On Some Aspects of the Response to Stochastic and Deterministic Forcings

TL;DR

This paper develops a mathematical framework using operator semigroup perturbation theory to analyze how stochastic and deterministic forcings influence the response of dynamical systems, including chaotic and stochastic backgrounds, with implications for understanding relaxation timescales and correlation changes.

Contribution

It introduces a unified response formula framework for stochastic and deterministic forcings, clarifies the impact of noise interpretation, and extends response analysis to two-point correlations in complex systems.

Findings

Response formulas decompose using Koopman eigenfunctions

Linear response to deterministic forcings in stochastic/chaotic systems

Second order response clarifies noise impact in chaotic dynamics

Abstract

The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation - Stratonovich vs. Ito - of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.