Harmonic analysis of additive Lévy processes

Abstract. Let X1,..., XN denote N independent d-dimensional Lévy processes, and con-sider the N-parameter random field X(t): = X1(t1) + · · ·+XN (tN). First we demonstrate that for all nonrandom Borel sets F ⊆ Rd, the Minkowski sum X(RN+) ⊕ F, of the range X(RN+) 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 by removing a certain condition of symmetry in [70]. 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 [70], the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. Presently, we highlight a few new consequences. 1