Linear response for random dynamical systems

TL;DR

This paper establishes the differentiability of the absolutely continuous stationary measure for a class of one-dimensional random maps with respect to changes in the distribution, deriving a linear response formula applicable to various systems.

Contribution

It introduces the first linear response results for random compositions of maps and provides explicit formulas for invariant densities in complex systems like continued fractions and Pomeau-Manneville maps.

Findings

Proves differentiability of acsm for a broad class of random maps.

Derives explicit linear response formulas for invariant measures.

Connects random and deterministic linear responses in specific systems.

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-R\'enyi 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.