Efficient generation of barrier crossing trajectories using approximate Brownian bridges

We examine continuous random walks that are conditioned to reach one region before another. These conditioned processes are used to generate stochastic trajectories for barrier crossing events, which are generally rare and difficult to sample. The processes are generated using a Brownian bridge technique, resulting in near perfect sampling efficiency without accruing error in the conditional statistics of the process. The construction requires the hitting probability or committer function, which is a solution to the backward Fokker-Planck equation, a partial-differential equation that can be difficult to solve through general means. Therefore, we derive a one-dimensional approximation through asymptotic methods for barrier crossing trajectories. We show that this approximation has a simple analytical representation and approaches the true solution as the barrier height increases. Brownian bridge trajectories generated with this approximate solution are then shown to result in accurate conditional statistics when used in conjunction with importance sampling, even in the case when potential energy barriers are not large. We show this idea's effectiveness by simulating rare events in a stochastic chemical reaction network (Schögl reaction) with multiple steady states. This methodology shows great promise for future implementation to simulate rare barrier crossing events for a wide variety of physical processes.