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

Consider the elliptic operator given by \[ L_ε f=b·∇ f+εΔ f \] for some smooth vector field b:Rᵈᵈ and ε>0, and the initial-valued problem on Rᵈ \[ \{{aligned}&∂_t u_ε=L_ε u_ε,\\ &u_ε(0,\,·)=u_0(·), {aligned} . \] for some bounded continuous function u₀. Under the hypothesis that the diffusion on Rᵈ induced by L_ε has a Gibbs invariant measure of the form exp \-U(x)/ε for some smooth Morse potential function U, we provide the complete characterization of the multi-scale behavior of the solution u_ε in the regime ε→0. More precisely, we find the critical time scales 1 θ_ε⁽¹⁾⋯ θ_ε(q) as ε→0, and the kernels Rₜ⁽ᵖ⁾:M₀× M₀₊, where M₀ denotes the set of local minima of U, such that \[ limε→0u_ε(tθ_ε⁽ᵖ⁾,\,x)=∑m'∈ M_0R_t⁽ᵖ⁾(m,\,m')u_0(m'), \] 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_ε by computing the exact asymptotic limit of the solution between time scales θ_ε⁽ᵖ⁾ and θ_ε⁽ᵖ⁺¹⁾ for each p, where θ_ε⁽⁰⁾=1 and θ_ε(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_ε. This result can be regarded as the precise refinement of Freidlin-Wentzell theory which was not known for more than a half century.