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

Abstract

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 $b:R\to R$ and the diffusion coefficient $a:R\to R$ are periodic $C^1(R)$ functions satisfying further conditions, and $\hat{I}\mu >0$. 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 $\hat{I}{¸}_{\hat{I}}{\mu }^{(1)},\dots ,\hat{I}{¸}_{\hat{I}}{\mu }^{(\mathfrak{q})}$ such that $\hat{I}{¸}_{\hat{I}}{\mu }^{(1)}\to \infty$, $\hat{I}{¸}_{\hat{I}}{\mu }^{(p+1)}/\hat{I}{¸}_{\hat{I}}{\mu }^{(p)}\to \infty$, $1\le p\le \mathfrak{q}-1$, 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_{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 $u_{\hat{I}}\mu$ asymptotic behavior description is completed by the characterisation of its behaviour in the intermediate time-scales ${\varrho }_{\hat{I}}\mu$ such that ${\varrho }_{\hat{I}}\mu /\hat{I}{¸}_{\hat{I}}{\mu }^{(p)}\to \infty$, ${\varrho }_{\hat{I}}\mu /\hat{I}{¸}_{\hat{I}}{\mu }^{(p+1)}\to 0$ for some $0\le p\le \mathfrak{q}$, where $\hat{I}{¸}_{\hat{I}}{\mu }^{(0)}=1$, $\hat{I}{¸}_{\hat{I}}{\mu }^{(\mathfrak{q}+1)}=+\infty$. The proof relies on the analysis of the diffusion $X_{\hat{I}}\mu (\cdot )$ induced by the generator $L_{\hat{I}}\mu$ based on the resolvent approach to metastability introduced in [21].