Mean-square random invariant manifolds for stochastic differential equations
type: Item
instance of: scholarly article
title: Mean-square random invariant manifolds for stochastic differential equations
zbMATH Open document ID: 1467.37053
author: Bixiang Wang
published in: Discrete and Continuous Dynamical Systems
publication date: 2021-02-24T00:00:00Z
review text: This paper is concerned with the existence of mean-square random stable and unstable invariant manifolds of the following stochastic differential equations defined in a separable Hilbert space \(H\) for \(t > 0\): \[du(t) = Au(t)dt + F(u(t))dt + \sigma (u(t))dW(t), \quad u(0) = u_0; \tag{1}\] where \(A\) is a densely defined closed linear operator in \(H\) that generates a strongly continuous semigroup \(e^{At}, F\) and \(\sigma\) are nonlinear functions, and \(W\) is a two-sided cylindrical Wiener process in a Hilbert space defined on a complete filtered probability space \((\Omega; \mathcal{F}; \{\mathcal{F}_t\} _{t\in \mathbb{R}}; P)\). In order to overcome the difficulty caused by the nonlinearity of white noise, the authors introduce the concept of mean-square random invariant manifold in \(L_2(\Omega;H)\) and prove its existence for the stochastic equation (1) driven by nonlinear white noise. The mean-square random unstable invariant manifolds and then the stable invariant sets for (1) in \(L_2(\Omega;H)\) are studied and then the Lyapunov-Perron method and the Banach fixed point theorem in the mean-square space are used to construct an random invariant manifold for (1).
reviewed by: Irina V. Konopleva
Mathematics Subject Classification ID: 37H30
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 37L55
zbMATH DE Number: 7314916
zbMATH Keywords: mean-square random invariant manifold
zbMATH Keywords: random stable manifold
zbMATH Keywords: random unstable manifold
zbMATH Keywords: stochastic equation
zbMATH Keywords: backward equation
MaRDI profile type: Publication
full work available at URL: dcds.2020324
OpenAlex ID: W3083600221
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DOI: 10.3934/DCDS.2020324
description: scientific article; zbMATH DE number 7314916