  The probability density is a fundamental quantity for characterizing
diffusion processes. However, it is seldom known except in a few renowned
cases, including Brownian motion and the Ornstein-Uhlenbeck process and their
bridges, geometric Brownian motion, Brownian excursion, or Bessel processes. In
this paper, we utilize Girsanov's theorem, along with a variation of the method
of images, to derive the exact expression of the probability density for
diffusions that have one entrance boundary. Our analysis encompasses numerous
families of conditioned diffusions, including the Taboo process and Brownian
motion conditioned on its growth behavior, as well as the drifted Brownian
meander and generalized Brownian excursion.
