From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics

Consider the one-dimensional elliptic operator given by \begin{equation*} (L_\epsilon f)(x) \;=\; b (x) \, f'(x) \,+\, \epsilon\, a (x)\, f''(x) \;, \end{equation*} where the drift $b\colon R \to R$ and the diffusion coefficient $a\colon R \to R$ are periodic $C^1(R)$ functions satisfying further conditions, and $\epsilon>0$. Consider the initial-valued problem \begin{equation*} \left\{ \begin{aligned} & \partial_{t}\,u_{\epsilon}\,=\,L_{\epsilon}\,u_{\epsilon}\;,\\ & u_{\epsilon}(0,\,\cdot)=u_{0}(\cdot)\;, \end{aligned} \right.\end{equation*} for some bounded continuous function $u_{0}$. We prove the existence of time-scales $\theta_{\epsilon}^{(1)},\,\dots,\,\theta_{\epsilon}^{(\mathfrak{q})}$ such that $\theta_{\epsilon}^{(1)}\to\infty$, $\theta_{\epsilon}^{(p+1)}/\theta_{\epsilon}^{(p)}\to\infty$, $1\le p\le\mathfrak{q}-1$, probability measures $p(x,\cdot)$, $x\in R$, and kernels $R_{t}^{(p)}(m_j,m_k)$, where $\{m_j:j\in Z\}$ represents the set of stable equilibrium of the ODE $\dot{x}(t) = b(x(t))$ such that \begin{equation*} \lim_{\epsilon\to0} u_{\epsilon}(t\theta_{\epsilon}^{(p)}, x) \;=\;\sum_{j,k\in Z} p(x,m_j)\, R_{t}^{(p)} (m_j,m_k) \,u_{0}(m_k)\;, \end{equation*} for all $t>0$ and $x\in R$. The solution $u_{\epsilon}$ asymptotic behavior description is completed by the characterisation of its behaviour in the intermediate time-scales $\varrho_{\epsilon}$ such that $\varrho_{\epsilon}/\theta_{\epsilon}^{(p)}\to\infty$, $\varrho_{\epsilon}/\theta_{\epsilon}^{(p+1)}\to0$ for some $0\le p\le\mathfrak{q}$, where $\theta_{\epsilon}^{(0)}=1$, $\theta_{\epsilon}^{(\mathfrak{q}+1)}=+\infty$. The proof relies on the analysis of the diffusion $X_\epsilon(\cdot)$ induced by the generator $L_\epsilon$ based on the resolvent approach to metastability introduced in [21].