of our theorem is similar to that of [[@hwang-sheu-90 Theorem 2.1, Part I]]{}, where similar results are established for convergence to the invariant measure for small noise diffusions.

The next question that we want to understand is the asymptotics of the second eigenvalue of the generator $L^N$ of the Markov process $\mu_N=(\mu_N(t), t \geq 0)$ when it is reversible with respect to its invariant measure. This study is motivated by the fact that, for a fixed $N$, the convergence speed of the process $\mu_N$ to its invariant measure (over time) can be understood by studying the modulus of the second eigenvalue of the generator of $\mu_N$. We show that, for the same constant $\Lambda$ above, the modulus of the second eigenvalue scales like $\exp\{-N\Lambda\}$ (see Theorem \[thm:eval\_problem\]). It turns out that $\Lambda$ can be positive only when there are metastable states in the limiting dynamics. In such situations, one expects slower convergence to the invariant measure for large values of $N$. On the other hand, $\Lambda$ can be $0$, for example, when the limiting dynamics (\[eqn:MVE\]) has a unique globally asymptotically stable equilibrium; in this special case, convergence of $\mu_N$ to its invariant measure does not suffer from the slowing 