  We have developed efficient techniques to solve the first-time problems of
Brownian motion. Based on a time-scale separation of recrossings, we show that
Eyring's transmission coefficient ($\kappa$) equals to the one
($\kappa_\mathrm{V}$) corresponding to an absorbing boundary consistent with
the transition state theory, which is greater than the one
($\kappa_\mathrm{K}$) derived by Kramers. We also propose methods for reaction
rate determination by analyzing short-time trajectories from the barrier
maximum, and discuss the relation to the reactive flux method and the
significance of reaction coordinates.
