Moderate deviations of inhomogeneous functionals of Markov processes and application to averaging

In this paper, we study the moderate deviation principle of an inhomogeneous integral functional of a Markov process (ξs) which is exponentially ergodic, i.e. the moderate deviations of1εh(ε)∫0.f(s,ξs/ε) ds,in the space of continuous functions from [0,1] to Rd, where f is some Rd-valued bounded function. Our method relies on the characterization of the exponential ergodicity by Down–Meyn–Tweedie (Ann. Probab. 25(3) (1995) 1671) and the regeneration split chain technique for Markov chain. We then apply it to establish the moderate deviations of Xtε given by the following randomly perturbed dynamic system in RdẊtε=b(Xtε,ξt/ε),around its limit behavior, usually called the averaging principle, studied by Freidlin and Wentzell (Random Perturbations of Dynamical Systems, Springer, New York, 1984).