Reactive trajectories and the transition path process

Abstract

© 2014, Springer-Verlag Berlin Heidelberg.We study the trajectories of a solution (formula presented) to an Itô stochastic differential equation in (formula presented), as the process passes between two disjoint open sets, (formula presented) and (formula presented). These segments of the trajectory are called transition paths or reactive trajectories, and they are of interest in the study of chemical reactions and thermally activated processes. In that context, the sets (formula presented) and (formula presented) represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process (formula presented), which is a strong solution to an auxiliary SDE having a singular drift term. We also show that statistics of the transition path process may be recovered by empirical sampling of the original process (formula presented). As an application of these ideas, we prove various representation formulas for statistics of the transition paths. We also identify the density and current of transition paths. Our results fit into the framework of the transition path theory by Weinan and Vanden-Eijnden.