Effective Langevin equations for constrained stochastic processes

We propose a novel stochastic method to exactly generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time t(f). These paths are weighted with a probability given by the overdamped Langevin dynamics. We show how these paths can be exactly generated by a local stochastic differential equation. The method is illustrated on the generation of Brownian bridges, Brownian meanders, Brownian excursions and constrained Ornstein-Uhlenbeck processes. In addition, we show how to solve this equation in the case of a general force acting on the particle. As an example, we show how to generate a constrained path joining the two minima of a double-well. Our method allows us to generate statistically independent paths and is computationally very efficient.