Harmonic analysis of additive Levy processes

Let X-1 , . . . , X-N denote N independent d-dimensional Levy processes, and consider the N-parameter random field (sic)(t) := X-1(t(1))+ . . . + X-N (t(N)). First we demonstrate that for all nonrandom Borel sets F subset of R-d, the Minkowski sum (sic)(R-+(N)) circle plus F, of the range (sic)(R-+(N)) of (sic) 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 Levy processes. Presently, we highlight a few new consequences.