Brownian Bridges for Stochastic Containment using a Self-Adjoint Formulation of the Backward Fokker-Planck Equation

We show that continuous random walks, processes which model a wide variety of chemical and physical systems, can be conditioned to efficiently generate rare events. Specifically, we use a Brownian bridge to examine contained trajectories, i.e. processes which stay within a given region of state-space for a set period time T, or in other words, paths whose survival time within that region is greater than T. The bridge acts as a conditioned process so that we generate only the subset of sample paths which meet this containment criteria. Bridges are constructed via the solution of the Backwards Fokker-Planck (BFP) equation. We derive a method which reformulates the BFP into a self-adjoint representation, effectively reducing its complexity. This representation shows that in the asymptotic limit, T>>1, the bridge is time-independent and is a function of only the dominant eigenfunction of the self-adjoint BFP operator. In this limit, we show that the subset of paths contained within a specified region is equivalent to the set of paths sampled from a modified potential energy landscape. We demonstrate that this idea is accurate for many systems, even for times T ~ O(1). Lastly, we discuss how this idea could be scaled to higher dimensions, using existing numerical techniques to approximate dominant eigenfunctions.