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

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 \({\mathfrak {X}}\) 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 \(q: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}.\)

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Correspondence to Anselmo Torresblanca-Badillo.

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Gutiérrez García, I., Torresblanca-Badillo, A. Strong Markov processes and negative definite functions associated with non-Archimedean elliptic pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 11, 345–362 (2020). https://doi.org/10.1007/s11868-019-00293-3

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Keywords

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