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

TL;DR

The paper develops a comprehensive multi-scale metastability framework for a one-dimensional diffusion with periodic coefficients ${\mathscr L}_ε f=b f'+ε a f''$. Using the resolvent approach, it constructs a hierarchy of time scales $\theta_ε^{(p)}$ and coarse-grained well-sets to describe the evolution of $u_ε$ on each scale via reduced Markov dynamics on the wells. Central to the analysis is the reduced resolvent equation $(λ−{\mathfrak L}_p)f= g$, which governs the limit behavior of the diffusion on every level and yields explicit formulas for the limits of $u_ε$ in terms of hitting probabilities ${\mathtt h}_p$, stationary weights $π$, and transition kernels $p_t^{(p)}$; this yields convergence of finite-dimensional distributions of the diffusion to a hierarchical sequence of Markov chains. The results extend prior metastability analyses by accommodating general periodic $a(x)$ and $b(x)$, and by providing a rigorous, constructive description of the full metastable hierarchy and its connection to parabolic asymptotics. The findings have broad implications for understanding long-time diffusion in multi-well landscapes with spatially varying diffusion and drift.

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 $b\colon R \to R$ and the diffusion coefficient $a\colon R \to R$ are periodic $C^1(R)$ functions satisfying further conditions, and $ε>0$. 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 $θ_ε^{(1)},\,\dots,\,θ_ε^{(\mathfrak{q})}$ such that $θ_ε^{(1)}\to\infty$, $θ_ε^{(p+1)}/θ_ε^{(p)}\to\infty$, $1\le p\le\mathfrak{q}-1$, 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 $u_ε$ asymptotic behavior description is completed by the characterisation of its behaviour in the intermediate time-scales $\varrho_ε$ such that $\varrho_ε/θ_ε^{(p)}\to\infty$, $\varrho_ε/θ_ε^{(p+1)}\to0$ for some $0\le p\le\mathfrak{q}$, where $θ_ε^{(0)}=1$, $θ_ε^{(\mathfrak{q}+1)}=+\infty$. The proof relies on the analysis of the diffusion $X_ε(\cdot)$ induced by the generator $L_ε$ based on the resolvent approach to metastability introduced in [21].

Figures (3)

• Figure 1: Here we represent a hierarchy structure with 5 different levels. For the level one, the states ${\mathscr M}_1(k)$, $-2\leq k\leq 4$ are given by ${\mathscr M}_1(k) = \{m_k\}$. Moreover, ${\mathscr M}_1(0)$ and ${\mathscr M}_1(1)$ belong to the same ${\mathbb X}_1(\cdot)-$recurrent class, while the remaining states that appear in the figure are absorbing. For the level 2, we have ${\mathscr M}_2(0) = \{m_0, m_1\}$ and ${\mathscr M}_2(k) = \{m_k\}$, for $k\in \{-2,-1,2,3,4\}$. At this level, ${\mathscr M}_2(0)$ is a ${\mathbb X}_2(\cdot)-$transient state, while the remaining states are absorbing. Turning to level 3, we see the first relabeling. Here ${\textsf{j}}_3=2$ and the metastable states of level 3 that appear in the figure are given by ${\mathscr M}_3(-2)=\{m_{-2}\}$, ${\mathscr M}_3(-1)=\{m_{-1}\}$, ${\mathscr M}_3(0) = \{m_2\}$, ${\mathscr M}_3(1)=\{m_3\}$ and ${\mathscr M}_3(2)=\{m_4\}$. At this level, ${\mathscr M}_3(-2)$ is ${\mathbb X}_3(\cdot)-$transient, ${\mathscr M}_3(-1)$ and ${\mathscr M}_3(2)$ are absorbing, and $\{{\mathscr M}_3(0), {\mathscr M}_3(1)\}$ is a recurrent class. At level 4 we have the states ${\mathscr M}_4(-1) = \{m_{-1}\}$, ${\mathscr M}_4(0) = \{m_2, m_3\}$ and ${\mathscr M}_4(1) = \{m_4\}$. Here ${\textsf{j}}_4 = 2$. At this level, the states ${\mathscr M}_4(-1)$ and ${\mathscr M}_4(1)$ are absorbing, while the class $\{{\mathscr M}_4(0), {\mathscr M}_4(1)\}$ is ${\mathbb X}_4(\cdot)-$recurrent. Finally, at the level 5, the only states appearing in the figure are ${\mathscr M}_5(0) = \{m_{-1},\; m_2,\; m_3\}$ and ${\mathscr M}_5(1) = \{m_4\}$, while ${\textsf{j}}_5=2$. Finally, as ${\mathfrak h}_5 < h_0^{5,-}$, then $R_5({\mathscr M}_5(0), {\mathscr M}_5(1))>0,$ while $R_5({\mathscr M}_5(0), {\mathscr M}_)5(-1)) = 0$. Similarly, as ${\mathfrak h}_5<h_1^{5,+}$, then $R_5({\mathscr M}_5(1), {\mathscr M}_5(0))>0,$ while $R_5({\mathscr M}_5(1), {\mathscr M}_5(2)) = 0$.
• Figure 2: The core and the binding sets.
• Figure 3: Here we illustrate some of the important definitions surrounding the structure of a higher order metastable state. Fixed $p>1$ and $k\in {\mathbb Z}$, the state ${\mathscr M}_p(k)$ is represented in the figure by the cyan circles, while the set $M_p(k)$ is composed by the green circles. The orange circles represent the local minima in between the left-most and right-most maxima separating the states ${\mathscr M}_p(k)$ and ${\mathscr M}_p(k+1)$. As stated in Proposition \ref{['l13']}, the minima contained in $M_p(k)$ are "higher" than the ones contained in ${\mathscr M}_p(k)$, which are all of the same height. Regarding the escape barriers of the well ${\mathscr E}({\mathscr M}_p(k))$, while we have only one global maxima separating ${\mathscr E}({\mathscr M}_p(k))$ and ${\mathscr E}({\mathscr M}_p(k-1))$, there are three between ${\mathscr E}({\mathscr M}_p(k))$ and ${\mathscr E}({\mathscr M}_p(k+1))$. Here, $\sigma_{k-1,k}^{p,+}=\sigma_{k-1,k}^{p,-}$, while $\sigma_{k,k+1}^{p,-}\,<\,\sigma_{k,k+1}^{p,+}$.

Theorems & Definitions (98)

• Theorem 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.8
• Remark 3.1
• Theorem 3.2: Convergence of the finite-dimensional distributions