Poisson Boundaries of Random Walks on Discrete Solvable Groups

Abstract

Let G be a topological group, and μ — a probability measure on G. A function f on G is called harmonic if it satisfies the mean value property f(g)=∫f(gx)dμ(x) for all g ∈ G. It is well known that under natural assumptions on the measure μ there exists a measure G-space Γ with a quasi-invariant measure v such that the Poisson formula f(g)= states an isometric isomorphism between the Banach space H ∞(G, μ) of bounded harmonic functions with sup-norm and the space X∞(Γ, μ). The space (Γ, v) is called the Poisson boundary of the pair (G, μ). Thus triviality of the Poisson boundary is equivalent to absence of non-constant bounded harmonic functions for the pair (G, μ) (the Liouville property).