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 $$dX_{t}=-\nabla F(X_{t})\,dt+\sqrt{2\varepsilon }\,dW_{t},\qquad \varepsilon >0,$$ and the spectrum near zero of its generator −L_ɛ≡ɛΔ−∇F⋅∇, where F:ℝ^d→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 ɛ↓0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L_ɛ 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.