Metastability of reversible random walks in potential fields

Let $\mathrm{\Xi }$ be an open and bounded subset of $\text{bb}{R}^d$, and let $F:\mathrm{\Xi }\to \text{bb}R$ be a twice continuously differentiable function. Denote by ${\mathrm{\Xi }}_N$ th discretization of $\mathrm{\Xi }$, ${\mathrm{\Xi }}_N=\mathrm{\Xi }\cap (N^{-1}\text{bb}{Z}^d)$, and denote by $X_N(t)$ the continuous-time, nearest-neighbor, random walk on ${\mathrm{\Xi }}_N$ which jumps from $\text{bs}x$ to $\text{bs}y$ at rate $e^{-(1\mathrm{/}2)N[F(\text{bs}y)-F(\text{bs}x)]}$. We examine in this article the metastable behavior of $X_N(t)$ among the wells of the potential $F$.