Stochastic \(p\)-adic equations of mathematical physics
type: Item
instance of: scholarly article
title: Stochastic \(p\)-adic equations of mathematical physics
zbMATH Open document ID: 0951.60073
DOI: 10.1007/BF02557352
published in: Theoretical and Mathematical Physics
publication date: 2000-12-13T00:00:00Z
review text: The author introduces several classes of complex-valued stochastic processes with \(p\)-adic time parameters. The basic process is the \(p\)-adic white noise, a Gaussian generalized stochastic process \(\Phi (\varphi)\), \(\varphi \in \mathcal D(Q_p)\) (the Schwartz-Bruhat space of test functions), with zero mean and the covariation \[ \mathbf E\Phi (\varphi)\overline{\Phi (\psi)}=\int\limits_{Q_p}\varphi (t)\overline{\psi (t)} dt. \] An analogue of the Wiener process is defined as solution of the Cauchy problem for the equation \(D_t\Psi =\frac{p}{\sqrt{2(p+1)}}\Phi\), where \(D\) is a pseudo-differential operator with the symbol \(|\xi |_p\). Properties of this process are studied. Similarly, \(p\)-adic counterparts of the Brownian sheet and the Lévy Brownian motion are connected with appropriate two-dimensional pseudo-differential operators.
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 81S99
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 1465871
zbMATH Keywords: \(p\)-adic white noise
zbMATH Keywords: \(p\)-adic Brownian sheet
zbMATH Keywords: \(p\)-adic Wiener process
zbMATH Keywords: \(p\)-adic Lévy Brownian motion
author: Albert Kh. Bikulov
reviewed by: Anatoly N. Kochubei
MaRDI profile type: Publication
cites work: Q5283909
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cites work: Q5607391
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full work available at URL: https://doi.org/10.1007/bf02557352
OpenAlex ID: W2170418089
description: scientific article; zbMATH DE number 1465871