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 TMF, 2011, Volume 168, Number 2, Pages 299–340 (Mi tmf6683)

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

V. R. Fatalov

Lomonosov Moscow State University, Moscow, Russia

Abstract: 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.

Keywords: Bogoliubov measure, Laplace method in Banach space, large deviation principle, action functional

DOI: https://doi.org/10.4213/tmf6683

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English version:
Theoretical and Mathematical Physics, 2011, 168:2, 1112–1149

Bibliographic databases:

Citation: V. R. Fatalov, “Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure”, TMF, 168:2 (2011), 299–340; Theoret. and Math. Phys., 168:2 (2011), 1112–1149

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf6683
• https://doi.org/10.4213/tmf6683
• http://mi.mathnet.ru/eng/tmf/v168/i2/p299

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. R. Fatalov, “Negative-order moments for $L^p$-functionals of Wiener processes: exact asymptotics”, Izv. Math., 76:3 (2012), 626–646
2. V. R. Fatalov, “Asymptotic behavior of small deviations for Bogoliubov's Gaussian measure in the $L^p$ norm, $2\le p\le\infty$”, Theoret. and Math. Phys., 173:3 (2012), 1720–1733
3. V. R. Fatalov, “Perturbation theory series in quantum mechanics: Phase transition and exact asymptotic forms for the expansion coefficients”, Theoret. and Math. Phys., 174:3 (2013), 360–385
4. V. R. Fatalov, “On the Laplace method for Gaussian measures in a Banach space”, Theory Probab. Appl., 58:2 (2014), 216–241
5. V. R. Fatalov, “The Laplace method for Gaussian measures and integrals in Banach spaces”, Problems Inform. Transmission, 49:4 (2013), 354–374
6. V. R. Fatalov, “Gaussian Ornstein–Uhlenbeck and Bogoliubov processes: asymptotics of small deviations for $L^p$-functionals, $0<p<\infty$”, Problems Inform. Transmission, 50:4 (2014), 371–389
7. V. R. Fatalov, “Exact Laplace-type asymptotic formulas for the Bogoliubov Gaussian measure: The set of minimum points of the action functional”, Theoret. and Math. Phys., 191:3 (2017), 870–885
8. V. R. Fatalov, “Functional integrals for the Bogoliubov Gaussian measure: Exact asymptotic forms”, Theoret. and Math. Phys., 195:2 (2018), 641–657
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