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.

Quadratic response of random and deterministic dynamical systems

“…By Sobolev embedding, F ∈ C 0 (S 1 ) ⊂ L ∞ (S 1 ), so that in fact f ∈ W −1,∞ (S 1 ). Thus it follows that (16) applies, so that L 0 f ∈ C ∞ (S 1 ) and…”

“…In this section, we consider a non-singular map T , defined on the circle S 1 , perturbed by composition with a C 3 diffeomorphism near identity (in a sense explained precisely in (28) and (31)) D δ , and an additive noise with Gaussian kernel (16) ρ…”

“…General Linear response results for random systems were proved in [19] where the technical framework was adapted to stochastic differential equations and in [6], where the authors consider random compositions of expanding or non-uniformly expanding maps. In the paper [16], like in the present paper in Section 5, general discrete time systems with additive noise are considered, i.e. systems where the dynamics map a point deterministically to another point and then some random perturbation is added independently at each iteration according to a certain bounded variation noise distribution kernel.…”

Arnold Maps with Noise: Differentiability and Non-monotonicity of the Rotation Number

“…In recent work [27], it was shown that, in the presence of additive noise, one can expect linear response, even for maps which are not expanding or hyperbolic. The Arnold maps with noise and strong nonlinearity and the kinds of perturbations we consider herein fit into this framework.…”

“…The Arnold maps with noise and strong nonlinearity and the kinds of perturbations we consider herein fit into this framework. In this paper, we will adapt the results of [27] to prove linear response for this class of systems and perturbations, along with smoothness of the rotation number. We remark that the results of [27] are not directly applicable to the kind of perturbations we need to consider here because these perturbations change the critical values of the deterministic part of the dynamics.…”