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 Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 1, Pages 197–224 (Mi izv738)

Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals

V. R. Fatalov

Abstract: We prove theorems on the exact asymptotic behaviour of the integrals
$$\mathsf{E}\exp\{u(\int_0^1|\xi(t)|^p dt)^{\alpha/p}\}, \quad \mathsf{E}\exp\{-u\int_0^1|\xi(t)|^p dt\}, \qquad u\to\infty,$$
for $p>0$ and $0<\alpha<2$ for two random processes $\xi(t)$, namely, the Wiener process and the Brownian bridge, and obtain other related results. Our approach is via the Laplace method for infinite-dimensional distributions, namely, Gaussian measures and the occupation time for Markov processes.

Keywords: large deviation, Gaussian process, Markov process, occupation time, covariance operator, generating operator, Schrödinger operator, hypergeometric function.

DOI: https://doi.org/10.4213/im738

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English version:
Izvestiya: Mathematics, 2010, 74:1, 189–216

Bibliographic databases:

UDC: 519.2
MSC: Primary 60H05; Secondary 28C20, 60F10, 60J65
Revised: 19.10.2007

Citation: V. R. Fatalov, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. RAN. Ser. Mat., 74:1 (2010), 197–224; Izv. Math., 74:1 (2010), 189–216

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv738
• https://doi.org/10.4213/im738
• http://mi.mathnet.ru/eng/izv/v74/i1/p197

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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, “Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$”, Problems Inform. Transmission, 46:1 (2010), 62–85
2. V. R. Fatalov, “Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 168:2 (2011), 1112–1149
3. V. R. Fatalov, “Integral Functionals for the Exponential of the Wiener Process and the Brownian Bridge: Exact Asymptotics and Legendre Functions”, Math. Notes, 92:1 (2012), 79–98
4. V. R. Fatalov, “Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. Math., 77:6 (2013), 1224–1259
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, “Brownian motion on $[0,\infty)$ with linear drift, reflected at zero: exact asymptotics for ergodic means”, Sb. Math., 208:7 (2017), 1014–1048
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