The Annals of Probability

Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime

Michael Eckhoff

Full-text: Open access


We investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation

\[dX_{t}=-\nabla F(X_{t})\,dt+\sqrt{2\varepsilon }\,dW_{t},\qquad \varepsilon >0,\]

and the spectrum near zero of its generator −Lɛ≡ɛΔ−∇F⋅∇, where F:ℝd→ℝ and W denotes Brownian motion on ℝd. For generic F to each local minimum of F there corresponds a metastable state. We prove that the distribution of its rescaled relaxation time converges to the exponential distribution as ɛ↓0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of Lɛ with eigenvalue which converges to zero exponentially fast in 1/ɛ. Modulo errors of exponentially small order in 1/ɛ this eigenvalue is given as the inverse of the expected metastable relaxation time. The eigenstate is highly concentrated in the basin of attraction of the corresponding trap.

Article information

Ann. Probab., Volume 33, Number 1 (2005), 244-299.

First available in Project Euclid: 11 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 31C15: Potentials and capacities 31C05: Harmonic, subharmonic, superharmonic functions 35P15: Estimation of eigenvalues, upper and lower bounds 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 58J37: Perturbations; asymptotics 60F10: Large deviations 60F05: Central limit and other weak theorems

Capacity eigenvalue problem exit problem exponential distribution diffusion process ground-state splitting large deviations metastability relaxation time reversibility potential theory Perron–Frobenius eigenvalues semiclassical limit Witten’s Laplace


Eckhoff, Michael. Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime. Ann. Probab. 33 (2005), no. 1, 244--299. doi:10.1214/009117904000000991.

Export citation


  • Agmon, S. (1982). Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schroedinger Operators. Princeton Univ. Press.
  • Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2001). Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Related Fields 119 99--161.
  • Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002). Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228 219--255.
  • Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002). Metastability in reversible diffusion processes I. Sharp asymptotics of capacities and exit times. Preprint.
  • Buslov, V. A. and Makarov, K. A. (1988). A time scale hierarchy with small diffusion. Teoret. Mat. Fiz. 76 219--230.
  • Buslov, V. A. and Makarov, K. A. (1992). Life spans and least eigenvalues of an operator of small diffusion. Mat. Zametki 51 20--31.
  • Cassandro, M., Galves, A., Olivieri, E. and Varés, M. E. (1988). Metastable behavior of stochastic dynamics: A pathwise approach. J. Statist. Phys. 35 603--634.
  • Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B. (1987). Schroedinger Operators. Springer, Berlin.
  • Davies, E. B. (1982). Metastable states of symmetric Markov semigroups I. Proc. London Math. Soc. (3) 45 133--150.
  • Davies, E. B. (1982). Metastable states of symmetric Markov semigroups II. J. London Math. Soc. (2) 26 541--556.
  • Davies, E. B. (1983). Spectral properties of metastable Markov semigroups. J. Funct. Anal. 52 315--329.
  • Davies, E. B. (1995). Spectral Theory and Differential Operators. Cambridge Univ. Press.
  • Day, M. V. (1983). On the exponential exit law in the small parameter exit problem. Stochastics 8 297--323.
  • Devinatz, A. and Friedman, A. (1978). Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem. Indiana Univ. Math. J. 27 143--157.
  • Donsker, D. and Varadhan, S. R. S. (1976). On the principal eigenvalue of second-order elliptic differential operators. Comm. Pure Appl. Math. 29 595--621.
  • Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin.
  • Eckhoff, M. (2000). Capacity and low lying spectra of attractive Markov chains. Ph.D. dissertation, Univ. Potsdam.
  • Eckhoff, M. (2002). The low lying spectrum of irreversible Markov chains with infinite state space. Preprint.
  • Fleming, W. H. and James, M. R. (1992). Asymptotic series and exit time probabilities. Ann. Probab. 20 1369--1384.
  • Freidlin, M. I. and Wentzell, A. D. (1996). Random Perturbations of Dynamical Systems. Springer, Berlin.
  • Friedman, A. (1973). Asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives. Indiana Univ. Math. J. 22 1005--1015.
  • Gaveau, B. and Moreau, M. (1998). Metastable relaxation times and absorption probabilities for multidimensional systems. J. Math. Phys. 39 1517--1533.
  • Gaveau, B. and Schulman, L. S. (2000). Theory of nonequilibrium first-order phase transitions for stochastic dynamics. J. Phys. 33 4837--4850.
  • Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. Springer, Berlin.
  • Helffer, B. (1980). Semi-Classical Analysis for the Schrödinger Operator and Applications. Springer, Berlin.
  • Helffer, B. and Sjöstrand, J. (1984). Multiple wells in the semi-classical limit. I. Comm. Partial Differential Equations 9 337--408.
  • Helffer, B. and Sjöstrand, J. (1985). Puits multiples en limite semi-classique. II: Interaction moleculaire. Symetries. Perturbation. Ann. Inst. H. Poincaré Phys. Théor. 42 127--212.
  • Jackson, J. D. (1975). Classical Electrodynamics. Wiley, New York.
  • Kipnis, C. and Newmann, C. M. (1985). The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes. SIAM J. Appl. Math. 45 972--982.
  • Knessl, C., Matkowsky, B. J., Schuss, Z. and Tier, C. (1985). An asymptotic theory of large deviations for Markov jump processes. SIAM J. Appl. Math. 46 1006--1028.
  • Kolokoltsov, V. N. (2000). Semiclassical Analysis for Diffusions and Stochastic Processes. Springer, Berlin.
  • Kolokoltsov, V. N. and Makarov, K. A. (1996). Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion process. Russian J. Math. Phys. 4 341--360.
  • Martinelli, F. and Scoppola, E. (1988). Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition. Comm. Math. Phys. 120 25--69.
  • Miclo, L. (1995). Comportement de spectres d'operateurs de Schroedinger a basse temperature. Bull. Sci. Math. 119 529--553.
  • Pinchover, Y. (1996). On positivity, criticality and the spectral radius of the Shuttle operator for elliptic operators. Duke Math. J. 85 431--445.
  • Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Univ. Press.
  • Reed, M. and Simon, B. (1970). Analysis of Operators. Academic Press, New York.
  • Schuss, Z. (1980). Theory and Applications of Stochastic Differential Equations. Wiley, New York.
  • Simon, B. (1984). Semiclassical analysis of low lying eigenvalues, II. Tunneling. Ann. of Math. 120 89--118.
  • Wentzell, A. D. (1972). On the asymptotic of the greatest eigenvalue of a second-order elliptic differential operator with a small parameter in the higher derivatives. Soviet Math. Dokl. 12 13--18.
  • Wentzell, A. D. (1973). Formulas for eigenfunctions and eigenmeasures that are connected with a Markov process. Theory Probab. Appl. 18 3--29.
  • Whittaker, E. T. and Watson, G. N. (1958). A Course of Modern Analysis. Cambridge Univ. Press.
  • Williams, M. (1982). Asymptotic one-dimensional exit time distributions. SIAM J. Appl. Math. 42 149--154.
  • Zhao, Z. (1992). Subcriticality and gaugeability of the Schroedinger operator. Trans. Amer. Math. Soc. 334 75--96.