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Probab. Theory Relat. Fields (2015) 161:195244 DOI 10.1007/s00440-014-0547-y
Reactive trajectories and the transition path process
Jianfeng Lu James Nolen
Received: 5 March 2013 / Revised: 18 November 2013 / Published online: 19 January 2014 Springer-Verlag Berlin Heidelberg 2014
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.
Keywords Transition path process Reactive trajectory Stochastic differential
equations
Mathematics Subject Classication 60H10 60H30
We are grateful to Weinan E, Jonathan Mattingly, and Eric Vanden-Eijnden for helpful discussions. The work of JL was supported in part by the Alfred P. Sloan foundation and the National Science Foundation under Grant No. DMS-1312659. The work of JN was supported by NSF Grant DMS-1007572.
J. Lu (B) J. Nolen
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, USA e-mail: [email protected]
J. Nolene-mail: [email protected]
J. LuDepartment of Physics, Duke University, Box 90320, Durham, NC 27708, USA
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1 Introduction
In this article we study solutions Xt
Rd of the It stochastic differential equation
dXt = b(Xt) dt + 2 (Xt) dWt, (1.1)
where (Wt, FWt ) is a standard Brownian motion in
Rd, dened on a probability space
( , F,
P). This diffusion process in Rd has generator
Lu = tr(a2u) + b u,
where a := T is a symmetric matrix. We suppose that (x) is smooth...