Invariant Parabolic equations and Markov process on Adéles

2018/05/29 by Victor A. Aguilar-Arteaga, Aguilar-Arteaga, V. A., Samuel Estala-Arias +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1805.11726

openalex publication_date 2018/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article a class of additive invariant positive selfadjoint pseudodifferential unbounded operators on L²(\mathbbA_f), where \mathbbA_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 \mathbbA_f. The fundamental solutions of these parabolic equations determines normal transition functions of Markov process on \mathbbA_f. Using the fractional Laplacian on the Archimedean place, ℝ, a class of parabolic--type equations on the complete adèle ring, \mathbbA, is obtained.

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