LAUSR

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 $$C_{0}$$ C 0 -transition function of some strong Markov processes $${\mathfrak {X}}$$ X with state space $${\mathbb {Q}}_{p}^{n}.$$ 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 constant $$c\ge 0,$$ c ≥ 0 , a continuous homomorphism $$l: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}$$ l : Q p n → R and a non-negative, continuous quadratic form $$q: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}.$$ q : Q p n → R .