Probability Theory and Related Fields

, Volume 161, Issue 1–2, pp 195–244 | Cite as

Reactive trajectories and the transition path process

  • Jianfeng Lu
  • James Nolen


We study the trajectories of a solution \(X_t\) to an Itô stochastic differential equation in \({\mathbb { R}}^d\), 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 \(Y_t\), 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 \(X_t\). 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.


Transition path process Reactive trajectory Stochastic differential equations 

Mathematics Subject Classification

60H10 60H30 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of PhysicsDuke UniversityDurhamUSA

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