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arXiv · 1809.06572

Convergence to a L\'evy process in the Skorohod $M_1$ and $M_2$ topologies for nonuniformly hyperbolic systems, including billiards with cusps

Ian Melbourne·Paulo Varandas
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Abstract

We prove convergence to a Levy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung & Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod J_1 topology. Our main results yield elementary geometric conditions for convergence (i) in M_1, (ii) in M_2 but not M_1. In general, we show for a large class of nonuniformly hyperbolic systems how to deduce functional limit laws once convergence to the corresponding stable law is known.

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Ian Melbourne, Paulo Varandas. 2018-09-18. Convergence to a L\'evy process in the Skorohod $M_1$ and $M_2$ topologies for nonuniformly hyperbolic systems, including billiards with cusps. https://doi.org/10.1007/s00220-019-03501-9

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