Exact Asymptotics of Large Deviations of Stationary Ornstein–Uhlenbeck Processes for $L^p$-Functional, $p>0$

We prove a general result on the exact asymptotics of the probability
$$ \mathbf P\biggl\{\int\limits_0^1|\eta_\gamma(t)|^p\,dt>u^p\biggr\} $$
as $u\to\infty$, where $p>0$, for a stationary Ornstein–Uhlenbeck process $\eta_\gamma(t)$, i.e., a Gaussian Markov process with zero mean and with the covariance function $\mathbf E\eta_\gamma(t)\eta_\gamma(s)=e^{-\gamma|t-s|}$, $t,s\in\mathbb R$, $\gamma>0$. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm–Liouville type. For $p=1$ and $p=2$, explicit formulas for the asymptotics are given.