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

Consider the one-dimensional elliptic operator given by \begin{equation*} (L_εf)(x) ;=; b (x) , f’(x) ,+, ε, a (x), f’'(x) ;, \end{equation*} where the drift $b\text{\hspace{0.17em}⁣}:R\to R$ and the diffusion coefficient $a\text{\hspace{0.17em}⁣}:R\to R$ 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_ε,=,L_ε,u_ε;,\ & u_ε(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 ${\theta }_{\epsilon }^{(1)}\to \mathrm{\infty }$, ${\theta }_{\epsilon }^{(p+1)}\mathrm{/}{\theta }_{\epsilon }^{(p)}\to \mathrm{\infty }$, $1\le p\le \mathfrak{q}-1$, 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_{ε\to0} u_ε(tθ_ε^{(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 ${\varrho }_{\epsilon }\mathrm{/}{\theta }_{\epsilon }^{(p)}\to \mathrm{\infty }$, ${\varrho }_{\epsilon }\mathrm{/}{\theta }_{\epsilon }^{(p+1)}\to 0$ for some $0\le p\le \mathfrak{q}$, where ${\theta }_{\epsilon }^{(0)}=1$, ${\theta }_{\epsilon }^{(\mathfrak{q}+1)}=+\mathrm{\infty }$. 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].