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

This article is cited in 18 scientific papers (total in 18 papers)

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.


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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
Received: 05.09.2003 and 24.08.2004

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
\by V.~R.~Fatalov
\paper The Laplace method for small deviations of Gaussian processes of Wiener type
\jour Mat. Sb.
\yr 2005
\vol 196
\issue 4
\pages 135--160
\jour Sb. Math.
\yr 2005
\vol 196
\issue 4
\pages 595--620

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    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  mathnet  crossref  mathscinet  zmath  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. V. R. Fatalov, “Some asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 157:2 (2008), 1606–1625  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  zmath  isi
    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  mathnet  crossref  mathscinet  isi
    8. V. R. Fatalov, “Large deviations for distributions of sums of random variables: Markov chain method”, Problems Inform. Transmission, 46:2 (2010), 160–183  mathnet  crossref  mathscinet  isi  elib
    9. V. R. Fatalov, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. Math., 74:1 (2010), 189–216  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. V. R. Fatalov, “Negative-order moments for $L^p$-functionals of Wiener processes: exact asymptotics”, Izv. Math., 76:3 (2012), 626–646  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. V. R. Fatalov, “On the Laplace method for Gaussian measures in a Banach space”, Theory Probab. Appl., 58:2 (2014), 216–241  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  isi
    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  mathnet  crossref  mathscinet  isi
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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