Mean-square random invariant manifolds for stochastic differential equations
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).
scientific article; zbMATH DE number 7314916
scientific article