Abstract
We propose a method for describing stationary Markov processes on the class of ultrametric spaces \(\mathbb{U}\) 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.
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Bikulov, A.K., Zubarev, A.P. Application of p-Adic analysis methods in describing Markov processes on ultrametric spaces isometrically embedded into ℚ p . P-Adic Num Ultrametr Anal Appl 7, 121–132 (2015). https://doi.org/10.1134/S2070046615020041
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DOI: https://doi.org/10.1134/S2070046615020041