Local Limit Theorem for Randomly Deforming Billiards

  • Mark F. DemersEmail author
  • Françoise Pène
  • Hong-Kun Zhang


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 piecewise Hölder 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.



This work was begun at the AIM workshop Stochastic Methods for Non-Equilibrium Dynamical Systems, in June 2015. Part of this work was carried out during visits by the authors to ESI, Vienna in 2016, to CIRM, Luminy in 2017 and 2018, and to BIRS, Canada in 2018, and by a visit of FP to the University of Massachusetts at Amherst in 2018. MD was supported in part by NSF Grant DMS 1800321. FP is grateful to the IUF for its important support.


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Authors and Affiliations

  1. 1.Department of MathematicsFairfield UniversityFairfieldUSA
  2. 2.Laboratoire de Mathématique de Bretagne Atlantique, LMBA, UMR CNRS 6205, Institut Universitaire de France, IUFUniv Brest, Université de BrestBrest CedexFrance
  3. 3.Department of Mathematics and StatisticsUniversity of Massachusetts AmherstAmherstUSA

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