Near Invariance and Local Transience for Random Diffeomorphisms

Nearly (or almost) invariant sets for random dynamical systems are subsets of the state space that are left only after long time and, maybe, are visited again after even longer times. The present paper takes up the approach developed in Colonius, Gayer, and Kliemann [8] for Markov diffusions. We develop an analogous theory for random diffeomorphisms and also use results from Zmarrou and Homburg [21] who studied bifurcation problems based on an eigenvalue analysis of the associated Perron-Frobenius operator. Related work includes approaches based on transfer operator theory combined with set oriented numerics in Dellnitz and Junge [11], [10], Froyland [15], Froyland and Dellnitz [16], and graph theoretic methods, Dellnitz et al. [12], as well as extensions of metastability in the classical Freidlin/Wentzell theory [14] in Schütte, Huisinga, and Meyn [20] and Bovier et al. [5], [6]; and the analysis of dominant eigenvalues of transfer operators (Schütte, Huisinga, and Deuflhard [19], Deuflhard et al. [13]). The paper [8] introduces the notion of nearly invariant sets as a way to formalize the idea of almost invariance. Here the maximal amplitudes of the noise process are varied and nearly invariant sets can be described by properties of an associated deterministic control system. The escape times from a nearly invariant set become