Semi-linear Cauchy problem and Markov process associated with a p-adic non-local ultradiffusion operator

Abstract

This work is dedicated to study the pseudodifferential operator $$(D^\alpha _{d_1,d_2} \varphi )(x)=-\int \limits _{{\mathbb {Q}}_p^n} {\mathcal {A}}^{-\alpha }_{d_1,d_2}(y) [\varphi (x+y)-\varphi (x)] d^ny$$ , which can be seen as a generalization of Taibleson operator; here $${\mathcal {A}}^{\alpha }_{d_1,d_2}(x)=\max \left\{ \left\| x\right\| _p^{d_1},\ \left\| x\right\| _p^{d_2}\right\} ^\alpha $$ . We show that semi-linear Cauchy problem is well-posed in $${\mathfrak {M}}_{\lambda }$$ [a Banach space containing functions that do not belong to $$L^1({\mathbb {Q}}_p^n)$$ ], assuming that semi-linear part f is a Lipschitz function. We associate to the corresponding homogeneous problem a Markov process, which is indeed a Feller process. Finally, we study the first passage time problem.