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Abstract Using linear operator techniques, we demonstrate an efficient method for investigating rare events in stochastic processes. Specifically, we examine contained trajectories, which are continuous random walks that only leave a specified region of phase space after a set period of time . We show that such trajectories can be efficiently generated through the use of a Brownian Bridge, derived via the solution to the Backward Fokker–Planck (BFP) equation. Using linear operator techniques, we place the BFP operator in self‐adjoint form and show that in the asymptotic limit , the set of paths contained in a specified region is equivalent to paths on a modified potential energy landscape that is related to the dominant eigenfunction of the self‐adjoint BFP operator. We demonstrate this idea on several example problems, one of which is the Graetz problem, where one is interested in the survival time of a particle diffusing in tube flow. 

A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths toward the desired region of phase space. This idea has many applications in chemical science 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 Backward 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, this paper uses 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 the BFP equation, we show that it drastically accelerates the generation of conditioned random walks. We also show that this approach offers reasonable improvement compared to other sampling approaches using simple bias potentials.