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

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.

14 pages