In this article we study certain ultradiffusion equations connected with energy landscapes of exponential type. These equations are connected with the p-adic models of complex systems introduced by Avetisov et al. We show that the fundamental solutions of these equations are transition density functions of Lévy processes with state space Q n p , we also study some aspects of these processes including the first passage time problem.2000 Mathematics Subject Classification. Primary 60J25, 82C41; Secondary 46S10. Key words and phrases. Markov processes, ultradiffusion, relaxation of complex systems, the first passage time problem, p-adic analysis.

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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 $${\mathfrak {X}}$$ X with state space the n-dimensional p-adic unit ball ($${\mathbb {Z}}_{p}^{n}$$ Z p n ). We introduce a family of operators $$\{T_{t}\}_{t\ge 0}$$ { T t } t ≥ 0 (obtained explicitly) that determine a Feller semigroup on $$C_{0}({\mathbb {Z}}_{p}^{n})$$ C 0 ( Z p n ) . Also, we study the asymptotic behavior of the survival probability of a strong Markov processes $${\mathfrak {X}}$$ X on a ball $$B_{-m}^{n}\subset {\mathbb {Z}}_{p}^{n}$$ B - m n ⊂ Z p n , $$m\in {\mathbb {N}}$$ 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.

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