Communications of the American Mathematical Society
Abstract
We consider deterministic fast-slow dynamical systems of the form (formula Presented) where (formula Presented). Here, T is a slowly mixing nonuniformly hyperbolic dynamical system and the process (formula Presented) converges weakly to a d-dimensional α-stable Lévy process Lα .(formula Presented) We are interested in convergence of the m-dimensional process (formula Presented) to the solution of a stochastic differential equation (SDE) (formula Presented) In the simplest cases considered in previous work, the limiting SDE has the Marcus interpretation. In particular, the SDE is Marcus if the noise coefficient B is exact or if the excursions for Wn converge to straight lines as n → ∞. Outside these simplest situations, it turns out that typically the Marcus interpretation fails. We develop a general theory that does not rely on exactness or linearity of excursions. To achieve this, it is necessary to consider suitable spaces of “decorated” càdlàg paths and to interpret the limiting decorated SDE. In this way, we are able to cover more complicated examples such as billiards with flat cusps where the limiting SDE is typically non-Marcus for m ≥ 2.
