By Mickaël D. Chekroun, Honghu Liu, Shouhong Wang

This first quantity is anxious with the analytic derivation of specific formulation for the leading-order Taylor approximations of (local) stochastic invariant manifolds linked to a huge classification of nonlinear stochastic partial differential equations. those approximations take the shape of Lyapunov-Perron integrals, that are additional characterised in quantity II as pullback limits linked to a few partly coupled backward-forward structures. This pullback characterization presents an invaluable interpretation of the corresponding approximating manifolds and ends up in an easy framework that unifies another approximation techniques within the literature. A self-contained survey is additionally integrated at the life and allure of one-parameter households of stochastic invariant manifolds, from the viewpoint of the idea of random dynamical systems.

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1 Stochastic Evolution Equations 19 where L(X, Y ) denotes the space of bounded linear operators from the Banach space X to the Banach space Y . 24b) accounts for the instantaneous smoothing effects of the semigroup et L λ for t > 0 from H to Hα where we recall that Hα has been imposed by the choice of the nonlinearity. 24c) that—because of our assumptions (L λ being not necessarily self-adjoint)—could be present in front of the exponential terms with ηc (resp. ηs) in place of η1 (resp. η2 ). 24c) depends on η∗ := min{ηc − η1 , η2 − ηs}, and may get larger as η∗ gets closer to zero in the non-self-adjoint case.

In particular, given γ > 0, if we denote by tσ∗ (ω) the minimal time after which u λ (t, ω) − u λ (t, ω) α gets smaller than γ , it can be shown that its expected value m σ increases with σ , while as σ tends to zero, m σ converges to the corresponding attraction time associated with σ = 0. Hence, although the critical rate of attraction is independent of σ , the expected attraction time to the inertial manifold (for a given precision γ ) is not. 3, we see that in the case where η > 0 the difference between the given solution u λ of Eq.

2 Random Dynamical Systems In this section, we recall the definitions of metric dynamical systems (MDSs) and random dynamical systems (RDSs), and specify—in a measure-theoretic sense—the canonical MDS associated with the Wiener process in Eq. 1) which will be used throughout this monograph. The interested readers are referred to [1, 47, 54] for more details, and to [39] for an intuitive and “physically-oriented” presentation of these concepts. Metric dynamical system. , (θt )∗ P = P for all t ∈ R, where (θt )∗ P is the push-forward measure of P by θt , defined by (θt )∗ (F) := P(θ−t (F)), for all F ∈ F .