Transition Path Theory for Markov Jump Processes

TLDR

Transition path theory (TPT) is formulated for continuous‑time Markov chains on discrete state spaces. This paper develops TPT for Markov chains in detail and relates it to electric resistor network theory and data‑analysis tools such as Laplacian eigenmaps and diffusion maps. Under ergodicity, TPT analyzes reactive trajectories between two subsets and introduces numerical algorithms to compute their statistical properties. TPT yields the probability distribution, current, flux, and dominant pathways of reactive trajectories, and the theory and algorithms are illustrated with examples.

Abstract

The framework of transition path theory (TPT) is developed in the context of continuous-time Markov chains on discrete state-spaces. Under assumption of ergodicity, TPT singles out any two subsets in the state-space and analyzes the statistical properties of the associated reactive trajectories, i.e., those trajectories by which the random walker transits from one subset to another. TPT gives properties such as the probability distribution of the reactive trajectories, their probability current and flux, and their rate of occurrence and the dominant reaction pathways. In this paper the framework of TPT for Markov chains is developed in detail, and the relation of the theory to electric resistor network theory and data analysis tools such as Laplacian eigenmaps and diffusion maps is discussed as well. Various algorithms for the numerical calculation of the various objects in TPT are also introduced. Finally, the theory and the algorithms are illustrated in several examples.