Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals

We prove theorems on the exact asymptotic behaviour of the integrals
$$\mathsf{E}\exp\biggl\{u\biggl(\int_0^1|\xi(t)|^p\,dt\biggr)^{\alpha/p}\biggr\}, \quad \mathsf{E}\exp\biggl\{-u\int_0^1|\xi(t)|^p\,dt\biggr\}, \qquad u\to\infty,$$
for $p>0$ and $0<\alpha<2$ for two random processes $\xi(t)$, namely, the Wiener process and the Brownian bridge, and obtain other related results. Our approach is via the Laplace method for infinite-dimensional distributions, namely, Gaussian measures and the occupation time for Markov processes.