Skip to main content


Monday 12 Mar 2018Dynamics Seminar: Anomalous diffusion in billiards with cusps

Ian Melbourne - University of Warwick

H103 14:30-15:30

Slowly mixing dynamical systems (with nonsummable decay of correlations) are associated with anomalous diffusion. For mean zero observables v, define the normalized Birkhoff sum n^{-c}(v + v o f + ... + v o f^n). It is often possible to prove convergence to a limit law Y. In the standard diffusive case, c=1/2 and Y is normally distributed. In the anomalous superdiffusive case, c>1/2 and Y belongs to a class of distributions known as stable laws.

It is also often possible to prove stronger results about convergence of processes W_n(t)=n^{-c}(v + v o f + ... + v o f^{[nt]}). In the diffusive case, W_n converges to Brownian motion; in the superdiffusive case, W_n converges to a Levy process.

Recently, Jung & Zhang proved convergence to a stable law for a class of dispersing billiards with cusps. Here we show how to obtain convergence to a Levy process. The method applies to a large class of nonuniformly hyperbolic systems.

This is joint work with Paulo Varandas. No prior knowledge of stable laws or Levy processes will be assumed in this talk.

Visit website

Add to calendar

Add to calendar (.ics)