Parabolic Type Equations and Markov Stochastic Processes on Adeles

TL;DR

This paper develops a framework for parabolic pseudodifferential equations over adeles, providing explicit fundamental solutions and connecting them to Markov processes analogous to Brownian motion.

Contribution

It introduces new classes of adelic parabolic equations, constructs explicit heat kernels, and establishes well-posedness and probabilistic interpretations.

Findings

Explicit fundamental solutions (heat kernels) for adelic parabolic equations.

Markov processes on adeles as analogues of Brownian motion.

Well-posedness of the Cauchy problem for these equations.

Abstract

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations.