Invariant Parabolic equations and Markov process on Ad\'eles

TL;DR

This paper develops a class of invariant parabolic equations on the ring of finite adèles and the adèle ring, linking pseudodifferential operators to Markov processes and extending classical fractional Laplacian concepts.

Contribution

It introduces a new class of additive invariant pseudodifferential operators on adèles and establishes their connection to Markov processes and parabolic equations on these structures.

Findings

Constructed invariant pseudodifferential operators on $\, ext{A}_f$ and $ ext{A}$.

Derived fundamental solutions that define Markov process transition functions.

Extended fractional Laplacian concepts to the adèle ring.

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.