Brownian motion on with linear drift, reflected at zero: exact asymptotics for ergodic means

Abstract

For the Brownian motion on the half-axis with linear drift , reflected at zero and for fixed numbers , , , , we calculate the exact asymptotics as of the mathematical expectations and probabilities as well as of their conditional versions. For 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 , where is the mass of the Brownian particle, is the gravitational acceleration, is the Boltzmann constant, 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.