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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
Article

Abstract

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.

Keywords

Transition path process Reactive trajectory Stochastic differential equations 

Mathematics Subject Classification

60H10 60H30 

References

  1. 1.
    Athreya, S.R., Barlow, M.T., Bass, R.F., Perkins, E.A.: Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Relat. Fields 123, 484–520 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Athanasopoulos, I., Caffarelli, L.A.: A theorem of real analysis and its application to free boundary problems. Commun. Pure Appl. Math. 38, 499–502 (1985)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Athreya, K.B., Ney, P.: A new approach to the limit thoery of recurrent markov chains. Trans. Am. Math. Soc. 245, 493–501 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bakhtin, Y.: Gumbel distribution in exit problems (2013, preprint) [arXiv:1307.7060]Google Scholar
  5. 5.
    Bauman, P.: Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22, 153–173 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bolhuis, P.G., Chandler, D., Dellago, C., Geissler, P.L.: Transition path sampling: throwing ropes over rough mountain passes, in the dark. Annu. Rev. Phys. Chem. 53, 291–318 (2002)CrossRefGoogle Scholar
  7. 7.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. 6, 399–424 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. 7, 69–99 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Berman, K.A., Konsowa, M.H.: Random paths and cuts, electrical networks, and reversible Markov chains. SIAM J. Discrete Math. 3, 311–319 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cerou, F., Guyader, A., Lelievre, T., Malrieu, F.: On the length of one-dimensional reactive paths (2012, preprint) [arXiv:1206.0949]Google Scholar
  11. 11.
    Caffarelli, L., Salsa, S.: Geometric Approach to Free Boundary Problems. American Mathematical Society, Providence (2005)Google Scholar
  12. 12.
    Dellago, C., Bolhuis, P.G., Geissler, P.L.: Transition path sampling. Adv. Chem. Phys. 123 (2002)Google Scholar
  13. 13.
    Dean, T., Dupuis, P.: The design and analysis of a generalized RESTART/DPR algorithm for rare event simulation. Ann. Oper. Res. 189, 63–102 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    DeBlassie, D.: Uniqueness for diffusions degenerating at the boundary of a smooth bounded set. Ann. Probab. 32, 3167–3190 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Weinan, E., Vanden-Eijnden, E.: Toward a theory of transition paths. J. Stat. Phys. 123, 503–523 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Weinan, E., Vanden-Eijnden, E.: Transition path theory and path-finding algorithms for the study of rare events. Annu. Rev. Phys. Chem. 61, 391–420 (2010)CrossRefGoogle Scholar
  17. 17.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gilberg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)Google Scholar
  19. 19.
    Haussmann, U.G., Pardoux, É.: Time reversal of diffusions. Ann. Probab. 14, 1188–1205 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hairer, M., Stuart, A.M., Voss, J.: Analysis of SPDEs arising in path sampling part II: the nonlinear case. Ann. Appl. Probab. 17, 1657–1706 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hairer, M., Stuart, A.M., Voss, J., Wiberg, P.: Analysis of SPDEs airisng in path sampling part I: the Gaussian case. Commun. Math. Sci. 3, 587–603 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Hummer, G.: From transition paths to transition states and rate coefficients. J. Chem. Phys. 120, 516–523 (2004)CrossRefGoogle Scholar
  23. 23.
    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)zbMATHGoogle Scholar
  24. 24.
    Metzner, P., Schütte, C., Vanden-Eijnden, E.: Illustration of transition path theory on a collection of simple examples. J. Chem. Phys. 125, 084110 (2006)Google Scholar
  25. 25.
    Meyer, P.A., Smythe, R.T., Walsh, J.B.: Birth and death of markov processes. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, pp. 295–305 (1972)Google Scholar
  26. 26.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  27. 27.
    Prinz, J.-H., Held, M., Smith, J.C., Noé, F.: Efficient computation, sensitivity, and error analysis of committor probabilities for complex dynamical processes. Multiscale Model. Simul. 9, 545–567 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Pinsky, R.G.: Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  29. 29.
    Reznikoff, M.G., Vanden-Eijnden, E.: Invariant measures of stochastic partial differential equations and conditioned diffusions. C. R. Acad. Sci. Paris Ser. I 340, 305–308 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  31. 31.
    Stuart, A.M., Voss, J., Wiberg, P.: Conditional path sampling of SDEs and the Langevin MCMC method. Commun. Math. Sci. 2, 685–697 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  33. 33.
    Vanden-Eijnden, E.: Transition path theory (2013, preprint)Google Scholar
  34. 34.
    Veretennikov, A.Yu.: On polynomial mixing bounds for stochastic differential equations. Stochastic Process Appl. 70, 115–127 (1997)Google Scholar
  35. 35.
    Vanden-Eijnden, E., Venturoli, M., Ciccotti, G., Elber, R.: On the assumptions underlying milestoning. J. Chem. Phys. 129, 174102 (2008)CrossRefGoogle Scholar
  36. 36.
    Vanden-Eijnden, E., Weare, J.: Rare event simulation of small noise diffusions. Commun. Pure Appl. Math. 65, 1770–1803 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

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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