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

This work is dedicated to study the pseudodifferential operator (Dd1,d2αφ)(x)=-∫QpnAd1,d2-α(y)[φ(x+y)-φ(x)]dny, which can be seen as a generalization of Taibleson operator; here Ad1,d2α(x)=max{∥x∥pd1,∥x∥pd2}α. We show that semi-linear Cauchy problem is well-posed in M_λ [a Banach space containing functions that do not belong to L1(Qpn)], 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.