The stable manifold theorem for non-linear stochastic systems with memory. I: Existence of the semiflow.
type: Item
instance of: scholarly article
title: The stable manifold theorem for non-linear stochastic systems with memory. I: Existence of the semiflow.
zbMATH Open document ID: 1039.60060
author: Salah Mohammed
author: Michael Scheutzow
published in: Journal of Functional Analysis
publication date: 2004-02-03T00:00:00Z
review text: Let \(M_2= \mathbb{R}^d\times L^2([-r, 0],\mathbb{R}^d)\) be the Hilbert space with the norm \[ \|(\nu,\eta)\|_{M_2}= (| \nu|^2+ \|\eta\|^2_{L^2})^{1/2},\quad \nu\in\mathbb{R}^d,\;\eta\in L^2([-r, 0],\mathbb{R}^d). \] The authors consider the stochastic functional equation \[ \begin{gathered} dx(t)= H(t,x(t),x_t)\,dt+ G(t,x(t))\,dW(t),\quad t\geq t_0\geq 0,\\ x(t_0)= \nu\in\mathbb{R}^d,\quad x_{t_0}= \eta\in L^2([-r,0],\mathbb{R}^d),\end{gathered}\tag{1} \] where the drift is a nonlinear functional \(H: \mathbb{R}^+\times M_2\to \mathbb{R}^d\), the noise coefficient is a mapping \(G: \mathbb{R}^+\times \mathbb{R}^d\to \mathbb{R}^{d\times p}\) and \(W\) is \(p\)-dimensional Brownian motion on some probability space \((\Omega,{\mathfrak I},P)\). A soluton of (1) is a process \(x:[t_0- r,\infty)\times \Omega\to \mathbb{R}^d\), where \[ x_t(\cdot,\omega)(s)= x(t+ s,\omega),\quad s\in [-r,0],\;\omega\in\Omega,\;t\geq t_0, \] and (1) holds a.s. It is show that under sufficient regularity conditions on the coefficients (1) admits a stochastic semiflow \(X:\Delta\times M_2\times \Omega\to M_2\) satisfying \(X(t_0, t,(\nu,\eta),\cdot)= (x(t), x_t)\) for all \((\nu,\eta)\in M_2\) and \(t\geq t_0\) a.s., where \(x\) is the solution of (1) and \(\Delta= \{(s,t)\in \mathbb{R}^2: 0\leq s\leq t\}\). It is also established a global locally compact semiflow for (1) and it is show that, if the coefficients in (1) are time-independent, then the global semiflow defines a cocycle on the space \(M_2\). The authors also give some extensions and generalizations of the equation (1).
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H20
zbMATH DE Number: 2037130
zbMATH Keywords: stochastic semiflow
zbMATH Keywords: stochastic functional differential equation
reviewed by: Maria Stolarczyk
MaRDI profile type: Publication
full work available at URL: j.jfa.2002.04.001
OpenAlex ID: W2068495549
cites work: Perfect cocycles through stochastic differential equations
cites work: Large deviations and stochastic flows of diffeomorphisms
cites work: A generalized formula of Ito and some other properties of stochastic flows
cites work: Q3660553
cites work: Q4850166
cites work: Stochastic flows for nonlinear second-order parabolic SPDE
cites work: Stochastic parabolic equations in bounded domains: random evolution operator and lyapunov exponents
cites work: Q3860165
cites work: Q3992729
cites work: On the spatial asymptotic behavior of stochastic flows in Euclidean space
cites work: Q3995203
cites work: Q3707054
cites work: The lyapunov spectrum and stable manifolds for stochastic linear delay equations
cites work: Q4010326
cites work: Q4379368
cites work: Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. I: The multiplicative ergodic theory
cites work: Lyapunov exponents of linear stochastic functional differential equations. II: Examples and case studies
cites work: Spatial estimates for stochastic flows in Euclidean space
cites work: The stable manifold theorem for stochastic differential equations
cites work: Semimartingales and measure preserving flows
cites work: Q3996259
cites work: Characteristic exponents and invariant manifolds in Hilbert space
DOI: 10.1016/J.JFA.2002.04.001
description: scientific article; zbMATH DE number 2037130