Preprint:
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 symmetrytype
condition there. Moreover, we show that under
mild regularity conditions, our necessary and
sufficient condition can be recast in terms of
onepotential densities. This rests on developing
results in classical [nonprobabilistic]
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.
EPrint.

First public version (posted: 6/28/07)

Final version (posted: 7/28/08)
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 841120090, U.S.A.
davar@math.utah.edu

Yimin Xiao
Department of Statistics and Probability
A413 Wells Hall
Michigan State University
East Lansing, MI 48824, U.S.A.
xiao@stt.msu.edu

Last Update: July 28, 2008