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 ∫₀^Tε Ψ(X_s^ε)g(ξ_s^ε)ds, ε ↘ 0, where Ψ and g are smooth functions, ξ_t ^ε is a "fast" ergodic diffusion while X _t^ε 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 ∫₀^t g(ξ_s ^ε)ds)_t≥0, ε ↘ 0, which has an independent interest as well. In addition, we give a preview for a vector case.