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

Abstract: Consider the diffusions $$dX_t<sup>\varepsilon=b(X_t<sup>\varepsilon)dt+\sqrt{2\varepsilon}σ(X_t<sup>\varepsilon)dB_t$$ on $\mathbb{R}<sup>d$, 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 $\mathcal{I}<em>\varepsilon$ 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 $$\mathcal{I}</em>\varepsilon=\frac{1}{\varepsilon}\mathcal{J}<sup>{(-1)}+\mathcal{J}<sup>{(0)}+\sum_{p=1}<sup>{\mathfrak{q}}\frac{1}{θ_\varepsilon<sup>{(p)}}\mathcal{J}<sup>{(p)}$$ in the sense of successive $Γ$-limits in the weak topology on probability measures. Here $θ_\varepsilon<sup>{(p)}$ are the metastable time scales, and each $\mathcal{J}<sup>{(p)}$, $p\ge1$, 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.