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Izvestiya: Mathematics, 2013, Volume 77, Issue 6, Pages 1224–1259
DOI: https://doi.org/10.1070/IM2013v077n06ABEH002675
(Mi im7938)
 

This article is cited in 5 scientific papers (total in 5 papers)

Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
References:
Abstract: We prove results on exact asymptotics as $T\to\infty$ for the means $\mathsf{E}_{a,c}\exp\bigl\{-\int_0^T g(\mathbf{w}(t))\,dt\bigr\}$ and probabilities $\mathsf{P}_{a,c}\bigl\{\frac1T\int_0^Tg(\mathbf{w}(t))\,dt<d\bigr\}$, where $\mathbf{w}(t)=(w_1(t),\dots,w_n(t))$, $t\geqslant 0$, is an $n$-dimensional Wiener process, $g(x)$ is a positive continuous function (potential) satisfying certain conditions, $d>0$, and $a,c\in\mathbb{R}^n$ are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities $\mathsf{P}_0\bigl\{\int_0^1\sum_{j=1}^n|w_j(t)|^p\,dt<\varepsilon^p\bigr\}$ as $\varepsilon\to 0$ with a fixed $p>0$.
Keywords: large deviations, Markov processes, Laplace method, action functional, occupation time, multi-dimensional Schrödinger operator.
Funding agency Grant number
Russian Foundation for Basic Research 07-01-00077
11-01-00050
Received: 22.11.2011
Revised: 18.12.2012
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2013, Volume 77, Issue 6, Pages 169–206
DOI: https://doi.org/10.4213/im7938
Bibliographic databases:
Document Type: Article
UDC: 519.2
MSC: 60F15, 60J25, 41A60
Language: English
Original paper language: Russian
Citation: V. R. Fatalov, “Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. RAN. Ser. Mat., 77:6 (2013), 169–206; Izv. Math., 77:6 (2013), 1224–1259
Citation in format AMSBIB
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\paper Ergodic means for large values of~$T$ and exact asymptotics of small deviations for a~multi-dimensional Wiener process
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  • https://doi.org/10.1070/IM2013v077n06ABEH002675
  • https://www.mathnet.ru/eng/im/v77/i6/p169
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Russian version PDF:179
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    References:61
    First page:21
     
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