Optimal transportation and stationary measures for Randomly Iterated Function Systems

Benoît Kloeckner (1, 2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 IF - Institut Fourier

In this article, we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measures of Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove moment bounds and generalized moment bounds, consider the convergence of the empirical measure of an associated Markov chain, prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a "linear response formula" at almost every parameter of the perturbation, and prove singularity of the stationary measure in some cases where the classical dimension bound coincides with the dimension of the ambient space.

Domaines: Analyse classique [math.CA]; Géométrie métrique [math.MG]

Benoît Kloeckner. Optimal transportation and stationary measures for Randomly Iterated Function Systems. 2019. ⟨hal-02276750v1⟩