On the affine recursion on $\mathbb R_+^d$

We fix $d \geq 2$ and denote $\mathcal S$ the semi-group of $d \times d$ matrices with non negative entries. We consider a sequence $(A_n, B_n)_{n \geq 1}$ of i. i. d. random variables with values in $\mathcal S\times \mathbb R_+^d$ and study the asymptotic behavior of the Markov chain $(X_n)_{n \geq 0}$ on $\mathbb R_+^d$ defined by: $$\forall n \geq 0, \qquad X_{n+1}=A_{n+1}X_n+B_{n+1},$$ 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 $(X_n)_{n \geq 0}$. 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 .