Occupation Time and Exact Asymptotics of Distributions of $L^p$-Functionals of the Ornstein–Uhlenbeck Processes, $p>0$

The paper proves the results on exact asymptotics of the probabilities $\mathbf{P}_\mu\{T^{-1}\times\int_0^T|\eta_\gamma(t)|^p\,dt<d\}$, $T\to\infty$, for $p>0$ for Gaussian Markov Ornstein–Uhlenbeck processes $\eta_\gamma$ and also for their conditional versions. The author uses the Laplace method for the occupation time of Markov processes with continuous time. The calculations are given for the case $p=2$ with the help of the solution of the extremal problem for the action functional.