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

Yves Guivarc'H (1) , Emile Le Page (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMAM - Laboratoire de Mathématiques et Applications des Mathématiques, EA 3885

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 Hölder 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 $V$.

Domains: Probability [math.PR]

Yves Guivarc'H, Emile Le Page. Spectral gap properties and asymptotics of stationary measures for affine random walks. 2012. ⟨hal-00691516v2⟩