Harmonic analysis of additive Lévy processes

Article

Abstract

Let X 1, . . . ,X N denote N independent d-dimensional Lévy 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 Lévy processes. Presently, we highlight a few new consequences.

Keywords

Additive Lévy processes Multiplicative Lévy processes Capacity Intersections of regenerative sets 

Mathematics Subject Classification (2000)

60G60 60J55 60J45 

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Authors and Affiliations

  1. 1.Department of MathematicsThe University of UtahSalt Lake CityUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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