  The fluctuation-dissipation theorem is a central result in statistical
mechanics and is usually formulated for systems described by diffusion
processes. In this paper, we propose a generalization for a wider class of
stochastic processes, namely the class of Markov processes that satisfy
detailed balance and a large-deviation principle. The generalized
fluctuation-dissipation theorem characterizes the deterministic limit of such a
Markov process as a generalized gradient flow, a mathematical tool to model a
purely irreversible dynamics via a dissipation potential and an entropy
function: these are expressed in terms of the large-deviation dynamic rate
function of the Markov process and its stationary distribution. We exploit the
generalized fluctuation-dissipation theorem to develop a new method of
coarse-graining and test it in the context of the passage from the diffusion in
a double-well potential to the jump process that describes the simple reaction
$A \rightleftarrows B$ (Kramers' escape problem).
