Necessary and sufficient condition for $\cM_2$-convergence to a L\'evy process for billiards with cusps at flat points

TL;DR

This paper establishes the necessary and sufficient conditions for $\\cM_2$-convergence to a Lévy process in planar dispersing billiards with cusps, extending previous results on convergence in different topologies.

Contribution

It provides a complete characterization of when $\\cM_2$-convergence occurs, filling a gap in the understanding of limit behaviors for billiards with cusps.

Findings

Identifies necessary conditions for $\\cM_2$-convergence.

Shows these conditions are also sufficient.

Extends previous convergence results to a broader setting.

Abstract

We consider a class of planar dispersing billiards with a cusp at a point of vanishing curvature. Convergence to a stable law and to the corresponding L\'evy process in the $\cM_1$ and $\cM_2$ Skorohod topologies has been studied in recent work. Here we show that certain sufficient conditions for $\cM_2$-convergence are also necessary.