Poisson boundaries of discrete groups of matrices (Q1063925)

Let G be a discrete subgroup of GL(d,\({\mathbb{R}})\) and let \(\mu\) be a probability measure on G. Let (\(\Omega\),P) be the product space of an infinite number of copies of (G,\(\mu)\). Let S be a compact metric space on which G acts continuously and let \(\rho\) be a probability measure on S, which is \(\mu\)-invariant, i.e. \(\int_{G}(x\cdot \rho)(f)d\mu (x)=\rho (f)\) for \(f\in C(S).\) (S,\(\rho)\) is a boundary (for the random walk defined by \(\mu)\) if for P- a.e. \(\omega =(x_ n)_{n\in {\mathbb{N}}}\in \Omega\) the sequence \(\rho_ n=x_ 1...x_ n\cdot \rho\) converges to a Dirac measure \(\delta_{Z(\omega)}\). (S,\(\rho)\) is a Poisson-boundary if \(L^{\infty}(S,\rho)\) is isometrically isomorphic to the space \({\mathcal H}\) of bounded \(\mu\)-harmonic functions. We suppose now that the entropy-conditions \(\sum_{g\in G}\log \| g\| \mu (g)<\infty\) and \(\sum \log \| g^{-1}\| \mu (g)<\infty\) hold and furthermore that G equals the semigroup generated by supp(\(\mu)\). By the sequence of exponents of the random walk [for the precise definition see e.g. the author: ''Quelques propriétés des exposants caractéristiques'', École d'été de probabilités de Saint-Flour XII-1982, Lect. Notes Math. 1097, 305-396 (1984)] there is defined a subgroup P of G and a \(\mu\)-invariant measure \(\nu\) on the homogeneous space \(B:=G/P\). (B,\(\nu)\) is called the natural boundary. The main results of the paper under review are the following: (B,\(\nu)\) is a Poisson boundary. If in addition all exponents coincide the bounded harmonic functions are constant. The proof is given in a sequence of propositions, too technical to go into details here.