A linear response for dynamical systems with additive noise

Abstract

We show a linear response statement for fixed points of a family of Markov operators, which are perturbations of mixing and regularizing operators. We apply the statement to random dynamical systems on the interval given by a deterministic map T with additive noise (distributed according to a bounded variation kernel). We prove linear response for these systems, also providing explicit formulas both for deterministic perturbations of the map T and for changes in the noise kernel. The response holds with mild assumptions on the system, allowing the map T to have critical points, contracting and expanding regions. We apply our theory to topological mixing maps with additive noise, to a model of the Belozuv–Zhabotinsky chemical reaction and to random rotations. In the final part of the paper we discuss the linear request problem for these kind of systems, determining which perturbations of T produce a prescribed response.