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In this article a class of additive invariant positive selfadjointpseudodifferential unbounded operators on $L^{2}(\mathbb{A}_{f})$, where$\mathbb{A}_{f}$ is the ring of finite ad\'eles of the rational numbers, isconsidered to state a Cauchy problem of parabolic--type equations. Theseoperators come from a set of additive invariant non-Archimedean metrics on$\mathbb{A}_{f}$. The fundamental solutions of these parabolic equationsdetermines normal transition functions of Markov process on $\mathbb{A}_{f}$.Using the fractional Laplacian on the Archimedean place, $\mathbb{R}$, a classof parabolic--type equations on the complete ad\`ele ring, $\mathbb{A}$, isobtained.

Invariant Parabolic equations and Markov process on Ad\'eles