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 Mat. Sb., 2005, Volume 196, Number 4, Pages 135–160 (Mi msb1289)

The Laplace method for small deviations of Gaussian processes of Wiener type

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Results on the exact asymptotics of the probabilities
$$\mathsf P\{ \int_0^1|\xi(t)|^p dt \le\varepsilon^p\},\qquad\varepsilon\to 0,$$
for $p>0$ are proved for two Gaussian processes $\xi(t)$: the Wiener process and the Brownian bridge. The method of study is the Laplace method in Banach spaces and the approach to the probabilities of small deviations based on the theory of large deviations for the occupation time. The calculations are carried out for the cases $p=1$ and $p=2$ as a result of solving the extremal problem for the action functional and studying the corresponding Schrödinger equations.

DOI: https://doi.org/10.4213/sm1289

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English version:
Sbornik: Mathematics, 2005, 196:4, 595–620

Bibliographic databases:

UDC: 519.2
MSC: Primary 60G15; Secondary 60J65, 60F05, 60F10, 60G60

Citation: V. R. Fatalov, “The Laplace method for small deviations of Gaussian processes of Wiener type”, Mat. Sb., 196:4 (2005), 135–160; Sb. Math., 196:4 (2005), 595–620

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1289
• https://doi.org/10.4213/sm1289
• http://mi.mathnet.ru/eng/msb/v196/i4/p135

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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, “Exact Asymptotics of Large Deviations of Stationary Ornstein–Uhlenbeck Processes for $L^p$-Functional, $p>0$”, Problems Inform. Transmission, 42:1 (2006), 46–63
2. V. R. Fatalov, “Occupation times and exact asymptotics of small deviations of Bessel processes for $L^p$-norms with $p>0$”, Izv. Math., 71:4 (2007), 721–752
3. V. R. Fatalov, “Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$”, Problems Inform. Transmission, 44:2 (2008), 138–155
4. V. R. Fatalov, “Occupation Time and Exact Asymptotics of Distributions of $L^p$-Functionals of the Ornstein–Uhlenbeck Processes, $p>0$”, Theory Probab. Appl., 53:1 (2009), 13–36
5. V. R. Fatalov, “Some asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 157:2 (2008), 1606–1625
6. Albeverio S., Fatalov V., Piterbarg V.I., “Asymptotic behavior of the sample mean of a function of the Wiener process and the Macdonald function”, J. Math. Sci. Univ. Tokyo, 16:1 (2009), 55–93
7. 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
8. V. R. Fatalov, “Large deviations for distributions of sums of random variables: Markov chain method”, Problems Inform. Transmission, 46:2 (2010), 160–183
9. V. R. Fatalov, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. Math., 74:1 (2010), 189–216
10. V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868
11. V. R. Fatalov, “Negative-order moments for $L^p$-functionals of Wiener processes: exact asymptotics”, Izv. Math., 76:3 (2012), 626–646
12. 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
13. 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
14. V. R. Fatalov, “On the Laplace method for Gaussian measures in a Banach space”, Theory Probab. Appl., 58:2 (2014), 216–241
15. 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
16. V. R. Fatalov, “Weighted $L^p$, $p\ge2$, for a wiener process: Exact asymptoties of small deviations”, Moscow University Mathematics Bulletin, 70:2 (2015), 68–73
17. 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
18. V. R. Fatalov, “Integrals of Bessel processes and multi-dimensional Ornstein–Uhlenbeck processes: exact asymptotics for $L^p$-functionals”, Izv. Math., 82:2 (2018), 377–406
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