Видеотека: A. Kh. Bikulov, Application of p-adic analysis methods in describing Markov processes on ultrametric spaces isometrically embeddable into $\mathbb{Q}

ВИДЕОТЕКА


Международная конференция «Современная математическая физика. Владимиров-95» 16 ноября 2018 г. 12:35, г. Москва, г. Москва, ул. Губкина, д. 8, конференц-зал МИАН (9 этаж)

Application of p-adic analysis methods in describing Markov processes on ultrametric spaces isometrically embeddable into $\mathbb{Q}_{p}$ A. Kh. Bikulov
Аннотация: 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. An orthonormal basis for $L^{2}\left(\mathbb{Q}_{p}, m\left(x\right)d_{p}x\right)$ is constructed from the eigenfunctions of this operator. Язык доклада: английский