Exact asymptotic behaviors for $L^p$-functionals of Bessel processes and multidimensional Ornstein-Uhlenbeck process

New results will be presented on exact asymptotic behaviors of means
$$E_a \exp \Bigl \{ - \int\limits_0^T \xi_q^p(t) \, dt \Bigr \}, \quad E_a \Bigl[ \exp \Bigl \{ - \int\limits_0^T \xi_q^p(t) \, dt \Bigr \} \Bigl | \, \xi_q(T) = b \Bigr]$$
as $T \to \infty$, where $\xi_q(t)$, $t \geq 0$, is Bessel process of order $q \geq - 1/2$, and $p > 0$, $a \geq 0$, $b \geq 0$ are arbitrary fixed. As well, a new formula will be given for exact asymptotic behavior of the probability
$$P \Bigl \{ \int\limits_0^1 \Bigl[ \sum\limits_{k=1}^n Y_k^2(t) \Bigr]^{p/2} \, dt \leq \varepsilon^p \Bigr \}$$
as $\varepsilon \to 0$, where $Y(t) = (Y_1(t), \dots, Y_n(t))$, $t \geq 0$, is a $n$-dimensional beginning at zero non-stationary Ornstein-Uhlenbeck process with parameter $\gamma = (\gamma_1, \dots, \gamma_n)$.