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

Dariusz Buraczewski, and Ewa Damek. "Regular behavior at infinity of stationary measures of stochastic recursion on NA groups." Colloquium Mathematicae 118.2 (2010): 499-523. <http://eudml.org/doc/286283>.

@article{DariuszBuraczewski2010,
abstract = {Let N be a simply connected nilpotent Lie group and let $S = N ⋊ (ℝ⁺)^\{d\}$ 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 $lim_\{t→∞\} t^\{χ₀\}ν\{x: |x| > t\} = C > 0$. In particular, this applies to classical Poisson kernels on symmetric spaces, bounded homogeneous domains in ℂⁿ or homogeneous manifolds of negative curvature.},
author = {Dariusz Buraczewski, Ewa Damek},
journal = {Colloquium Mathematicae},
keywords = {solvable Lie groups; stationary measure; Poisson kernel},
language = {eng},
number = {2},
pages = {499-523},
title = {Regular behavior at infinity of stationary measures of stochastic recursion on NA groups},
url = {http://eudml.org/doc/286283},
volume = {118},
year = {2010},
}

TY - JOUR
AU - Dariusz Buraczewski
AU - Ewa Damek
TI - Regular behavior at infinity of stationary measures of stochastic recursion on NA groups
JO - Colloquium Mathematicae
PY - 2010
VL - 118
IS - 2
SP - 499
EP - 523
AB - Let N be a simply connected nilpotent Lie group and let $S = N ⋊ (ℝ⁺)^{d}$ 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 $lim_{t→∞} t^{χ₀}ν{x: |x| > t} = C > 0$. In particular, this applies to classical Poisson kernels on symmetric spaces, bounded homogeneous domains in ℂⁿ or homogeneous manifolds of negative curvature.
LA - eng
KW - solvable Lie groups; stationary measure; Poisson kernel
UR - http://eudml.org/doc/286283
ER -