the sojourn times in the basin of attraction of an $\omega$-limit set are of the order $\exp\{O(N)\}$, as we shall soon see. One of the aims of our paper is to quantify such metastable phenomena, namely

1.  the mean time spent by the process near an $\omega$-limit set,

2.  the probability of reaching a given $\omega$-limit set before reaching another one,

3.  the probability of traversing a given set of $\omega$-limit sets in a particular order,

and so on. These quantifications help predict the performance of engineering systems, some of which we will describe in Section \[section:examples\]. We study the aforementioned metastability questions in Section \[section:large\_time\_behaviour\]. Such large time phenomena for diffusion processes with a small noise parameter have been studied in the past by Freidlin and Wentzell [@freidlin-wentzell] under the “general position condition" (see [[@freidlin-wentzell Sections 6.4-6.6]]{}). Hwang and Sheu [@hwang-sheu-90] studied large time behaviour for diffusion processes under a more general setup. The key in both these works is the large deviation properties of the small noise diffusion processes over finite time durations, which have been established in [[@freidlin-wentzell Chapter 5]]{}. In this paper, we extend the analysis to Markov mean-field jump processes, specifically $(\mu_N(\cdot))_{N \geq 1}$. The 