PRECISE ASYMPTOTICS OF SMALL EIGENVALUES OF REVERSIBLE DIFFUSIONS IN THE METASTABLE REGIME

We investigate the close connection between metastability of the reversible diu- sion process X dened and W denotes Brownian motion on R d . 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 # 0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L with eigenvalue which converges to zero exponentially fast in 1= . Modulo errors of exponentially small order in 1= 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.

[1]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[2]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[3]  Mirosław Krzyżański.  Partial differential equations of second order , 1971 .

[4]  S. Varadhan,et al.  On the principal eigenvalue of second‐order elliptic differential operators , 1976 .

[5]  Avner Friedman,et al.  Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem , 1978 .

[6]  Zeev Schuss,et al.  Theory and Applications of Stochastic Differential Equations , 1980 .

[7]  E. Davies,et al.  Metastable States of Symmetric Markov Semigroups II , 1982 .

[8]  Shmuel Agmon,et al.  Lectures on exponential decay of solutions of second order elliptic equations : bounds on eigenfunctions of N-body Schrödinger operators , 1983 .

[9]  E. Davies,et al.  Metastable States of Symmetric Markov Semigroups I , 1982 .

[10]  E. Davies.  Spectral properties of metastable Markov semigroups , 1983 .

[11]  M. V. Day,et al.  On the exponential exit law in the small parameter exit problem , 1983 .

[12]  J. Doob.  Classical potential theory and its probabilistic counterpart , 1984 .

[13]  Barry Simon,et al.  Semiclassical analysis of low lying eigenvalues, II. Tunneling* , 1984 .

[14]  Bernard Helffer,et al.  Multiple wells in the semi-classical limit I , 1984 .

[15]  Antonio Galves,et al.  Metastable behavior of stochastic dynamics: A pathwise approach , 1984 .

[16]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[17]  S. R. S. Varadhan.  RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS (Grundlehren der mathematischen Wissenschaften, 260) , 1985 .

[18]  C. Knessl,et al.  An Asymptotic Theory of Large Deviations for Markov Jump Processes , 1985 .

[19]  B. Helffer,et al.  Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation , 1985 .

[20]  Charles M. Newman,et al.  The Metastable Behavior of Infrequently Observed, Weakly Random, One-Dimensional Diffusion Processes , 1985 .

[21]  M. Brelot.  Classical potential theory and its probabilistic counterpart , 1986 .

[22]  Fabio Martinelli,et al.  Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition , 1988 .

[23]  Bernard Helffer,et al.  Semi-Classical Analysis for the Schrödinger Operator and Applications , 1988 .

[24]  M. R. James,et al.  Asymptotic Series and Exit Time Probabilities , 1992 .

[25]  Subcriticality and gaugeability of the Schrodinger operator , 1992 .

[26]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[27]  Laurent Miclo.  Comportement de spectres d'opérateurs de Schrödinger à basse température , 1995 .

[28]  R. Pinsky.  Positive Harmonic Functions and Diffusion: References , 1995 .

[29]  Y. Pinchover.  On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators , 1996 .

[30]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[31]  B. Gaveau,et al.  Theory of nonequilibrium first-order phase transitions for stochastic dynamics , 1998 .

[32]  A. Bovier,et al.  Metastability in stochastic dynamics of disordered mean-field models , 1998, cond-mat/9811331.

[33]  B. Gaveau,et al.  Metastable relaxation times and absorption probabilities for multidimensional stochastic systems , 2000 .

[34]  A. Bovier,et al.  Metastability and Low Lying Spectra¶in Reversible Markov Chains , 2000, math/0007160.

[35]  V. Kolokoltsov.  Semiclassical Analysis for Diffusions and Stochastic Processes , 2000 .

[36]  A. Bovier,et al.  Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times , 2004 .

[37]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.