Q‐matrices as pseudo‐differential operators with negative definite symbols

Abstract

Abstract Operators induced by Q‐matrices on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$l_p(\mathbb {Z}^m)$\end{document} are shown to satisfy the positive maximum principle and to have a representation as a pseudo‐differential operator with symbol \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$q:\mathbb {Z}^m\times T^m\rightarrow \mathbb {C}$\end{document} which is with respect to the co‐variable negative definite (in the sense of Schoenberg). This observation leads already towards a more geometric interpretation of the transition matrix of the associated Markov chain.