A convergent discretisation method for transition path theory for diffusion processes

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.

arXiv comment25 pages, 4 figures; minor change in wording, title modification

References in corpus (6)

• Reaction Pathways Based on the Gradient of the Mean First-Passage Time
• Flows in Complex Networks: Theory, Algorithms, and Application to Lennard-Jones Cluster Rearrangement
• Dominant transport pathways in an atmospheric blocking event
• Efficient estimation of rare-event kinetics
• Statistical analysis of the first passage path ensemble of jump processes
• On a degeneracy ratio for bounded convex polytopes