Asymptotics and spectral results for random walks on p-adics

Markov semigroups on the field Q(p) of p-adic numbers which are symmetric w ith respect to a measure p(x) dx absolutely continuous relative to the Haar measure dx on Q(p) are constructed. The corresponding Dirichlet forms and associated Markov processes are exhibited and shown to be of the jump type. A detailed description of the spectrum of the generator (eigenvalues and e igenfunctions) is provided. Necessary and sufficient conditions for reducib ility, recurrence and transience are found. Results about exit times from b alls are also presented, as well as a proof of null recurrence. (C) 1999 El sevier Science B.V. All rights reserved.