A Decomposition Theorem for Lévy Processes on Local Fields

Abstract

The study of Levy processes on local fields has been initiated by Albeverio et al. (1985)–(1998) and Evans (1989)–(1998). In this paper, a decomposition theorem for Levy processes on local fields is given in terms of a structure result for measures on local fields and a Levy–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 Levy measure on a local field is locally constant, the Levy process is the sum of a spherically symmetric random walk, a finite or countable set of totally non, spherically symmetric Levy processes with single balls as support of their Levy measure, end a singular Levy process. These processes are independent. Explicit formulae for the transition function are obtained.