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

Abstract

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

62 pages, 1 figure