Abstract
In this article we define and study various generalizations non-archimedean pseudo-differential operators and its connections with probability theory. We provide explicit and sufficient conditions to ensure that these operators can satisfy the positive maximum principle, the heat Kernel or fundamental solution of the Cauchy problem (or heat equation) naturally associated with these operators determine a Feller semigroup and a transition function of some strong Markov processes X with state space the p-adic numbers. Also, we will study the property of Conservation of mass, the Comparison principle and some aspects of these processes including the first passage time problem and the probability of survival. These equations are proposed as mathematical models (connected with energy landscapes) to study the spread of an infectious or contagious disease that takes into account social clusters in a situation of extreme social isolation.
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