Convergence to decorated L\'evy processes in non-Skorohod topologies for dynamical systems

TL;DR

This paper develops a framework for weak convergence to decorated Lévy processes in enriched function spaces, capturing detailed excursion behavior in deterministic dynamical systems where traditional topologies fail.

Contribution

It introduces a novel approach to convergence in non-Skorohod topologies, enabling analysis of complex dynamical systems with unbounded or irregular observables.

Findings

Convergence often fails in classical Skorohod topologies for these systems.

Enriched spaces capture detailed excursion information missed by traditional topologies.

Framework applies to a wide range of systems, including billiards and intermittent maps.

Abstract

We present a general framework for weak convergence to decorated L\'evy processes in enriched spaces of c\`adl\`ag functions for vector-valued processes arising in deterministic systems. Applications include uniformly expanding maps and unbounded observables as well as nonuniformly expanding/hyperbolic maps with bounded observables. The latter includes intermittent maps and dispersing billiards with flat cusps. In many of these examples, convergence fails in all of the Skorohod topologies. Moreover, the enriched space picks up details of excursions that are not recorded by Skorohod or Whitt topologies.