Regular behavior at infinity of stationary measures of stochastic recursion on NA groups

Abstract

Let N be a simply connected nilpotent Lie group and let S = N o (R+)d be a semidirect product, (R+)d acting on N by diagonal automorphisms. Let (Qn, Mn) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xn = MnXn−1 + Qn. We prove that for an appropriate homogeneous norm on N there is χ0 such that lim t→∞ t χ0ν{x : |x| > t} = C. In particular, this applies to classical Poisson kernels on symmetric spaces, bounded homogeneous domains in Cn or homogeneous manifolds of negative curvature.