(578g) Brownian Bridges for Stochastic Containment Using Self-Adjoint Formulation of the Backward Fokker-Planck Equation

We develop an efficient sampling technique to investigate rare events using linear operator techniques. Specifically, we look at contained trajectories, which are continuous random walks whose first passage time to a specified region is large. We show that such trajectories can be efficiently generated through the use of a Brownian Bridge, derived via the solution to the Backwards Fokker-Planck (BFP) equation. Using linear operator techniques, we derive a method to place the BFP operator in self-adjoint form. This representation shows that in the asymptotic limit T >> 1, the set of paths with first passage time greate than T are equivalent to the paths on a modified potential energy landscape. The modified energy landscape is related to the dominant eigenfunction of the self-adjoint BFP operator. We demonstrate that this idea is accurate for many systems, even for times T ~ O(1). One such examples is modelled after the Graetz problem, where we are interested in the survival time of a particle diffusing in tube flow. We show that the containment methodology greatly increases the efficiency of simulations where one is interested in only the paths where the particle does not contact the tube surface.