  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 .
