We consider a Markov jump process on a general state space to which we apply
a time-dependent weak perturbation over a finite time interval. By
martingale-based stochastic calculus, under a suitable exponential moment bound
for the perturbation we show that the perturbed process does not explode almost
surely and we study the linear response (LR) of observables and additive
functionals. When the unperturbed process is stationary, the above LR formulas
become computable in terms of the steady state two-time correlation function
and of the stationary distribution. Applications are discussed for birth and
death processes, random walks in a confining potential, random walks in a
random conductance field. We then move to a Markov jump process on a finite
state space and investigate the LR of observables and additive functionals in
the oscillatory steady state (hence, over an infinite time horizon), when the
perturbation is time-periodic. As an application we provide a formula for the
complex mobility matrix of a random walk on a discrete $d$-dimensional torus,
with possibly heterogeneous jump rates.
