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

TL;DR

This paper demonstrates that heat kernels of non-archimedean elliptic pseudodifferential operators generate Feller semigroups and Markov processes, with explicit formulas and a characterization of their symbols as negative definite functions.

Contribution

It explicitly constructs the Feller semigroup and Markov transition function from the heat kernel and characterizes the symbols as negative definite functions with specific representations.

Findings

Heat kernels determine Feller semigroups and Markov processes.

Symbols are negative definite functions with explicit representations.

Explicit formulas for semigroups and transition functions are provided.

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.