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 Uspekhi Mat. Nauk, 2003, Volume 58, Issue 4(352), Pages 89–134 (Mi umn643)

Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields

V. R. Fatalov

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

Abstract: This paper presents a survey of results on computing the small deviation asymptotics for Gaussian measures, that is, the asymptotics of the probabilities
$$\mu(\varepsilon D), \qquad \varepsilon\to0,$$
where $D$ is a bounded domain in a Banach space $(B,{\|\cdot\|})$ (for example, $D=\{x\in B:\|x\|\leqslant 1\}$) and $\mu$ a Gaussian measure on $B$.
The main attention is focused on calculating the values of constants in the exact or logarithmic asymptotics. The survey contains new numerical results; some erroneous assertions in previous papers on this topic are also noted.
The following classes of Gaussian processes and fields are studied in detail: Wiener processes and related processes, Brownian bridges, Bessel processes, vector Wiener processes, Gaussian Markov processes, Gaussian processes with stationary increments, fractional Ornstein–Uhlenbeck processes, $n$-parameter fractional Brownian motion, $n$-parameter Wiener–Chentsov fields, and the Wiener pillow. Results on small deviations are presented in diverse norms, namely, the sup-norm, Hilbert norms, $L^p$-norms, Hölder norms, Orlicz norms, and weighted sup-norms.
About 30 problems concerned with finding exact constants in asymptotic expressions for small deviations are posed.
The relation to Chung's law of the iterated logarithm is also considered, and a number of other results are presented.

DOI: https://doi.org/10.4213/rm643

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English version:
Russian Mathematical Surveys, 2003, 58:4, 725–772

Bibliographic databases:

UDC: 519.21
MSC: Primary 60G15, 60B11; Secondary 60J65, 60J25, 60F10, 60F15, 28C20, 46T12

Citation: V. R. Fatalov, “Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields”, Uspekhi Mat. Nauk, 58:4(352) (2003), 89–134; Russian Math. Surveys, 58:4 (2003), 725–772

Citation in format AMSBIB
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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, “Asymptotics of large deviations of Gaussian processes of Wiener type for $L^p$-functionals, $p>0$, and the hypergeometric function”, Sb. Math., 194:3 (2003), 369–390
2. V. R. Fatalov, “Large deviations for Gaussian processes in Hölder norm”, Izv. Math., 67:5 (2003), 1061–1079
3. A. I. Nazarov, Ya. Yu. Nikitin, “Logarithmic $L_2$-small ball asymptotics for some fractional Gaussian processes”, Theory Probab. Appl., 49:4 (2005), 645–658
4. A. I. Nazarov, “Logarithmic $L_2$-small ball asymptotics with respect to self-similar measure for some Gaussian processes”, J. Math. Sci. (N. Y.), 133:3 (2006), 1314–1327
5. V. R. Fatalov, “The Laplace method for small deviations of Gaussian processes of Wiener type”, Sb. Math., 196:4 (2005), 595–620
6. 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
7. 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
8. 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
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, “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
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. Ya. Yu. Nikitin, R. S. Pusev, “The exact asymptotic of small deviations for a series of Brownian functionals”, Theory Probab. Appl., 57:1 (2013), 60–81
14. A.N.. Frolov, “Small deviations of iterated processes in the space of trajectories”, centr.eur.j.math, 11:12 (2013), 2089
15. 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
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. Ai X., “A Distributional Identity For the Bivariate Brownian Bridge: a Nontensor Gaussian Field”, Math. Probl. Eng., 2018, 9687039
18. MacKay A., Melnikov A., Mishura Yu., “Optimization of Small Deviation For Mixed Fractional Brownian Motion With Trend”, Stochastics, 90:7 (2018), 1087–1110
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