Palestrante: Ian Melbourne
Insituição: University of Warwick, Reino Unido
I will survey results over the last 10 years on convergence to Levy processes for nonuniformly hyperbolic dynamical systems. Convergence to a Levy process often holds when the central limit theorem fails. The limiting process is superdiffusive (growing like (time)^a with a>1/2) and sample paths have dense sets of discontinuities. Classical treatments of convergence to Levy processes use Skorokhod topologies from 1956. In the 1990s, Whitt recognised that convergence may fail in such topologies, and that important information may be lost even when convergence holds. Accordingly, Whitt introduced ``decorated'' Skorokhod-type topologies. However, there was a lack of examples to illustrate how best to proceed. It turns out that nonuniformly hyperbolic systems provide a wealth of examples where decorated topologies are needed. Moreover, their analysis leads to the correct (we claim!) definition of decorated Skorokhod topology. The precise definitions are technical. Instead I'll provide examples and pictures to illustrate the theory. This is joint work with Chevyrev & Korepanov and with Freitas, Freitas & Todd.