Invariant Parabolic equations and Markov process on Adéles

Abstract

In this article a class of additive invariant positive selfadjoint pseudodifferential unbounded operators on $L^2({\mathbb{A}}_f)$, where ${\mathbb{A}}_f$ is the ring of finite adéles of the rational numbers, is considered to state a Cauchy problem of parabolic--type equations. These operators come from a set of additive invariant non-Archimedean metrics on ${\mathbb{A}}_f$. The fundamental solutions of these parabolic equations determines normal transition functions of Markov process on ${\mathbb{A}}_f$. Using the fractional Laplacian on the Archimedean place, $\mathbb{R}$, a class of parabolic--type equations on the complete adèle ring, $\mathbb{A}$, is obtained.

20 pages