Using straightforward linear algebra we derive response operators describing the
impact of small perturbations to finite state Markov processes. The results can be used
for studying empirically constructed—e.g. from observations or through coarse graining
of model simulations—finite state approximation of statistical mechanical systems. Recent
results concerning the convergence of the statistical properties of finite state Markov approximation
of the full asymptotic dynamics on the SRB measure in the limit of finer and finer
partitions of the phase space are suggestive of some degree of robustness of the obtained
results in the case of Axiom A system. Our findings give closed formulas for the linear and
nonlinear response theory at all orders of perturbation and provide matrix expressions that can
be directly implemented in any coding language, plus providing bounds on the radius of convergence
of the perturbative theory. In particular, we relate the convergence of the response
theory to the rate of mixing of the unperturbed system. One can use the formulas derived
for finite state Markov processes to recover previous findings obtained on the response of
continuous time Axiom A dynamical systems to perturbations, by considering the generator
of time evolution for the measure and for the observables. A very basic, low-tech, and computationally
cheap analysis of the response of the Lorenz ’63 model to perturbations provides
rather encouraging results regarding the possibility of using the approximate representation
given by finite state Markov processes to compute the system’s response.