Probability Theory and Related Fields

, 145:459

Harmonic analysis of additive Lévy processes

Authors

    • Department of MathematicsThe University of Utah
  • Yimin Xiao
    • Department of Statistics and ProbabilityMichigan State University
Article

DOI: 10.1007/s00440-008-0175-5

Cite this article as:
Khoshnevisan, D. & Xiao, Y. Probab. Theory Relat. Fields (2009) 145: 459. doi:10.1007/s00440-008-0175-5

Abstract

Let X1, . . . ,XN 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 processesMultiplicative Lévy processesCapacityIntersections of regenerative sets

Mathematics Subject Classification (2000)

60G6060J5560J45
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© Springer-Verlag 2008