Metastability and time scales for parabolic equations with drift 2: the general time scale

Consider the elliptic operator given by \ L_ f=b f+ f \ for some smooth vector field b: Rᵈᵈ and >0, and the initial-valued problem on Rᵈ \ \aligned&ₜ u_=L_ u_, \\ &u_ (0, \, ) =u₀ (), aligned. \ for some bounded continuous function u₀. Under the hypothesis that the diffusion on Rᵈ induced by L_ has a Gibbs invariant measure of the form \{-U (x) /\dx for some smooth Morse potential function U, we provide the complete characterization of the multi-scale behavior of the solution u_ in the regime 0. More precisely, we find the critical time scales 1 _^ (1) _^ (q) as 0, and the kernels Rₜ^ (p): M₀ M₀_+, where M₀ denotes the set of local minima of U, such that \ ₀u_ (t_^ (p), \, x) =₌' ₌䃐Rₜ^ (p) (m, \, m') u₀ (m'), \ for all t>0 and x in the domain of attraction of m for the dynamical system ẋ (t) =b (x (t) ). We then complete the characterization of the solution u_ by computing the exact asymptotic limit of the solution between time scales _^ (p) and _^ (p+1) for each p, where _^ (0) =1 and _^ (q+1) =. Our proof relies on the full tree-structure characterization of the metastable behavior in different time-scales of the diffusion induced by L_. This result can be regarded as the precise refinement of Freidlin-Wentzell theory which was not known for more than a half century.