New classes of p-adic evolution equations and their applications

Abstract

AbstractIn this article, we study new classes of evolution equations in thep-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$${\mathfrak {X}}$$Xwith state space then-dimensionalp-adic unit ball ($${\mathbb {Z}}_{p}^{n}$$Zpn). We introduce a family of operators$$\{T_{t}\}_{t\ge 0}$${Tt}t≥0(obtained explicitly) that determine a Feller semigroup on$$C_{0}({\mathbb {Z}}_{p}^{n})$$C0(Zpn). Also, we study the asymptotic behavior of the survival probability of a strong Markov processes$${\mathfrak {X}}$$Xon a ball$$B_{-m}^{n}\subset {\mathbb {Z}}_{p}^{n}$$B-mn⊂Zpn,$$m\in {\mathbb {N}}$$m∈N. Moreover, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the mentioned above Feller semigroup.