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

Abstract

Consider the elliptic operator given by \[ \mathscr{L}_εf=b\cdot\nabla f+εΔf \] for some smooth vector field $b:\mathbb{R}^d\to\mathbb{R}^d$ and $ε>0$, and the initial-valued problem on $\mathbb{R}^d$ \[ \left\{\begin{aligned}&\partial_t u_ε=\mathscr{L}_εu_ε,\\ &u_ε(0,\,\cdot)=u_0(\cdot), \end{aligned} \right. \] for some bounded continuous function $u_0$. Under the hypothesis that the diffusion on $\mathbb{R}^d$ induced by $\mathscr{L}_ε$ has a Gibbs invariant measure of the form $\exp \{-U(x)/ε\}dx$ 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 $ε\to0$. More precisely, we find the critical time scales $1\ll θ_ε^{(1)}\ll\cdots\ll θ_ε^{(q)}$ as $ε\to0$, and the kernels $R_t^{(p)}:M_0\times M_0\to\mathbb{R}_+$, where $M_0$ denotes the set of local minima of $U$, such that \[ \lim_{ε\to0}u_ε(tθ_ε^{(p)},\,x)=\sum_{m'\in M_0}R_t^{(p)}(m,\,m')u_0(m'), \] for all $t>0$ and $x$ in the domain of attraction of $m$ for the dynamical system $\dot{x}(t)=b(x(t))$. 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 $θ_ε^{(q+1)}=\infty$. Our analysis makes essential use of the hierarchical tree structure underlying the metastable behavior in different time-scales of the diffusion induced by $\mathscr{L}_ε$. This result can be regarded as the precise refinement of Freidlin-Wentzell theory which was not known for more than a half century.