Sampling constrained stochastic trajectories using Brownian bridges

We present a new method to sample conditioned trajectories of a system evolving under Langevin dynamics based on Brownian bridges. The trajectories are conditioned to end at a certain point (or in a certain region) in space. The bridge equations can be recast exactly in the form of a non-linear stochastic integro-differential equation. This equation can be very well approximated when the trajectories are closely bundled together in space, i.e., at low temperature, or for transition paths. The approximate equation can be solved iteratively using a fixed point method. We discuss how to choose the initial trajectories and show some examples of the performance of this method on some simple problems. This method allows us to generate conditioned trajectories with a high accuracy.