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

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