Conditioned Langevin Dynamics enables efficient sampling of transition paths

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.