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

Abstract

Fix a smooth Morse function $U\colon \mathbb{R}^{d}\to\mathbb{R}$ with finitely many critical points, and consider the solution of the stochastic differential equation \[ d\boldsymbol{x}_{\epsilon}(t)=-\nabla U(\boldsymbol{x}_{\epsilon}(t))\,dt \,+\,\sqrt{2\epsilon}\, d\boldsymbol{w}_{t}\,, \] where $(\boldsymbol{w}_{t})_{t\ge0}$ 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 $\mathcal{I}_{\epsilon} \colon \mathcal{P}(\mathbb{R}^{d})\to[0,\,\infty]$ the Donsker--Varadhan level two large deviations rate functional. We express $\mathcal{I}_\epsilon$ as $\mathcal{I}_\epsilon = \epsilon^{-1} \mathcal{J}^{(-1)} + \mathcal{J}^{(0)} + \sum_{1\le p\le \mathfrak{q}} (1/\theta^{(p)}_\epsilon) \, \mathcal{J}^{(p)}$, where $\mathcal{J}^{(p)}\colon \mathcal{P}(\mathbb{R}^d) \to [0,+\infty]$ stand for rate functionals independent of $\epsilon$ and $\theta^{(p)}_\epsilon$ for sequences such that $\theta^{(1)}_\epsilon \to\infty$, $\theta^{(p)}_\epsilon / \theta^{(p+1)}_\epsilon \to 0$ for $1\le p< \mathfrak{q}$. The speeds $\theta^{(p)}_\epsilon$ correspond to the time-scales at which the diffusion $\boldsymbol{x}_{\epsilon}(\cdot)$ 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 $\boldsymbol{x}_{\epsilon}(\cdot)$ among the wells in the time-scale $\theta^{(p)}_\epsilon$.