MDP for integral functionals of fast and slow processes with averaging

We establish the moderate deviation principle (MDP) for the family of X t ɛ = 1 ɛ κ ∫ 0 t H ( ξ s ɛ , Y s ɛ ) d s , ɛ ↓ 0 , where 0 < κ < 0.5 and ( ξ t ɛ , Y t ɛ ) are slow and fast diffusion processes. We embed the original problem in the MDP study for the pair ( X t ɛ , Y t ɛ ) . The main tool for the MDP analysis is the Poisson equation technique, borrowed from the recent papers of Pardoux and Veretennikov, (Ann. Probab. 29 (3) (2001) 1061; Ann. Probab. 31 (3) (2003) 1166), and a new approach to the large deviation analysis, proposed by Puhalskii, (Large Deviations and Idempotent Probability, 2001), which exploits “fast homogenization” of the drift and diffusion parameters instead of the traditional Laplace transform technique. The obtained MDP for ( X t ɛ , Y t ɛ ) has a typical structure of the Freidlin–Wentzell-type large deviation principle.