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We give a description of stationary probability measures on projective spaces for
an iid random walk on GLd(R) without any algebraic assumptions. This is done
in two parts. In a first part, we study the case (non-critical or block-dominated
case) where the random walk has distinct deterministic exponents in the sense of
Furstenberg-Kifer-Hennion. In a second part (critical case), we show that if the
random walk has only one deterministic exponent, then any stationary probability
measure on the projective space lives on a subspace on which the ambient group of
the random walk acts completely reducibly. This connects the critical setting with
the work of Guivarc’h-Raugi and Benoist-Quint. Combination of all these works
allow to get a description of stationary probability measures. Joint works with Richard Aoun.