}^N_+)\oplus F\), of the range \({\bf X}({\bf R}^N_+)\) of \({\bf X}\) with \(F\), can have positive \(d\)-dimensional Lebesgue measure if and only if a certain capacity of \(F\) is positive. This improves our earlier joint effort with Yuquan Zhong (2003) by removing a symmetry-type condition there. Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical [non-probabilistic] harmonic analysis that might be of independent interest. As was shown in Khoshnevisan, Xiao, and Zhong (2003), the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. We present a few new consequences here.

Keywords. Additive Lévy processes, multiplicative Lévy processes, capacity, intersections of regenerative sets.

AMS Classification (2000). 60G60, 60J55, 60J45.

Support. Research supported in part by grants from the U.S. National Science Foundation.


Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
Yimin Xiao
Department of Statistics and Probability
A-413 Wells Hall
Michigan State University
East Lansing, MI 48824, U.S.A.

Last Update: July 28, 2008