  Let $\langle x\rangle^{\alpha}=(\max\{|x|_{p},p^r\})^{\alpha}$ 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
$H^{\alpha}$.
