EN
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 = MₙX_{n-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.