Normally hyperbolic invariant manifolds for random dynamical systems. I: Persistence (Q2849036)

The authors consider a \(C^r\) flow \(\psi(t)\) in \(\mathbb R^n\) with a compact and connected \(C^r\) normally hyperbolic invariant manifold \(\mathcal M\), and a random dynamical system \(\phi(t,\omega)\) over a probabilistic space \((\Omega,\mathcal F, P)\) to be its randomly perturbed counterpart. The main result states that if the random flow is close to the deterministic one, i.e., there exists \(\rho>0\) such that if \(\|\phi(t,\omega)-\psi(t)\|_{C^1}< \rho\) for all \(\omega\in\Omega\) and \(t\in[0,1]\), then it also has a \(C^1\) normally hyperbolic ``random invariant manifold'' \(\mathcal{\tilde M} (\omega)\), which, if the hyperbolicity of the deterministic flow is strong enough, is a \(C^r\) manifold diffeomorphic to \(\mathcal M\). Moreover, the existence of stable and unstable manifolds for \(\phi(\omega,t)\) at \(\mathcal{\tilde M} (\omega)\) is proved.NEWLINENEWLINEIn addition, the authors formulate a similar result on the persistence for flows with normally hyperbolic overflowing or inflowing invariant manifolds and its random counterpart.NEWLINENEWLINEThe authors announce the paper as the first step in a program to build geometric singular perturbation theory under stochastic perturbations, by providing the persistence theory for random normally hyperbolic invariant manifolds. This work was already followed by [``Invariant foliations for random dynamical systems'', Discrete Contin. Dyn. Syst., Ser. A, 34 (9), 3639--3666 (2014)], where the same authors prove the existence of invariant foliations of stable and unstable manifolds of a normally hyperbolic random invariant manifold.