Christian Maura | Search arXiv

arXiv2026

Full $Γ-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions

Consider the diffusion process \begin{equation*} dX_ε(t) = \mss b(X_ε(t)) \, dt + \sqrt{2\, ε\, \mss a(X_ε(t))} \, dW_{t}, \end{equation*} on the one-dimensional torus $\bb T = [0,1)$. Here $ε$ is the temperature, $W_{t}$ a Brownian motion on $\bb T$ and $\mss a$, $\mss b$ functions of class $C^{2}(\bb T)$ satisfying further conditions. Denote by $\mss P(\bb T)$ the set of probability measures on $\bb T$ equipped with the weak topology, and by $\ms I_ε\colon \mss P(\bb T)\to [0,+\infty)$ the level two large deviation rate functional of the diffusion $X_ε(\cdot)$. We derive a full $Γ-$expansion of $\ms I_ε$, as $ε\to 0$, expressing it as \begin{equation*} \ms I_ε = \frac{1}ε \;\ms J^{(-1)} \; +\; \ms J^{(0)} \;+\; \sum_{p=1}^{\widehat{\mf q}}\frac{1}{θ^{(p)}_ε}\;\ms J^{(p)}\,, \end{equation*} where $\ms J^{(-1)}$, $\ms J^{(0)}$, $\ms J^{(p)} \colon \mss P(\bb T)\to [0,+\infty]$ represent rate functionals, independent of $ε$, and $θ^{(p)}_ε$ are the time-scales at which the Markov process $X_ε(\cdot)$ exhibits a metastable behaviour.

math.PR↗

arXiv2025

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 $b\colon R \to R$ 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].

math.PR↗