Brownian motion on $[0,\infty)$ with linear drift, reflected at zero: exact asymptotics for ergodic means

V. R. Fatalov

Sbornik. Mathematics, Tome 208 (2017) no. 7, pp. 1014-1048

Résumé

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.

Détail

Keywords: Brownian motion with drift, reflected at zero, ergodicity, sojourn time, large deviations, Airy function, Schrödinger operator.