On the uniqueness of the solution to the Wiener-Hopf equation with probability kernel
type: Item
instance of: scholarly article
title: On the uniqueness of the solution to the Wiener-Hopf equation with probability kernel
zbMATH Open document ID: 1485.45004
publication date: 2021-12-08T00:00:00Z
review text: The inhomogeneous generalised Wiener-Hopf equation is \[ z(x) = \int_{-\infty}^x z(x-y)\, F(dy) + f(x), \qquad x\geq 0, \] where \(z\) is the unknown function, \(F\) is a given probability distribution and \(f\) is a known function. A distribution \(F\) is called arithmetic if it is concentrated at a set \(\lambda \mathbb{Z}\) (for some non-zero \(\lambda\in\mathbb{R}\)), otherwise it is called nonarithmetic. For a nonarithmetic \(F\) with a finite positive mean \(\mu=\int_{\mathbb{R}} x \,F(dx)\) the uniqueness of the solution is proven. The method is based on an elaborated von Neumann series expansion \((I-\mathcal{F})^{-1}=\sum_0^{\infty} \mathcal{F}^n\) for convolutions. An analogous result for a discrete case with arithmetic \(F\) is discussed as well.
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45R05
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 41A58
zbMATH DE Number: 7440268
zbMATH Keywords: integral equation
zbMATH Keywords: inhomogeneous equation
zbMATH Keywords: Wiener-Hopf equation
zbMATH Keywords: probability distribution
zbMATH Keywords: positive mean
author: M. S. Sgibnev
reviewed by: Vladimir V. Kisil
MaRDI profile type: Publication
full work available at URL: semi.2021.18.086
OpenAlex ID: W3212671875
cites work: Homogeneous conservative Wiener-Hopf equation
cites work: Q3241504
cites work: The discrete Wiener-Hopf equation with probability kernel of oscillating type
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cites work: The uniqueness of a solution to the renewal type system of integral equations on the line
cites work: Q3261612
DOI: 10.33048/SEMI.2021.18.086
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published in: Sibirskie Elektronnye Matematicheskie Izvestiya
description: scientific article; zbMATH DE number 7440268