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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
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.
Received: 22.11.2011 Revised: 18.12.2012
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
Linking options:
https://www.mathnet.ru/eng/im7938https://doi.org/10.1070/IM2013v077n06ABEH002675 https://www.mathnet.ru/eng/im/v77/i6/p169
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Abstract page: | 434 | Russian version PDF: | 179 | English version PDF: | 15 | References: | 61 | First page: | 21 |
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