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This article is cited in 4 scientific papers (total in 4 papers)
Brownian motion on $[0,\infty)$ with linear drift, reflected at zero: exact asymptotics for ergodic means
V. R. Fatalov Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
For the Brownian motion $X_\mu(t)$ on the half-axis $[0,\infty)$ with linear drift $\mu$, reflected at zero and for fixed numbers $p>0$, $\delta>0$, $d>0$, $a \geqslant 0$, we calculate the exact asymptotics as $T\to\infty$ of the mathematical expectations and probabilities
$$
\mathsf E\biggl[\exp\biggl\{-\delta\!\!\int_0^T \!\!X_\mu^p(t)\,dt\biggr\}\biggm| X_\mu(0)=a\biggr],
\mathsf P\biggl\{\frac1 T\!\int_0^T \!\!X_\mu^p(t)\,dt\!<\!d\biggm| X_\mu(0)=a\biggr\},
$$
as well as of their conditional versions. For $p=1$ we give explicit formulae for the emerging constants via the Airy function. We consider an application of the results obtained to the problem of studying the behaviour of a Brownian particle in a gravitational field in a container bounded below by an impenetrable wall when $\mu=-mg/(2kT_{\mathrm K})$, where $m$ is the mass of the Brownian particle, $g$ is the gravitational acceleration, $k$ is the Boltzmann constant, $T_{\mathrm K}$ is the temperature in the Kelvin scale. The analysis is conducted by the Laplace method for the sojourn time of homogeneous Markov processes.
Bibliography: 31 titles.
Keywords:
Brownian motion with drift, reflected at zero, ergodicity, sojourn time, large deviations, Airy function, Schrödinger operator.
Received: 04.03.2016 and 14.11.2016
Citation:
V. R. Fatalov, “Brownian motion on $[0,\infty)$ with linear drift, reflected at zero: exact asymptotics for ergodic means”, Mat. Sb., 208:7 (2017), 109–144; Sb. Math., 208:7 (2017), 1014–1048
Linking options:
https://www.mathnet.ru/eng/sm8692https://doi.org/10.1070/SM8692 https://www.mathnet.ru/eng/sm/v208/i7/p109
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Abstract page: | 485 | Russian version PDF: | 53 | English version PDF: | 20 | References: | 53 | First page: | 13 |
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