Functional integrals for the Bogoliubov Gaussian measure: Exact asymptotic forms

We prove theorems on the exact asymptotic forms as $u\to\infty$ of two functional integrals over the Bogoliubov measure $\mu_{{\mathrm B}}$ of the forms
$$ \int_{C[0,\beta]}\biggl[\,\int_0^\beta |x(t)|^p\,dt\biggr]^{u}\,d\mu_{{\mathrm B}}(x),\qquad \int_{C[0,\beta]}\exp\biggl\{u\biggl(\,\int_0^\beta |x(t)|^p\,dt\biggr)^{\!\alpha/p}\,\biggr\}\,d\mu_{{\mathrm B}}(x) $$
for $p=4,6,8,10$ with $p>p_0$, where $p_0=2+4\pi^2/\beta^2\omega^2$ is the threshold value, $\beta$ is the inverse temperature, $\omega$ is the eigenfrequency of the harmonic oscillator, and $0<\alpha<2$. As the method of study, we use the Laplace method in Hilbert functional spaces for distributions of almost surely continuous Gaussian processes.