Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$

We prove results on exact asymptotics of the probabilities
$$ \mathrm{P}\biggl\{\int_0^1|\eta(t)|^p dt\leq\varepsilon^p\biggr\},\quad\varepsilon\to 0, $$
where $2\leq p\leq\infty$, for two types of Gaussian processes $\eta(t)$, namely, a stationary Ornstein–Uhlenbeck process and a Gaussian diffusion process satisfying the stochastic differential equation
\begin{gather*} dZ(t)=dw(t)+g(t)Z(t)dt,\quad t\in[0,1], \\ Z(0)=0. \end{gather*}
Derivation of the results is based on the principle of comparison with a Wiener process and Girsanov's absolute continuity theorem.