Sampling constrained stochastic trajectories using Brownian bridges

TL;DR

This paper introduces a novel method for sampling conditioned Langevin trajectories using Brownian bridges, enabling high-accuracy generation of paths ending at specific points or regions, especially effective at low temperatures or for transition paths.

Contribution

The paper develops an exact reformulation of bridge equations into a nonlinear stochastic integro-differential equation and proposes an iterative fixed point method for efficient approximation.

Findings

Method accurately samples conditioned trajectories.

Effective at low temperatures and for transition paths.

Demonstrated performance on simple example problems.

Abstract

We present a new method to sample conditioned trajectories of a system evolving under Langevin dynamics, based on Brownian bridges. The trajectories are conditioned to end at a certain point (or in a certain region) in space. The bridge equations can be recast exactly in the form of a non linear stochastic integro-differential equation. This equation can be very well approximated when the trajectories are closely bundled together in space, i.e. at low temperature, or for transition paths. The approximate equation can be solved iteratively, using a fixed point method. We discuss how to choose the initial trajectories and show some examples of the performance of this method on some simple problems. The method allows to generate conditioned trajectories with a high accuracy.