Strong Markov processes and negative definite functions associated with non-Archimedean elliptic pseudo-differential operators

  • Ismael Gutiérrez García
  • Anselmo Torresblanca-BadilloEmail author


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 \({\mathfrak {X}}\) with state space \({\mathbb {Q}}_{p}^{n}.\) 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 \(c\ge 0,\) a continuous homomorphism \(l: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}\) and a non-negative, continuous quadratic form \(q: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}.\)


Pseudo-differential operators Feller semigroups Markov transition function Convolution semigroup Negative definite function Non-Archimedean analysis 



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Authors and Affiliations

  • Ismael Gutiérrez García
    • 1
  • Anselmo Torresblanca-Badillo
    • 1
    Email author
  1. 1.Department of Mathematic and StatisticUniversidad del NorteBarranquillaColombia

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