Functional integrals for the~Bogoliubov Gaussian measure: Exact asymptotic forms

V. R. Fatalov

Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 2, pp. 171-189

Résumé

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 $0\alpha2$. As the method of study, we use the Laplace method in Hilbert functional spaces for distributions of almost surely continuous Gaussian processes.

Détail

Keywords: Bogoliubov measure, almost surely continuous Gaussian process, Laplace method in a functional Hilbert space, manifold of minimum values.