  We propose a novel stochastic method to generate Brownian paths conditioned
to start at an initial point and end at a given final point during a fixed time
$t_{f}$ under a given potential $U(x)$. These paths are sampled with a
probability given by the overdamped Langevin dynamics. We show that these paths
can be exactly generated by a local Stochastic Partial Differential Equation
(SPDE). This equation cannot be solved in general. We present several
approximations that are valid either in the low temperature regime or in the
presence of barrier crossing. We show that this method warrants the generation
of statistically independent transition paths. It is computationally very
efficient. We illustrate the method on the two dimensional Mueller potential as
well as on the Mexican hat potential.
