New classes of p-adic evolution equations and their applications

Abstract

In this article, we study new classes of evolution equations in the p-adic context. We establish rigorously that the fundamental solutions of the homogeneous Cauchy problem, naturally associated to these equations, are transition density functions of some strong Markov processes {mathfrak {X}} with state space the n-dimensional p-adic unit ball ({mathbb {Z}}_{p}^{n}). We introduce a family of operators {T_{t}}_{tge 0} (obtained explicitly) that determine a Feller semigroup on C_{0}({mathbb {Z}}_{p}^{n}). Also, we study the asymptotic behavior of the survival probability of a strong Markov processes {mathfrak {X}} on a ball B_{-m}^{n}subset {mathbb {Z}}_{p}^{n}, min {mathbb {N}}. Moreover, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the mentioned above Feller semigroup.