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

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

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
\Bibitem{Fat06} \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
2. V. R. Fatalov, “Some asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 157:2 (2008), 1606–1625
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, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. Math., 74:1 (2010), 189–216
5. V. R. Fatalov, “Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 168:2 (2011), 1112–1149
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
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
8. Nickelsen D., Touchette H., “Anomalous Scaling of Dynamical Large Deviations”, Phys. Rev. Lett., 121:9 (2018), 090602
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