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 $$S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH(X_s)ds, \ t\to\infty$$ 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 ``$\frac{1}{t^\kappa}M_t, t\to\infty.$'' Starting from Bayer and Freidlin, \cite{BF}, and finishing by Wu's papers \cite{Wu1}-\cite{WuH}, in the MD study Laplace's 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 ...''.