Linear response for random dynamical systems

2017/10/10 by Bahsoun, Wael, Ruziboev, Marks, Saussol, Benoît · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1710.03706

Abstract

We study for the first time linear response for random compositions of maps, chosen independently according to a distribution \PP. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when \PP changes smoothly to \PP_\eps? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to \eps; moreover, we obtain a linear response formula. We apply our results to iid compositions, with respect to various distributions \PP_\eps, of uniformly expanding circle maps, Gauss-Rényi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

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• Annealed and quenched representations of the Gauss-Rényi measure by "periodic points"
• Divergence-Kernel method for scores of random systems

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