Functional Integrals for the Bogoliubov Gaussian Measure: Exact Asymptotic Forms

Abstract

We prove theorems on the exact asymptotic forms as u → ∞ of two functional integrals over the Bogoliubov measure μ_B of the forms

$$\int_{C[0,\beta ]} {[\int_0^\beta {|x(t){|^p}dt{]^u}d{\mu _B}(x)} } ,\;\int_{C(0,\beta )} {\exp \left\{ {\mu {{(\int_0^\beta {|x(t){|^p}dt} )}^{a/p}}} \right\}d{\mu _B}(x)} $$

for p = 4, 6, 8, 10 with p > p0, where p0 = 2+4π²/β²ω² is the threshold value, β is the inverse temperature, ω is the eigenfrequency of the harmonic oscillator, and 0 < α < 2. As the method of study, we use the Laplace method in Hilbert functional spaces for distributions of almost surely continuous Gaussian processes.