The Gamma Expansion of the Level Two Large Deviation Rate Functional for Reversible Diffusion Processes

Fix a smooth Morse function $U\text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\to \mathbb{R}$ with finitely many critical points, and consider the solution of the stochastic differential equation $$d{\mathit{x}}_{\epsilon }(t)=-\mathrm{\nabla }U({\mathit{x}}_{\epsilon }(t))dt+\sqrt{2\epsilon }d{\mathit{w}}_t,$$ where $({\mathit{w}}_t{)}_{t\ge 0}$ represents a $d$-dimensional Brownian motion, and $\epsilon >0$ a small parameter. Denote by $\mathcal{P}({\mathbb{R}}^d)$ the space of probability measures on ${\mathbb{R}}^d$, and by ${\mathcal{I}}_{\epsilon }\text{\hspace{0.17em}⁣}:\mathcal{P}({\mathbb{R}}^d)\to [0,\mathrm{\infty }]$ the Donsker–Varadhan level two large deviations rate functional. We express ${\mathcal{I}}_{\epsilon }$ as ${\mathcal{I}}_{\epsilon }={\epsilon }^{-1}{\mathcal{J}}^{(-1)}+{\mathcal{J}}^{(0)}+{\sum }_{1\le p\le \mathfrak{q}}(1\mathrm{/}{\theta }_{\epsilon }^{(p)}){\mathcal{J}}^{(p)}$, where ${\mathcal{J}}^{(p)}\text{\hspace{0.17em}⁣}:\mathcal{P}({\mathbb{R}}^d)\to [0,+\mathrm{\infty }]$ stand for rate functionals independent of $\epsilon$ and ${\theta }_{\epsilon }^{(p)}$ for sequences such that ${\theta }_{\epsilon }^{(1)}\to \mathrm{\infty }$, ${\theta }_{\epsilon }^{(p)}\mathrm{/}{\theta }_{\epsilon }^{(p+1)}\to 0$ for $1\le p<\mathfrak{q}$. The speeds ${\theta }_{\epsilon }^{(p)}$ correspond to the time-scales at which the diffusion ${\mathit{x}}_{\epsilon }(\cdot )$ exhibits a metastable behaviour, while the functional ${\mathcal{J}}^{(p)}$ represent the level two, large deviations rate functionals of the finite-state, continuous-time Markov chains which describe the evolution of the diffusion ${\mathit{x}}_{\epsilon }(\cdot )$ among the wells in the time-scale ${\theta }_{\epsilon }^{(p)}$.