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.

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