Gaussian Ornstein–Uhlenbeck and Bogoliubov processes: asymptotics of small deviations for $L^p$-functionals, $0<p<\infty$

We prove results on sharp asymptotics of probabilities
$$ \mathbf P\Biggl\{\int_0^1|X(t)|^p\,dt<\varepsilon^p\Biggr\},\qquad\varepsilon\to0, $$
where $0<p<\infty$, for three Gaussian processes $X(t)$, namely the stationary and nonstationary Ornstein–Uhlenbeck process and the Bogoliubov process. The analysis is based on the Laplace method for sojourn times of a Wiener process.