Asymptotics and spectral results for random walks on p-adics

Markov semigroups on the field Qp of p-adic numbers which are symmetric with respect to a measure ρ(x) dx absolutely continuous relative to the Haar measure dx on Qp 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 eigenfunctions) is provided. Necessary and sufficient conditions for reducibility, recurrence and transience are found. Results about exit times from balls are also presented, as well as a proof of null recurrence.