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

Dariusz Buraczewski¹ ; Ewa Damek²

¹ Institute of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland
² Institute of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wroclaw, Poland

Colloquium Mathematicum, Tome 118 (2010) no. 2, pp. 499-523.

Résumé

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.

Détail

DOI : 10.4064/cm118-2-8

Keywords: simply connected nilpotent lie group rtimes mathbb semidirect product mathbb acting diagonal automorphisms sequence random variables values nbsp under natural conditions including contractivity mean there unique stationary measure markov process n prove appropriate homogeneous norm there chi lim infty chi particular applies classical poisson kernels symmetric spaces bounded homogeneous domains mathbb homogeneous manifolds negative curvature