Metastability and time scales for parabolic equations with drift 2: the general time scale (2402.07695v3)

Abstract: Consider the elliptic operator given by [ \mathscr{L}\epsilon f=b\cdot\nabla f+\epsilon\Delta f ] for some smooth vector field $b:\mathbb{R}^d\to\mathbb{R}^d$ and $\epsilon>0$, and the initial-valued problem on $\mathbb{R}^d$ [ \left{\begin{aligned}&\partial_t u\epsilon=\mathscr{L}\epsilon u\epsilon,\ &u_\epsilon(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 $\mathscr{L}\epsilon$ has a Gibbs invariant measure of the form $\exp {-U(x)/\epsilon}dx$ 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 $\epsilon\to0$. More precisely, we find the critical time scales $1\ll \theta_\epsilon^{(1)}\ll\cdots\ll \theta_\epsilon^{(q)}$ as $\epsilon\to0$, 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{\epsilon\to0}u_\epsilon(t\theta_\epsilon^{(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 $u_\epsilon$ 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 $\theta_\epsilon^{(q+1)}=\infty$. Our analysis makes essential use of the hierarchical tree structure underlying the metastable behavior in different time-scales of the diffusion induced by $\mathscr{L}_\epsilon$. This result can be regarded as the precise refinement of Freidlin-Wentzell theory which was not known for more than a half century.