Local limit theorem for randomly deforming billiards. (English) Zbl 1473.37033
Summary: 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.
MSC:
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 37A40 | Nonsingular (and infinite-measure preserving) transformations |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
Keywords:
random perturbations; dispersing billiards; Hölder continuous observables; mixing local limit theorem; decorrelation estimatesReferences:
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