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The perturbation theory of operator semigroups is used to derive responseformulas for a variety of combinations of acting forcings and referencebackground dynamics. In the case of background stochastic dynamics, wedecompose the response formulas using the Koopman operator generatoreigenfunctions and the corresponding eigenvalues, thus providing a functionalbasis towards identifying relaxation timescales and modes in physicallyrelevant systems. To leading order, linear response gives the correction toexpectation values due to extra deterministic forcings acting on eitherstochastic or chaotic dynamical systems. When considering the impact of weaknoise, the response is linear in the intensity of the (extra) noise forbackground stochastic dynamics, while the second order response given theleading order correction when the reference dynamics is chaotic. In this lattercase we clarify that previously published diverging results can be brought tocommon ground when a suitable interpretation - Stratonovich vs. Ito - of thenoise is given. Finally, the response of two-point correlations toperturbations is studied through the resolvent formalism via a perturbativeapproach. Our results allow, among other things, to estimate how thecorrelations of a chaotic dynamical system changes as a results of addingstochastic forcing.

On Some Aspects of the Response to Stochastic and Deterministic Forcings