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hal-00691516, version 1

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

Yves Guivarc'H () 1, Emile Le Page 2

(2012)

Résumé : 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 $V$.

• 1 :  Institut de Recherche Mathématique de Rennes (IRMAR)
• CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
• 2 :  Laboratoire de Mathématiques et Applications des Mathématiques, EA 3885 (LMAM)
• Université de Bretagne Sud

• hal-00691516, version 1
• oai:hal.archives-ouvertes.fr:hal-00691516
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• Soumis le : Jeudi 26 Avril 2012, 14:20:44
• Dernière modification le : Jeudi 26 Avril 2012, 21:00:50