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) $.