Stationary probability measures on projective spaces

We give a description of stationary probability measures
on projective spaces for an iid random walk on $\GL_d(\R)$
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 semisimply.
This connects the critical setting with the work of Guivarc’h–Raugi
and Benoist–Quint. Combination of all these works allow to get
a complete description. Joint works with Richard Aoun.