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

Abstract

We propose a method for describing stationary Markov processes on the class of ultrametric spaces 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.

On one real basis for $L^2(Q_p)$

“…Proof. Since Q p may be looked upon as the limit case of B r as r → ∞, repeating the argument of the previous section it is easily seen that functions (25) are eigenfunctions of operator (3) with eigenvalues (26). Next, since the [[[set of characteristic functions]]] of all balls forms a basis for L 2 (Q p ), it suffices to show that the characteristic function…”

“…From formulas (32) and (33), which express the relation of the overcomplete system of functions f γ,n,a (x) and the orthonormal basis ϕ γ,n,b (x) in terms of the functions of the wavelet basis ψ γ,n,j (x), it follows that functions (25) and (30) are also eigenfunctions of the pseudo-differential operator (36) with kernel (37) and eigenvalues (38).…”

“…In this connection, such types of bases can be the starting point in solving the spectral problem for the analogues of the Vladimirov operator in an ultrametric space not featuring a group structure by reducing this problem to a spectral problem of some modification of the Vladimirov operator in Q p . The last problem will be addressed in a forthcoming paper [25] by the authors.…”