Some aspects of the $m$-adic analysis and its applications to $m$-adic stochastic processes

TL;DR

This paper extends $p$-adic analysis to $m$-adic numbers, exploring their properties, integration, Fourier analysis, and applications to stochastic processes like $m$-adic random walks and Levy processes.

Contribution

It generalizes $p$-adic analysis to $m$-adic numbers and introduces new classes of $m$-adic stochastic processes, including fractional-time random walks.

Findings

Established properties of $m$-adic numbers and analysis.

Introduced classes of $m$-adic infinitely divisible distributions and Levy processes.

Analyzed asymptotic behavior of fractional-time $m$-adic random walk.

Abstract

In this paper we consider a generalization of analysis on $p$-adic numbers field to the $m$ case of $m$-adic numbers ring. The basic statements, theorems and formulas of $p$-adic analysis can be used for the case of $m$-adic analysis without changing. We discuss basic properties of $m$-adic numbers and consider some properties of $m$-adic integration and $m$-adic Fourier analysis. The class of infinitely divisible $m$-adic distributions and the class of $m$-adic stochastic Levi processes were introduced. The special class of $m$-adic CTRW process and fractional-time $m$-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time $m$-adic random walk.