On the affine recursion on ${\mathbb{R}}_{+}^d$

We fix $d\ge 2$ and denote $\mathcal{S}$ the semi-group of $d\times d$ matrices with non negative entries. We consider a sequence $(A_n, B_n){n \geq 1} $ofi\mathrm{.}i\mathrm{.}d\mathrm{.}randomvariableswithvaluesin$\mathcal S\times \mathbb R+^d$andstudytheasymptoticbehavioroftheMarkovchain$(X_n){n \geq 0}$on$ \mathbb R+^d$ defined by: $$\mathrm{\forall }n\ge 0,X_{n+1}=A_{n+1}{X}_n+B_{n+1},$$ where $X_0$ is a fixed random variable. We assume that the Lyapunov exponent of the matrices $A_n$ equals $0$ and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure $\lambda$ on $({\mathbb{R}}^{+}{)}^d$ which is invariant for the chain $(X_n{)}_{n\ge 0}$. The existence of $\lambda$ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices . Its unicity is a consequence of a general property, called “local contractivity”, highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion .