The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations

Abstract

This article is a sequel to [M.Z.Z.1] aimed at completing the characterization of the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Stationary solution are viewed as random points in the infinite-dimensional state space, and the characterization is expressed in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see's and spde's (Theorems 4.1-4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments (Theorem 2.1).

Anticipating random periodic solutions—I. SDEs with multiplicative linear noise

“…In the case of SPDEs, this property does not hold. In [16], applying the unstable manifold theorem ( [23]), we can still deduce the IHSIEs in the infinite dimensional case.…”

Pathwise Properties of Random Quadratic Mapping

“…In 1945, Ulam and Neumann [40] pointed out the importance of random dynamical systems. In the last thirty years, the research of random dynamical systems is further expanded especially in the field of stochastic differential equations and stochastic partial differential equations in a series of work such as [4,23,24,28,31]. Pathwise stationary solution and random periodic solution are two central concepts in the study of random dynamical systems [4,13,14,21,20,29,31,36,37,42,43,44].…”

Invariant Manifolds for Stochastic Models in Fluid Dynamics

“…We will only sketch the proof. Details may be found in [19]. Using the Oseledec integrability condition (2.2) and the Ruelle-Oseledec theorem ([19, Theorem 2.1.1]), we obtain a random family of compact self-adjoint positive operators Λ(ω) ∈ L(H), defined perfectly in ω, and satisfies…”

“…These manifolds are random objects and are perfectly defined for ω ∈ Ω. Using interpolation between discrete times and the (continuous-time) integrability condition (2.10), it can be shown that the above manifolds for the discrete-time cocycle (U (n, ·, ω), θ(n, ω)), n ≥ 1, also serve as perfectly defined local stable/unstable manifolds for the continuous-time cocycle (U (t, ·, ω), θ(t, ω)), t ≥ 0, near Y (ω) (see [16,19,22]). …”