Harmonic analysis of additive Lévy processes

Abstract

Let X 1, . . . ,X N denote N independent d-dimensional Levy processes, and consider the N-parameter random field $$\mathfrak{X}(t) := X_1(t_1)+\cdots+ X_N(t_N).$$ First we demonstrate that for all nonrandom Borel sets $${F\subseteq{{\bf R}^d}}$$ , the Minkowski sum $${\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}$$ , of the range $${\mathfrak{X}({{\bf R}^{N}_{+}})}$$ of $${\mathfrak{X}}$$ 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 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.