Moderate deviations type evaluation for integral functionals of diffusion processes

We establish a large deviations type evaluation for the family of integral functionals ε^-κ √^Tε₀ Ψ (X^ε_s)g (ξ^ε_s) ds, ε ↘ 0, where Ψ and g are smooth functions, ξ^ε_t is a "fast" ergodic diffusion while X^ε_t is a "slow" diffusion type process, κ ∈ (0, 1/2). Under the assumption that g has zero barycenter with respect to the invariant distribution of the fast diffusion, we derive the main result from the moderate deviation principle for the family ε^-κ √^Tε₀ Ψ (X^ε_s)g (ξ^ε_s) ds, ε ↘ 0 which has an independent interest as well. In addition, we give a preview for a vector case.