  We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics
from a mathematical perspective. We formalize the concept of "linear response
function" in the general framework of Markov processes. We show that for
processes out of equilibrium it depends not only on the given Markov process
X(s) but also on the chosen perturbation of it. We characterize the set of all
possible response functions for a given Markov process and show that at
equilibrium they all satisfy the FDT. That is, if the initial measure is
invariant for the given Markov semi-group, then for any pair of times s<t and
nice functions f,g, the dissipation, that is, the derivative in s of the
covariance of g(X(t)) and f(X(s)) equals the infinitesimal response at time t
and direction g to any Markovian perturbation that alters the invariant measure
of X(.) in the direction of f at time s. The same applies in the so called FDT
regime near equilibrium, i.e. in the limit s going to infinity with t-s fixed,
provided X(s) converges in law to an invariant measure for its dynamics. We
provide the response function of two generic Markovian perturbations which we
then compare and contrast for pure jump processes on a discrete space, for
finite dimensional diffusion processes, and for stochastic spin systems.
