Application of p-Adic analysis methods in describing Markov processes on ultrametric spaces isometrically embedded into ℚp

• Published in: P-adic numbers, ultrametric analysis, and applications Vol. 7; no. 2; pp. 121 - 132
• Main Authors: Bikulov, A. Kh, Zubarev, A. P.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 2015
• Subjects: Algebra; Mathematics; Mathematics and Statistics; isometric embedding; ultrametric spaces; p-adic analysis; Markov processes
• ISSN: 2070-0466; 2070-0474
• DOI: 10.1134/S2070046615020041

We propose a method for describing stationary Markov processes on the class of ultrametric spaces isometrically embedded in the field ℚ p of p -adic numbers. This method is capable of reducing the study of such processes to the investigation of processes on 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 ℚ p with non-translation-invariant measure m ( x ) d p x . The spectrum of the pseudo-differential operator of the Kolmogorov-Feller equation on ℚ p with measure m ( x ) d p x is found. Orthonormal basis of real valued functions for L 2 (ℚ p , m ( x ) d p x ) is constructed from the eigenfunctions of this operator.