A Decomposition Theorem for Levy Processes on Local Fields

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 Lévy 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.