Gamma Expansions of Level Two Large Deviation Rate Functionals for Metastable Diffusions

Abstract

Consider the diffusions $dX_t^\varepsilon=b(X_t^\varepsilon)dt+\sqrt{2\varepsilon}σ(X_t^\varepsilon)dB_t$ on $\mathbb{R}^d$, where $b$ and $σ$ are smooth coefficients and $B_t$ is an $r$-dimensional Brownian motion, with $r\le d$. We allow nonreversible dynamics and a degenerate diffusion matrix. Let $\mathcal{I}_\varepsilon$ be the level-two large deviation rate functional of the empirical measure. Under our assumptions, including a complete metastable hierarchy described by resolvent convergence, we establish the full $Γ$-expansion $\mathcal{I}_\varepsilon=\frac{1}{\varepsilon}\mathcal{J}^{(-1)}+\mathcal{J}^{(0)}+\sum_{p=1}^{\mathfrak{q}}\frac{1}{θ_\varepsilon^{(p)}}\mathcal{J}^{(p)}$ in the sense of successive $Γ$-limits in the weak topology on probability measures. Here $θ_\varepsilon^{(p)}$ are the metastable time scales, and each $\mathcal{J}^{(p)}$, $p\ge1$, is the lift of the level-two rate functional of the corresponding finite-state Markov chain. We establish the pre-metastable $Γ$-limits for generalized overdamped, underdamped, and third-order Langevin dynamics. For the generalized overdamped model, existing metastability results yield the full expansion under additional conditions. For the underdamped and third-order models, the full expansion remains conditional on verification of the complete metastable hierarchy.