Sampling rare trajectories using stochastic bridges

TL;DR

This paper introduces a sampling method using stochastic bridges to efficiently analyze rare events in stochastic processes, comparing its results with WKB optimal paths and assessing approximation accuracy.

Contribution

The paper presents a novel sampling approach with stochastic bridges that captures full process statistics and evaluates WKB approximation validity at finite noise levels.

Findings

The method effectively focuses on rare events by constraining start and end points.

Produced paths converge to WKB optimal paths as noise decreases.

The approach can assess WKB approximation accuracy at finite noise.

Abstract

The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rate trajectories. Further, we also compare the stochastic bridges produced using our method to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the paths produced using our method, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that our method can be used to judge the accuracy of the WKB approximation at finite levels of noise.