Symbols of non-archimedean elliptic pseudo-differential operators, Feller semigroups, Markov transition function and negative definite functions

Abstract

In this article we prove that the heat kernel attached to the non-archimedean elliptic pseudodifferential operators determine a Feller semigroup and a uniformly stochastically continuous C_0 transition function of some strong Markov processes X with state space (Q_p)^n. We explicitly write the Feller semigroup and the Markov transition function associated with the heat kernel. Also, we show that the symbols of these pseudo-differential operators are a negative definite function and moreover, that this symbols can be represented as a combination of a positive constant, a continuous homomorphism l : (Q_p)^n to R and a non-negative, continuous quadratic form q : (Q_p)^n to R.