ar X iv : 0 70 6 . 41 64 v 1 [ m at h . PR ] 2 8 Ju n 20 07 HARMONIC ANALYSIS OF ADDITIVE LÉVY PROCESSES

Let X1, . . . , XN denote N independent d-dimensional Lévy processes, and consider the N -parameter random field X(t) := X1(t1) + · · ·+XN (tN ). First we demonstrate that for all nonrandom Borel sets F ⊆ R, the Minkowski sum X(R+ ) ⊕ F , of the range X(R+ ) of 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.