Martin boundaries of random walks on Fuchsian groups
- 88 Downloads
Let Γ be a finitely generated non-elementary Fuchsian group, and let μ be a probability measure with finite support on Γ such that supp μ generates Γ as a semigroup. If Γ contains no parabolic elements we show that for all but a small number of co-compact Γ, the Martin boundaryM of the random walk on Γ with distribution μ can be identified with the limit set Λ of Γ. If Γ has cusps, we prove that Γ can be deformed into a group Γ', abstractly isomorphic to Γ, such thatM can be identified with Λ', the limit set of Γ'. Our method uses the identification of Λ with a certain set of infinite reduced words in the generators of Γ described in . The harmonic measure ν (ν is the hitting distribution of random paths in Γ on Λ) is a Gibbs measure on this space of infinite words, and the Poisson boundary of Γ, μ can be identified with Λ, ν.
Unable to display preview. Download preview PDF.
- 2.R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes470 (1975).Google Scholar
- 6.E. B. Dynkin and A. A. Yushkevich,Markov Processes: Theorems and Problems, New York, Plenum Press, 1969.Google Scholar
- 10.H. Furstenberg,Random walks and discrete subgroups of Lie groups, Advances in Probability and Related Topics, Vol. 1, Dekker, New York, 1971, pp. 1–63.Google Scholar
- 13.D. Ruelle,Characteristic exponents and invariant manifolds in Hilbert space, I.H.E.S., preprint, 1980.Google Scholar