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Abstract
We study the trajectories of a solution Xt to an It stochastic differential equation in Rd, as the process passes between two disjoint open sets, A and B. 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 A and B represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process Yt, 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 Xt. 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 t into the framework of the transition path theory by Weinan and Vanden-Eijnden.