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A Decomposition Theorem for Lévy Processes on Local Fields

Author

Listed:
  • Sergio Albeverio

    (Universität Bonn
    SFB 237 Essen–Bochum–Düsseldorf; BiBoS Research Centre
    CERFIM
    Acc. Arch. (USI))

  • Xuelei Zhao

    (Universität Bonn)

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.

Suggested Citation

  • Sergio Albeverio & Xuelei Zhao, 2001. "A Decomposition Theorem for Lévy Processes on Local Fields," Journal of Theoretical Probability, Springer, vol. 14(1), pages 1-19, January.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:1:d:10.1023_a:1007878412949
    DOI: 10.1023/A:1007878412949
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    References listed on IDEAS

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    1. Albeverio, Sergio & Karwowski, Witold, 1994. "A random walk on p-adics--the generator and its spectrum," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 1-22, September.
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    Cited by:

    1. Hiroshi Kaneko & Xuelei Zhao, 2006. "Transition Semi–groups on a Local Field Induced by Galois Group and their Representation," Journal of Theoretical Probability, Springer, vol. 19(1), pages 221-234, January.

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