Abstract
Let \(\Gamma\) be a geometrically finite Kleinian group, relative to the hyperbolic space \(\mathbb{H} = \mathbb{H}^{d + 1}\), and let \(\delta\) denote the Hausdorff dimension of its limit set. Denote by \(\Phi\) the eigenfunction of the hyperbolic Laplacian \(\Delta\), associated with its first eigenvalue \(2{\lambda}_0 = \delta(\delta-d)\), and by \(Z^{\Phi}_t\) the associated diffussion on \(\mathbb{H}\), whose generator is \(\frac{1}{2}{\Delta}^{\Phi}: = \frac{1}{2}{\Phi}^{-1}{\Delta}\circ{\Phi}-{\lambda}_0\). We give a simple construction of \(Z^{\Phi}_{t}\) through its canonical lift to the frame bundle \({\mathcal{O}}{\mathbb{H}}\), that allows to determine directly its asymptotic behaviour.
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© 2001 Springer-Verlag Berlin/Heidelberg
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Enriquez, N., Franchi, J., Jan, Y.L. (2001). Canonical Lift and Exit Law of the Fundamental Diffusion Associated with a Kleinian Group. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXV. Lecture Notes in Mathematics, vol 1755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44671-2_16
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DOI: https://doi.org/10.1007/978-3-540-44671-2_16
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