Regular behavior at infinity of stationary measures of stochastic recursion on NA groups
Volume 118 / 2010
Colloquium Mathematicum 118 (2010), 499-523
MSC: Primary 60B15; Secondary 60J50, 22E25.
DOI: 10.4064/cm118-2-8
Abstract
Let $N$ be a simply connected nilpotent Lie group and let $S=N\rtimes (\mathbb R ^+)^d$ be a semidirect product, $(\mathbb R ^+)^d$ acting on $N$ by diagonal automorphisms. Let $(Q_n, M_n)$ 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 $\nu $ on $N$ for the Markov process $X_n=M_nX_{n-1}+Q_n$. We prove that for an appropriate homogeneous norm on $N$ there is $\chi _0$ such that $$ \lim _{t\to \infty}t^{\chi _0} \nu \{ x: |x| >t\}=C>0. $$ In particular, this applies to classical Poisson kernels on symmetric spaces, bounded homogeneous domains in $\mathbb C ^n$ or homogeneous manifolds of negative curvature.
