A Scaling Limit of Random Walks in the Rational Adeles

TL;DR

This paper proves that adele-valued random walks converge to an adelic Lévy process under scaling, using p-adic random walks and survival time analysis to establish almost sure properties and weak convergence.

Contribution

It introduces a novel approach to construct and analyze adelic random walks, demonstrating their convergence to a limiting process in the Skorokhod topology.

Findings

Adele-valued random walks converge to an adelic Lévy process.

Random walks on p-adic numbers can be extended to the adelic setting.

Weak convergence is established in the J1 Skorokhod topology.

Abstract

This paper shows the convergence of adele-valued random walks to an adelic L\'evy process under scaling limits. We use random walks on the $p$-adic numbers to construct random walks initially on the infinite product space, and use survival time analysis to prove that the random walks are almost surely adelic for all time. The adelic random walks are shown to be small perturbations of processes that are supported on a finite product of path spaces. Weak convergence to an adelic L\'evy process is established in the $J_1$ Skorokhod topology.