Convergence to a Lévy process in the Skorohod M₁ and M₂ topologies for nonuniformly hyperbolic systems, including billiards with cusps

Commun. Math. Phys. 375 (2020) 653-678.

Ian Melbourne and Paulo Varandas

Abstract

We prove convergence to a Lévy 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 J₁ topology. Our main results yield elementary geometric conditions for convergence (i) in M₁, (ii) in M₂ but not M₁. As far as we know, this is the first instance where the M₂ topology arises naturally.

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.