Approximation of Stochastic Hyperbolic Invariant Manifolds

Abstract

In this chapter, we consider situations where the subspace $\fancyscript{H}^{\mathfrak {c}}$ spanned by the low modes contains a combination of critical modes and modes that remain stable as $\lambda$ varies in some interval $\varLambda$. We derive similar results to Theorem 6.1 and Corollary 6.1 concerning the leading-order approximation of the corresponding local stochastic invariant manifolds under suitable conditions. A possible relaxation of these conditions is also discussed. In particular, when the linear operator, $L_\lambda$, is self-adjoint, some non-resonance (NR) conditions arise naturally. As explained in Volume II [37], these NR conditions play an important role in the construction of certain type of manifolds which may not necessarily approximate an invariant manifold, but still convey useful dynamical information; see [37, Chaps. 6–7]. This is formulated in Volume II via the concept of stochastic parameterizing manifolds; see [37, Sect. 4.2].