Spectral gap properties and asymptotics of stationary measures for affine random walks

Details Spectral gap properties and asymptotics of stationary measures for affine random walks Spectral gap properties and asymptotics of stationary measures for affine random walks

2012-04-26

arxiv.org/abs/1204.6004v1

Mathematics

Probability

Scientific paper

Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $\mu$ (resp $\lambda$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp $H= Aff (V))$ and assume that $\mu$ is the projection of $\lambda$ on $G$. We study asymptotic properties of the convolutions $\mu^n *\delta_{v}$ (resp $\lambda^n*\delta_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $\mu$ (resp $\lambda$), if the subsemigroup $T\subset G$ (resp $\Sigma \subset H$) generated by the support of $\mu$ (resp $\lambda$) is "large". We show spectral gap properties for the convolution operator defined by $\mu$ on spaces of homogeneous functions of degree $s\geq 0$ on $V$, which satisfy H\"older type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $\displaystyle\mathop{\Sigma}_{0}^{\infty} \mu^k * \delta_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $\lambda$-boundary ; then we use the above results to show that the unique $\lambda$-stationary measure on $V$ is "homogeneous at infinity" with respect to dilations $v\rightarrow t v (t>0)$. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependant random matrices, on the use of a general renewal theorem, and on the dynamical properties of a conditional $\lambda$-boundary dual to $V$.

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