From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918034941149184 |
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| author | Landim, Claudio Maura, Christian |
| author_facet | Landim, Claudio Maura, Christian |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20217 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics Landim, Claudio Maura, Christian Probability Analysis of PDEs 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]. |
| title | From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2505.20217 |