A Decomposition Theorem for Lévy Processes on Local Fields

Sergio Albeverio () and Xuelei Zhao
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Sergio Albeverio: Universität Bonn
Xuelei Zhao: Universität Bonn

Journal of Theoretical Probability, 2001, vol. 14, issue 1, 1-19

Abstract: Abstract The study of Lévy processes on local fields has been initiated by Albeverio et al. (1985)–(1998) and Evans (1989)–(1998). In this paper, a decomposition theorem for Lévy processes on local fields is given in terms of a structure result for measures on local fields and a Lévy–Khinchine representation. It is shown that a measure on a local field can be decomposed into three parts: a spherically symmetric measure, a totally non-spherically symmetric measure and a singular measure. We show that if the Radon–Nikodym derivative of the absolutely continuous part of a Lévy measure on a local field is locally constant, the Lévy process is the sum of a spherically symmetric random walk, a finite or countable set of totally non, spherically symmetric Lévy processes with single balls as support of their Lévy measure, end a singular Lévy process. These processes are independent. Explicit formulae for the transition function are obtained.

Keywords: Lévy processes; local fields; Lévy–Khinchine representation; jump processes (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1007878412949

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