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 ad\'eles 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 ad`ele ring, $\mathbb{A}$, is obtained.