  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.
