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Probl. Peredachi Inf., 2006, Volume 42, Issue 1, Pages 52–71 (Mi ppi37)  

This article is cited in 8 scientific papers (total in 8 papers)

Large Systems

Exact Asymptotics of Large Deviations of Stationary Ornstein–Uhlenbeck Processes for $L^p$-Functional, $p>0$

V. R. Fatalov

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

Abstract: We prove a general result on the exact asymptotics of the probability
$$ \mathbf P\{\int\limits_0^1|\eta_\gamma(t)|^p dt>u^p\} $$
as $u\to\infty$, where $p>0$, for a stationary Ornstein–Uhlenbeck process $\eta_\gamma(t)$, i.e., a Gaussian Markov process with zero mean and with the covariance function $\mathbf E\eta_\gamma(t)\eta_\gamma(s)=e^{-\gamma|t-s|}$, $t,s\in\mathbb R$, $\gamma>0$. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm–Liouville type. For $p=1$ and $p=2$, explicit formulas for the asymptotics are given.

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English version:
Problems of Information Transmission, 2006, 42:1, 46–63

Bibliographic databases:

UDC: 621.391.1:519.2
Received: 25.05.2005

Citation: V. R. Fatalov, “Exact Asymptotics of Large Deviations of Stationary Ornstein–Uhlenbeck Processes for $L^p$-Functional, $p>0$”, Probl. Peredachi Inf., 42:1 (2006), 52–71; Problems Inform. Transmission, 42:1 (2006), 46–63

Citation in format AMSBIB
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\by V.~R.~Fatalov
\paper Exact Asymptotics of Large Deviations of Stationary Ornstein--Uhlenbeck
Processes for $L^p$-Functional, $p>0$
\jour Probl. Peredachi Inf.
\yr 2006
\vol 42
\issue 1
\pages 52--71
\mathnet{http://mi.mathnet.ru/ppi37}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2214512}
\zmath{https://zbmath.org/?q=an:1104.60011}
\elib{http://elibrary.ru/item.asp?id=9200313}
\transl
\jour Problems Inform. Transmission
\yr 2006
\vol 42
\issue 1
\pages 46--63
\crossref{https://doi.org/10.1134/S0032946006010054}
\elib{http://elibrary.ru/item.asp?id=13524962}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33645970429}


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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 Distributions of Integral Functionals of the Geometric Brownian Motion and Other Related Formulas”, Problems Inform. Transmission, 43:3 (2007), 233–254  mathnet  crossref  mathscinet  zmath  isi  elib
    2. 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
    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, “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
    5. V. R. Fatalov, “Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 168:2 (2011), 1112–1149  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. 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
    8. Nickelsen D., Touchette H., “Anomalous Scaling of Dynamical Large Deviations”, Phys. Rev. Lett., 121:9 (2018), 090602  crossref  isi  scopus
  • Проблемы передачи информации Problems of Information Transmission
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