Chapter

The central limit theorem for geodesics

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion

Published in print August 2012 | ISBN: 9780199654109
Published online January 2013 | e-ISBN: 9780191745676 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199654109.003.0008

Series: Oxford Mathematical Monographs

The central limit theorem for geodesics

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This chapter provides a proof of the Sinai central limit theorem, generalized to the case of a cofinite and geometrically finite Kleinian group. This theorem shows that, asymptotically, geodesics behave chaotically, and yields a quantitative expression for this phenomenon. The method used is to establish such a result first for Brownian trajectories, taking advantage of their strong independence properties. Then the chapter compares geodesics with Brownian trajectories, by means of a change of contour. This requires, in particular, considering diffusion paths on the stable foliation and deriving the existence of a key potential kernel, using the commutation relation of Chapter 2 and the spectral gap presented in Chapter 5.

Keywords: foliated diffusions; contour deformation; potential kernel; Sinai central limit theorem

Chapter.  15475 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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