Pinned Brownian Bridges in the Continuous-Time Limit

TL;DR

This paper critically examines the mathematical foundations of the Ito-Girsanov measure used in modeling pinned Brownian bridges, revealing issues with its convergence and physical validity in chemical transition simulations.

Contribution

It clarifies the formalism behind the IG measure and demonstrates its conditional convergence problems, challenging its use in transition path algorithms.

Findings

IG measure originates from a conditionally convergent expression

Unphysical results occur when using IG measure in algorithms

Highlights need for rigorous mathematical foundation in path sampling

Abstract

The current understanding of pinned Brownian bridges is based on the Onsager-Machlup (OM) functional. The continuous-time limit of the OM functional can be expressed either by using the Fokker-Planck equation or by using the Radon-Nikodym derivative with the help of the Girsanov theorem and Ito's lemma. The resulting expression, called here, the Ito-Girsanov (IG) measure, has been used as a basis of algorithms designed to create ensembles of transition paths, paths that are constrained to start in one free energy basin and end in another. Here we explore the underlying formalism and show that the IG measure originates in an expression that is only conditionally convergent. Thus without a sound mathematical foundation, the IG measure produces unphysical results when used in computer algorithms that are designed to elucidate chemical transitions.