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

Consider the diffusions $d{X}_t^{\epsilon }=b(X_t^{\epsilon })dt+\sqrt{2\epsilon }\sigma (X_t^{\epsilon })d{B}_t$ on ${\mathbb{R}}^d$, where $b$ and $\sigma$ 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 ${\mathcal{I}}_{\epsilon }$ 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 $\mathrm{\Gamma }$-expansion ${\mathcal{I}}_{\epsilon }=\frac{1}{\epsilon }{\mathcal{J}}^{(-1)}+{\mathcal{J}}^{(0)}+{\sum }_{p=1}^{\mathfrak{q}}\frac{1}{{\theta }_{\epsilon }^{(p)}}{\mathcal{J}}^{(p)}$ in the sense of successive $\mathrm{\Gamma }$-limits in the weak topology on probability measures. Here ${\theta }_{\epsilon }^{(p)}$ are the metastable time scales, and each ${\mathcal{J}}^{(p)}$, $p\ge 1$, is the lift of the level-two rate functional of the corresponding finite-state Markov chain. We establish the pre-metastable $\mathrm{\Gamma }$-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.