  In a recent article, Krapivsky and Redner (J. Stat. Mech. 093208 (2018))
established that the distribution of the first hitting times for a diffusing
particle subject to hitting an absorber is independent of the direction of the
external flow field. In the present paper, we build upon this observation and
investigate when the conditioning on the diffusion leads to a process that is
totally independent of the flow field. For this purpose, we adopt the Langevin
approach, or more formally the theory of conditioned stochastic differential
equations. This technique allows us to derive a large variety of stochastic
processes: in particular, we introduce a new kind of Brownian bridge ending at
two different final points and calculate its fundamental probabilities. This
method is also very well suited for generating statistically independent paths.
Numerical simulations illustrate our findings.
