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- Title
Harmonic analysis of additive Lévy processes.
- Authors
Khoshnevisan, Davar; Xiao, Yimin
- Abstract
Let X1, . . . , X N denote N independent d-dimensional Lévy processes, and consider the N-parameter random field 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.
- Subjects
RANDOM fields; BOREL sets; HARMONIC analysis (Mathematics); RINGS of integers; WIENER processes; PROBABILITY theory
- Publication
Probability Theory & Related Fields, 2009, Vol 145, Issue 3/4, p459
- ISSN
0178-8051
- Publication type
Article
- DOI
10.1007/s00440-008-0175-5