Harmonic analysis of additive Lévy processes

Abstract

Let X1, . . ., XN denote N independent d-dimensional Lévy processes, and consider the N-parameter random field, First we demonstrate that for all nonrandom Borel sets, the Minkowski sum, of the range of 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 Khoshnevisan et al. (Ann Probab 31(2):1097-1141, 2003). 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 et al. (Ann Probab 31(2):1097-1141, 2003), 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. © Springer-Verlag 2008.