Brownian bridges for contained random walks

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 $T$. 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 $T\gg 1$, 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.