# Local limit theorem for randomly deforming billiards

## TL;DR

This paper establishes limit theorems for randomly deformed dispersing billiards, including CLT and local limit theorems, using anisotropic Banach spaces, in both finite and infinite measure settings.

## Contribution

It extends limit theorems to randomly perturbed billiard systems, introducing new techniques for the random setting with anisotropic Banach spaces.

## Key findings

- Proved a Central Limit Theorem for cell index in finite horizon billiards.
- Established a mixing local limit theorem for observables.
- Derived limit theorems for visits to obstacles and self-intersections in infinite measure systems.

## Abstract

We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of movements and deformations of scatterers. We prove a Central Limit Theorem for the cell index of planar motion, as well as a mixing Local Limit Theorem for the cell index with piecewise H\"older continuous observables. In the context of the infinite measure random system, we prove limit theorems regarding visits to new obstacles and self-intersections, as well as decorrelation estimates. The main tool we use is the adaptation of anisotropic Banach spaces to the random setting.