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

Abstract

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+\sqrt2εdW_t,\qquad ε>0, and the spectrum near zero of its generator -L_ε\equiv εΔ-\nabla F\cdot\nabla, where F:R^d\to R 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 ε\downarrow 0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L_ε with eigenvalue which converges to zero exponentially fast in 1/ε. Modulo errors of exponentially small order in 1/εthis 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.