P-adic Markov process and the problem of the first return over balls

Abstract

Let $⟨x{⟩}^{\alpha }=(\max \{|x{|}_p,p^r\}{)}^{\alpha }$ and $H^{\alpha }\phi ={\mathcal{F}}^{-1}[(⟨\xi {⟩}^{\alpha }-p^{r\alpha })\mathcal{F}\phi ]$, in this article we study the Markov process associated to this operator and the first passage time problem associated to $H^{\alpha }$.