• Anton BovierEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 1970)


In these lectures we will discuss Markov processes with a particular interest for a phenomenon called metastability. Basically this refers to the existence of two or more time-scales over which the system shows very different behaviour: on the short time scale, the systems reaches quickly a “pseudo-equilibrium” and remains effectively in a restricted subset of the available phase space; the particular pseudo-equilibrium that is reached will depend on the initial conditions. However, when observed on the longer time scale, one will occasionally observe transitions from one such pseudo-equilibrium to another one. In many cases (as we will see) there exists one particular time scale for each such pseudo-equilibrium; in other cases of interest, several, or even many, such distinct pseudo-equilibria exist having the same time scale of exit. Mathematically speaking, our interest is to derive the (statistical) properties of the process on these long time scales from the given description of the process on the microscopic time scale. In principle, our aim should be an effective model for the motion at the long time scale on a coarse grained state space; in fact, disregarding fast motion leads us naturally to consider a reduced state space that may be labeled in some way by the quasi-equilibria.

The type of situation we sketched above occurs in many situations in nature. The classical example is of course the phenomenon of metastability in phase transitions: if a (sufficiently pure) container of water is cooled below freezing temperature, it may remain in the liquid state for a rather long period of time, but at some moment the entire container freezes extremely rapidly. In reality, this moment is of course mostly triggered by some slight external perturbation. Another example of the same phenomenon occurs in the dynamics of large bio-molecules, such as proteins. Such molecules frequently have several possible spatial conformations, transitions between which occur sporadically on often very long time scales. Another classical example is metastability in chemical reactions. Here reactants oscillate between several possible chemical compositions, sometimes nicely distinguished by different colours. This example was instrumental in the development of stochastic models for metastability by Eyring, Kramers and others [21, 30]. Today, metastable effects are invoked to explain a variety of diverse phenomena such as changes in global climate systems both on earth (ice-ages) and on Mars (liquid water presence), structural transitions on eco- and oeco systems, to name just a few examples.


Markov Chain Markov Process Invariant Measure Dirichlet Problem Small Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Ben Arous and R. Cerf. Metastability of the three-dimensional Ising model on a torus at very low temperatures. Electron. J. Probab., 1:no. 10 (electronic), 1996.Google Scholar
  2. 2.
    K. A. Berman and M. H. Konsowa. Random paths and cuts, electrical networks, and reversible Markov chains. SIAM J. Discrete Math., 3:311–319, 1990.MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Bovier. Metastability: a potential theoretic approach In (International Congress of Mathematicians. Vol. III), pages 499–518, Eur. Math. Soc., Zürich, 2006.Google Scholar
  4. 4.
    A. Bovier. Metastability and ageing in stochastic dynamics. In Dynamics and randomness II, Nonlinear Phenom. Complex Systems), pages 17–79, Kluwer Acad. Publ., Dordrecht, 2004.CrossRefGoogle Scholar
  5. 5.
    A. Bianchi, A. Bovier, and D. Ioffe. Sharp estimates for metastable exists in the random field curie-weiss model. WIAS-preprint, 2008.Google Scholar
  6. 6.
    A. Bovier, F. den Hollander, and F.R. Nardi. Sharp asymptotics for kawasaki dynamics on a finite box with open boundary conditions. Probab. Theor. Rel. Fields., 135:265–310, 2006.CrossRefGoogle Scholar
  7. 7.
    A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein. Metastability in stochastic dynamics of disordered mean-field models. Probab. Theor. Rel. Fields, 119:99–161, 2001.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein. Metastability and low-lying spectra in reversible Markov chains. Commun. Math. Phys., 228:219–255, 2002.MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein. Metastability in reversible diffusion processes i. sharp asymptotics for capacities and exit times.  J. Europ. Math. Soc. (JEMS), 6:399–424, 2004.MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Bovier, V. Gayrard, and M. Klein. Metastability in reversible diffusion processes ii. precise asymptotics for small eigenvalues. J. Europ. Math. Soc. (JEMS), 7:69–99, 2005.MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Bovier and F. Manzo. Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics. J. Statist. Phys., 107:757–779, 2002.MathSciNetCrossRefGoogle Scholar
  12. 12.
    M. Cassandro, A. Galves, E. Olivieri, and M. E. Vares. Metastable behavior of stochastic dynamics: a path-wise approach. J. Statist. Phys., 35(5-6):603–634, 1984.CrossRefGoogle Scholar
  13. 13.
    O. Catoni and R. Cerf. The exit path of a Markov chain with rare transitions. ESAIM Probab. Statist., 1:95–144 (electronic), 1995/97.MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. Cerf. A new genetic algorithm. Ann. Appl. Probab., 6(3):778–817, 1996.MathSciNetzbMATHGoogle Scholar
  15. 15.
    D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Springer Series in Statistics. Springer-Verlag, New York, 1988.zbMATHGoogle Scholar
  16. 16.
    E.B. Davies. Metastable states of symmetric Markov semi-groups. I. Proc. Lond. Math. Soc. III, Ser., 45:133–150, 1982.CrossRefGoogle Scholar
  17. 17.
    E.B. Davies. Metastable states of symmetric Markov semi-groups II. J. Lond. Math. Soc. II, Ser., 26:541–556, 1982.CrossRefGoogle Scholar
  18. 18.
    E.B. Davies. Spectral properties of metastable Markov semi-groups. J. Funct. Anal., 52:315–329, 1983.MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. V. Day. On the exit law from saddle points. Stochastic Process. Appl., 60(2):287–311, 1995.MathSciNetCrossRefGoogle Scholar
  20. 20.
    W.Th.F. den Hollander. Three lectures on metastability under stochastic dynamics. In this volume. Springer, Berlin, 2008.Google Scholar
  21. 21.
    H. Eyring. The activated complex in chemical reactions. J. Chem. Phys., 3:107–115, 1935.CrossRefGoogle Scholar
  22. 22.
    L. R. Fontes, P. Mathieu, and P. Picco. On the averaged dynamics of the random field Curie-Weiss model. Ann. Appl. Probab., 10(4):1212–1245, 2000.MathSciNetzbMATHGoogle Scholar
  23. 23.
    M. I. Freidlin and A. D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1998.Google Scholar
  24. 24.
    B. Gaveau and L. S. Schulman. Theory of non-equilibrium first-order phase transitions for stochastic dynamics. J. Math. Phys., 39(3):1517–1533,1998.CrossRefGoogle Scholar
  25. 25.
    B. Helffer, M. Klein, and F. Nier. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Mat. Contemp., 26:41–85, 2004.MathSciNetzbMATHGoogle Scholar
  26. 26.
    B. Helffer and F. Nier. Hypo-elliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, volume 1862 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2005.zbMATHGoogle Scholar
  27. 27.
    R. A. Holley, S. Kusuoka, and D. W. Stroock. Asymptotics of the spectral gap with applications to the theory of simulated annealing.J. Funct. Anal., 83(2):333–347, 1989.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Y. Kifer. Random perturbations of dynamical systems: a new approach. In Mathematics of random media (Blacksburg, VA, 1989), volume 27 of Lectures in Appl. Math., pages 163–173. Amer. Math. Soc., Providence, RI, 1991.Google Scholar
  29. 29.
    Vassili N. Kolokoltsov. Semiclassical analysis for diffusions and stochastic processes, volume 1724 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2000.Google Scholar
  30. 30.
    H.A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7:284–304, 1940.MathSciNetCrossRefGoogle Scholar
  31. 31.
    P. Mathieu. Spectra, exit times and long times asymptotics in the zero white noise limit. Stoch. Stoch. Rep., 55:1–20, 1995.MathSciNetCrossRefGoogle Scholar
  32. 32.
    P. Mathieu and P. Picco. Metastability and convergence to equilibrium for the random field Curie-Weiss model. J. Statist. Phys., 91(3-4):679–732, 1998.MathSciNetCrossRefGoogle Scholar
  33. 33.
    L. Miclo. Comportement de spectres d'opérateurs de Schrödinger à basse température. Bull. Sci. Math., 119(6):529–553, 1995.MathSciNetzbMATHGoogle Scholar
  34. 34.
    F. Nier. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. In Journées “Équations aux Dérivées Partielles”, pages Exp. No. VIII, 17. École Polytech., Palaiseau, 2004.Google Scholar
  35. 35.
    E. Olivieri and E. Scoppola. Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case. J. Statist. Phys., 79(3-4):613–647, 1995.MathSciNetCrossRefGoogle Scholar
  36. 36.
    E. Olivieri and E. Scoppola. Markov chains with exponentially small transition probabilities: first exit problem from a general domain. II. The general case. J. Statist. Phys., 84(5-6):987–1041, 1996.MathSciNetCrossRefGoogle Scholar
  37. 37.
    R. H. Schonmann and S. B. Shlosman. Wulff droplets and the metastable relaxation of kinetic Ising models. Comm. Math. Phys., 194(2):389–462, 1998.MathSciNetCrossRefGoogle Scholar
  38. 38.
    E. Scoppola. Renormalization and graph methods for Markov chains. In Advances in dynamical systems and quantum physics (Capri, 1993), pages 260–281. World Sci. Publishing, River Edge, NJ, 1995.Google Scholar
  39. 39.
    A. D. Ventcel'. The asymptotic behavior of the largest eigenvalue of a second order elliptic differential operator with a small parameter multiplying the highest derivatives. Dokl. Akad. Nauk SSSR, 202:19–22, 1972.MathSciNetGoogle Scholar
  40. 40.
    A. D. Ventcel'. Formulas for eigenfunctions and eigenmeasures that are connected with a Markov process. Teor. Verojatnost. i Primenen., 18:3–29, 1973.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Institut f¨ur Angewandte MathematikRheinische Friedrich-Wilhelms-Universit¨at BonnGermany

Personalised recommendations