Brownian bridge method for stochastic chemical processes – An approximation method under diffusion limit

A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths towards the desired region of phase space. This idea has many applications in chemical physics where one wants to control the endpoint of a stochastic process – e.g., polymer physics, chemical reaction pathways, heat/mass transfer, and Brownian dynamics simulations. Despite its broad applicability, the biggest limitation of the Brownian bridge technique is that it is often difficult to determine the effective drift, as it comes from a solution of a Backwards Fokker Planck (BFP) equation that is infeasible to compute for complex or high-dimensional systems. This paper introduces a fast approximation method to generate a Brownian bridge process without solving the BFP equation explicitly. Specifically, the paper will use the asymptotic properties of the BFP equation to generate an approximate drift, and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. Because such a procedure avoids the solution of BFP equation, we show that it drastically accelerates the generation of conditioned random walks and allows the generation of such processes to be scaled to higher dimensions and complex systems.