Application of $p$-adic analysis methods in describing Markov processes on ultrametric spaces isometrically embeddable into $\mathbb{Q}_{p}$

TL;DR

This paper introduces a method to analyze stationary Markov processes on ultrametric spaces by embedding them into the field of p-adic numbers, enabling the use of p-adic physics tools for their study.

Contribution

It develops a framework to reduce the study of Markov processes on certain ultrametric spaces to pseudo-differential equations on p-adic fields, with explicit spectral analysis.

Findings

Reduction of the Cauchy problem to pseudo-differential equations on Q_p

Explicit spectrum of the associated pseudo-differential operator

Construction of an orthonormal basis from eigenfunctions

Abstract

We propose a method for describing stationary Markov processes on the class of ultrametric spaces $\mathbb{U}$ isometrically embeddable in the field $\mathbb{Q}_{p}$ of $p$-adic numbers. This method is capable of reducing the study of such processes to the investigation of processes on $\mathbb{Q}_{p}$. Thereby the traditional machinery of $p$-adic mathematical physics can be applied to calculate the characteristics of stationary Markov processes on such spaces. The Cauchy problem for the Kolmogorov-Feller equation of a~stationary Markov process on such spaces is shown as being reducible to the Cauchy problem for a pseudo-differential equation on $\mathbb{Q}_{p}$ with non-translation-invariant measure $m\left(x\right)d_{p}x$. The spectrum of the pseudo-differential operator of the Kolmogorov-Feller equation on $\mathbb{Q}_{p}$ with measure $m\left(x\right)d_{p}x$ is found. Orthonormal basis of real valued functions for $L^{2}\left(\mathbb{Q}_{p},m\left(x\right)d_{p}x\right)$ is constructed from the eigenfunctions of this operator.