Pseudodifferential Operators and Markov Processes on Adèles

Abstract

In this article a class of Markov processes on the ring of finite adèles of the rational numbers are introduced. A class of non-Archimedean metrics on \(\mathbb{A}_{f}\) are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on \({L^2}(\mathbb{A}_{f})\). The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on \(\mathbb{A}_{f}\) which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adèles.