Examples of moderate deviation principle for diffusion processes

Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family $$ S^kappa_t=frac{1}{t^kappa}int_0^tH(X_s)ds, ttoinfty $$ for an ergodic diffusion process $X_t$ under $0.5<kappa<1$ and appropriate $H$. We mean a decomposition with ``corrector: $$ frac{1}{t^kappa}int_0^tH(X_s)ds={rm corrector}+frac{1}{t^kappa}underbrace{M_t}_{rm martingale}. $$ and show that, as in the CLT analysis, the corrector is negligible but in the MD scale, and the main contribution in the MD brings the family ``$ frac{1}{t^kappa}M_t, ttoinfty. $ Starting from Bayer and Freidlin, cite{BF}, and finishing by Wus papers cite{Wu1}-cite{WuH}, in the MD study Laplaces transform dominates. In the paper, we replace the Laplace technique by one, admitting to give the conditions, providing the MD, in terms of ``drift-diffusion parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named ``Examples ....