Periodic solutions of Fokker-Planck equations
type: Item
instance of: scholarly article
title: Periodic solutions of Fokker-Planck equations
zbMATH Open document ID: 1372.35014
author: Feng Chen
author: Yuecai Han
author: Xue Yang
author: Yong Li
published in: Journal of Differential Equations
publication date: 2017-05-08T00:00:00Z
review text: This paper deals with problem of existence of periodic solution in distribution to the stochastic differential equation \(dx(t) = f (t, x(t))dt + \sigma(t, x(t))dW(t)\), where \(W(t)\) is an \(l\)-dimensional standard Brownian motion, \(f: \mathbb R^{+}\times \mathbb R^{d}\to \mathbb R^{d}\), \(\sigma: \mathbb R^{+}\times \mathbb R^{d}\to\mathbb R^{d\times l}\) are Borel-measurable, \(\theta\)-periodic on \(t\) functions. The Fokker-Planck equation for the probability density \(p(t, x)\) of Ito diffusion \(x(t)\) satisfying considered stochastic differential equations is \(\partial p(t, x)/\partial t=-\sum_{i}\partial(f_{i} (t, x)p(t, x))/\partial x_{i} + {1\over 2}\sum_{ij}\partial^2 ((\sigma\sigma^{T})_{ij}(t, x)p(t, x))/\partial x_{i}\partial x_{j}\). The authors prove that if \(f(t,x)\) and \(\sigma(t,x)\) are continuous, \(\theta\)-periodic in the \(t\) functions satisfying the Lipschitz condition and Lyapunov conditions hold, then the Fokker-Plank equation admits a \(\theta\)-periodic solution. It is proved existence of \(L^2\)-bounded \(\theta\)-periodic in distribution solution to the considered Ito equation. The theorem on existence of unique uniformly asymptotically stable, \(\theta\)-periodic in distribution solution to the considered Ito equation is proved.
reviewed by: Aleksandr D. Borisenko
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35Q84
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 6712344
zbMATH Keywords: Halanay's criterion
zbMATH Keywords: Lyapunov's functions
MaRDI profile type: Publication
full work available at URL: j.jde.2017.02.032
OpenAlex ID: W2593124194
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DOI: 10.1016/J.JDE.2017.02.032
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description: scientific article; zbMATH DE number 6712344