# A convergent discretisation method for transition path theory for diffusion processes

## TL;DR

This paper introduces a Monte Carlo discretisation method for transition path theory in diffusion processes that avoids Markovian assumptions, providing rigorous error bounds and convergence guarantees based on Voronoi tessellations.

## Contribution

The authors develop a novel Monte Carlo approach for TPT that does not rely on Markovian approximations and establish theoretical error bounds and convergence results.

## Key findings

- Method accurately approximates TPT objects from trajectory data.
- Error bounds are rigorously proven and demonstrate convergence.
- Application to a triple-well potential illustrates the method's effectiveness.

## Abstract

Transition path theory (TPT) for diffusion processes is a framework for analysing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretisation of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretisation. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.