  Predicting the response of a system to perturbations is a key challenge in
mathematical and natural sciences. Under suitable conditions on the nature of
the system, of the perturbation, and of the observables of interest, response
theories allow to construct operators describing the smooth change of the
invariant measure of the system of interest as a function of the small
parameter controlling the intensity of the perturbation. In particular,
response theories can be developed both for stochastic and chaotic
deterministic dynamical systems, where in the latter case stricter conditions
imposing some degree of structural stability are required. In this paper we
extend previous findings and derive general response formulae describing how
n-point correlations are affected by perturbations to the vector flow. We also
show how to compute the response of the spectral properties of the system to
perturbations. We then apply our results to the seemingly unrelated problem of
coarse graining in multiscale systems: we find explicit formulae describing the
change in the terms describing parameterisation of the neglected degrees of
freedom resulting from applying perturbations to the full system. All the terms
envisioned by the Mori-Zwanzig theory - the deterministic, stochastic, and
non-Markovian terms - are affected at 1st order in the perturbation. The
obtained results provide a more comprehesive understanding of the response of
statistical mechanical systems to perturbations and contribute to the goal of
constructing accurate and robust parameterisations and are of potential
relevance for fields like molecular dynamics, condensed matter, and geophysical
fluid dynamics. We envision possible applications of our general results to the
study of the response of climate variability to anthropogenic and natural
forcing and to the study of the equivalence of thermostatted statistical
mechanical systems.
