Local contraction of stochastic affine recursions of $\mathbb{R}^d$

Abstract

We consider affine stochastic recursions $X^x_{n+1}=A_{n+1}X^x_n+B_{n+1}$ in $\mathbb{R}^d$, where $(B_n,A_n)$ are i.i.d. in $\operatorname{Aff}(\mathbb{R}^d)$ with strongly irreducible and proximal linear part. In the critical regime where the top Lyapunov exponent vanishes, we show that the recursion is locally contractive in every direction, yielding uniqueness, up to a positive multiplicative constant, of the $μ$-invariant non-zero Radon measure. This extends to every dimension $d\geq 2$ a result of Babillot-Bougerol-Élie (1997) in dimension one. The proof is based on the study of the asymptotic behaviour of the "joint process" $(X^x_n,X^y_n)$, using geometric and measure-theoretic properties of the diagonal affine action. A key ingredient is an abstract criterion for the total dissipativity of semigroup actions, which we apply to this process.