A note on an absorption problem for a Brownian particle moving in a harmonic potential

TL;DR

This paper analyzes the absorption time and position moments of a Brownian particle in a harmonic potential with an absorbing boundary, providing explicit formulas and insights into their asymptotic behavior.

Contribution

It offers explicit derivations of the moments of absorption time and position for a Brownian particle in a harmonic potential with an absorbing boundary.

Findings

Central moments of absorption time tend to finite constants for large initial displacement.

Position moments at the most probable absorption time also tend to finite constants.

Explicit formulas for these moments are derived.

Abstract

An analysis is presented of a Brownian particle moving on the half-line, subject to a restoring force proportional to its displacement and an absorbing boundary at the origin. When the initial displacement is large, the central moments of the time to be absorbed tend to finite constants, as do the position moments when evaluated at the most probable absorption time. These quantities are derived explicitly.