Stochastic oscillators
type: Item
instance of: scholarly article
title: Stochastic oscillators
zbMATH Open document ID: 0651.60067
DOI: 10.1016/0022-0396(88)90029-0
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: The authors consider the nonlinear stochastic differential equation \[ x''+k(x,x',t)=h w'(t),\quad t\geq 0, \] where w(t) is a Wiener process, \(h>0\), and k is a smooth, bounded real function. It is shown that this equation is equivalent to the linear stochastic differential equation \[ (1)\quad x''+x=h \hat w'(t),\quad t\geq 0, \] where \(\hat w(t)\) is a Wiener process. Then it is proved that the solution x(t) almost surely has simple zeros. For equation (1), theorems are proved which establish upper and lower bounds for the expected value and the cumulative distribution function of the first positive zero of x(t). The paper concludes with a theorem giving an asymptotic lower bound for the stochastic winding angle of the solution of the system of first order stochastic differential equations corresponding to equation (1) and with some conjectures concerning this stochastic winding angle.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 4062290
zbMATH Keywords: nonlinear stochastic differential equation
zbMATH Keywords: Wiener process
zbMATH Keywords: first order stochastic differential equations
zbMATH Keywords: stochastic winding angle
author: L. Markus
author: Ananda Weerasinghe
reviewed by: Melvin D. Lax
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90029-0
OpenAlex ID: W4210695383
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description: scientific article; zbMATH DE number 4062290