Consider a diffusion $X_t$ in an energy landscape $U(x)$, i.e., a solution to the stochastic differential equation $d X_t = - \nabla U(X_t) d t + \sqrt{\eps} d B_t$. In the small-noise limit $\eps \searrow 0$, the diffusion started in a local energy minimum exhibits metastable behavior - it takes a long time to reach the global minimum. The answer to the question "how long" is provided by the Eyring-Kramers law. The talk addresses similar questions for a Markov birth and death process of points in $\mathbb R^d$, where the energy landscape is replaced with the rate function of some suitable large deviations principle and the analogue of the Eyring-Kramers law brings in functional central limit theorems and infinite-dimensional Gaussians. Based on joint work in progress with Frank den Hollander, Roman Kotecky, and Elena Pulvirenti.