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

Consider the elliptic operator given by $${\mathcal{L}}_{\epsilon }f=b\cdot \mathrm{\nabla }f+\epsilon \mathrm{\Delta }f$$ for some smooth vector field $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and $\epsilon >0$, and the initial-valued problem on ${\mathbb{R}}^d$ $$\{\begin{array}{rl}{\displaystyle }& {\displaystyle {\mathrm{\partial }}_t{u}_{\epsilon }={\mathcal{L}}_{\epsilon }{u}_{\epsilon },}\\ {\displaystyle }& {\displaystyle u_{\epsilon }(0,\cdot )=u_0(\cdot ),}\end{array}$$ for some bounded continuous function $u_0$. Under the hypothesis that the diffusion on ${\mathbb{R}}^d$ induced by ${\mathcal{L}}_{\epsilon }$ has a Gibbs invariant measure of the form $\exp \{-U(x)\mathrm{/}\epsilon \}dx$ for some smooth Morse potential function $U$, we provide the complete characterization of the multi-scale behavior of the solution $u_{\epsilon }$ in the regime $\epsilon \to 0$. More precisely, we find the critical time scales $1\ll {\theta }_{\epsilon }^{(1)}\ll \cdots \ll {\theta }_{\epsilon }^{(q)}$ as $\epsilon \to 0$, 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 $$\underset{\epsilon \to 0}{\lim }{u}_{\epsilon }(t{\theta }_{\epsilon }^{(p)},x)=\sum _{m^{\mathrm{\prime }}\in M_0}{R}_t^{(p)}(m,m^{\mathrm{\prime }})u_0(m^{\mathrm{\prime }}),$$ 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 $u_{\epsilon }$ by computing the exact asymptotic limit of the solution between time scales ${\theta }_{\epsilon }^{(p)}$ and ${\theta }_{\epsilon }^{(p+1)}$ for each $p$, where ${\theta }_{\epsilon }^{(0)}=1$ and ${\theta }_{\epsilon }^{(q+1)}=\mathrm{\infty }$. Our analysis makes essential use of the hierarchical tree structure underlying the metastable behavior in different time-scales of the diffusion induced by ${\mathcal{L}}_{\epsilon }$. This result can be regarded as the precise refinement of Freidlin-Wentzell theory which was not known for more than a half century.