Linear Response for dynamical systems with additive noise

TL;DR

This paper establishes a linear response theory for Markov operators associated with noisy dynamical systems, providing explicit formulas and broad applicability including systems with critical points and various real-world models.

Contribution

It introduces a linear response framework for systems with additive noise, accommodating complex dynamics and offering explicit formulas for perturbation effects.

Findings

Proves linear response for systems with additive noise and critical points.

Derives explicit formulas for response to perturbations.

Applies theory to chemical reactions and random rotations.

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.