Abstract
We consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.
Highlights
1 Introduction In many scientific applications, a systematic determination of the response of a complex nonlinear dynamical system to time-dependent perturbations is of key importance; topical examples in high-dimensional, non-autonomous and/or stochastic settings include climate models (e.g. [1,16,34,36,59,62]), statistical physics and non-equilibrium thermodynamics (e.g. [47,50,75,82,83]), and even neural networks (e.g. [18,22,76])
The classical fluctuation–dissipation theorem (FDT) is of fundamental importance in statistical physics (e.g. [5,28,51]), and it roughly states that for systems of identical particles in statistical equilibrium, the average response to small external perturbations can be calculated through the knowledge of suitable correlation functions of the unperturbed time-asymptotic dynamics; see, for example, [14,52] for some of the many applications of the FDT in the statistical physics setting
The derivation of the linear response formula in the time-periodic setting is followed by the derivation of two classes of fluctuation–dissipation relationships: the first one applies to perturbations of dynamics with time-periodic ergodic probability measures required only to exist in the unperturbed dynamics, the second one involves sim
Summary
A systematic determination of the response of a complex nonlinear dynamical system to time-dependent perturbations is of key importance; topical examples in high-dimensional, non-autonomous and/or stochastic settings include climate models (e.g. [1,16,34,36,59,62]), statistical physics and non-equilibrium thermodynamics (e.g. [47,50,75,82,83]), and even neural networks (e.g. [18,22,76]). A systematic determination of the response of a complex nonlinear dynamical system to time-dependent perturbations is of key importance; topical examples in high-dimensional, non-autonomous and/or stochastic settings include climate models [1,16,34,36,59,62]), statistical physics and non-equilibrium thermodynamics [47,50,75,82,83]), and even neural networks The sought response is usually quantified in terms of a change in an ‘observable’ expressed as a statistical/ensemble average of some functional defined on the trajectories of the underlying
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