Stochastic Lyapunov method
type: Item
instance of: scholarly article
title: Stochastic Lyapunov method
zbMATH Open document ID: 0839.34067
DOI: 10.1007/BF01210621
author: Jean-Pierre Aubin
author: Guiseppe Da Prato
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1996-06-16T00:00:00Z
review text: Let \((\Omega, {\mathcal F}, P)\) be a complete probability space, and let \(\{{\mathcal F}_t \subset {\mathcal F}\}\) be an increasing family of \(\sigma\)-sub-algebras adopted to a standard \(m\)-dimensional Wiener process \(W\). The authors consider solutions to the stochastic differential equation (*) \(dx = f(x(t)) dt + \sigma (x(t)) dW(t)\), \(x \in \mathbb{R}^n\), starting from \(x(0)\). They study conditions under which the inequality \(\forall t \geq 0\) \(V(x(t)) \leq \omega (t)\) holds almost everywhere. Here \(\omega (\cdot)\) is a solution to the differential equation \(\omega' = - \varphi (\omega)\) and \(V : \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}\). On the basis of such inequalities a lot of asymptotic behaviors of \(V\) along the solutions to (*) is obtained.
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 822781
zbMATH Keywords: stochastic viability
zbMATH Keywords: stochastic Lyapunov functions
zbMATH Keywords: absorbing set
zbMATH Keywords: stochastic differential equation
reviewed by: Dobiesław Bobrowski
MaRDI profile type: Publication
cites work: Stochastic nagumo's viability theorem
cites work: Set-valued analysis
cites work: Q3929443
cites work: Q4199298
cites work: Pathwise global attractors for stationary random dynamical systems
cites work: Q3727031
cites work: Attractors for random dynamical systems
cites work: Stochastic Equations in Infinite Dimensions
cites work: Q4730506
cites work: Q4284180
cites work: On multi–dimensional diffusion process living in a bounded region
cites work: Invariance for stochastic equations with regular coefficients
cites work: Attractors of probability measures for semilinear stochastic evolution equations
description: scientific article; zbMATH DE number 822781