Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the $L^p$ functionals with $0<p<\infty$ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the $L^p$ norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value $p_0=2+4\pi^2/\beta^2\omega^2$, where $\beta>0$ is the inverse temperature and $\omega>0$ is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for $0<p<p_0 $ and $p>p_0$. We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.