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Local limit theorem for the Lorentz process and its recurrence in the plane

Published online by Cambridge University Press:  02 February 2004

DOMOKOS SZÁSZ and
TAMÁS VARJÚ
DOMOKOS SZÁSZ
Affiliation:
Budapest University of Technology, Mathematical Institute and Center of Applied Mathematics, Budapest, Egry J. u. 1 Hungary H-1111 (e-mail: szasz@renyi.hu, kanya@math.bme.hu)
TAMÁS VARJÚ
Affiliation:
Budapest University of Technology, Mathematical Institute and Center of Applied Mathematics, Budapest, Egry J. u. 1 Hungary H-1111 (e-mail: szasz@renyi.hu, kanya@math.bme.hu)
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Abstract

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For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Young's axioms (Lai-Sang Young, Ann. Math.147 (1998), 585–650), which imply exponential decay of correlations and the central limit theorem (CLT), a local CLT is proven. In fact, a unified version of the local CLT is found, covering, among others, the absolutely continuous and arithmetic cases. For planar Lorentz process with a finite horizon, this result implies (a) a local CLT and (b) recurrence. For the latter case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt (C. R. Acad. Sci. Paris Ser. 1 Math. 372(9) (1998), 837–842) and J.-P. Conze (Ergod. Th. & Dynam. Sys.19(5) (1999), 1233–1245) could already establish recurrence.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 1 , February 2004, pp. 257 - 278
Copyright
2004 Cambridge University Press