tools that we use in our paper are similar to those used by Hwang and Sheu, along with the large deviation properties of our process $\mu_N$ over finite time durations, which have been established by L[é]{}onard [@leonard-95] and Borkar and Sundaresan [@borkar-sundaresan-12].

Our study of the large time behaviour of the process $\mu_N$ (i.e., (i)-(iii) above) enables us to show our first main result: there exists a constant $\Lambda \geq 0$ such that, if time is of the order $\exp\{N(\Lambda+O(1))\}$, then the process $\mu_N$ is very close to its invariant measure (see Theorem \[thm:conv\]). Conversely, when $\Lambda > 0$ and when time is of the order $\exp\{N(\Lambda - \delta) \}$ with $\delta > 0$ arbitrarily small, the process may not have reached a neighbourhood of the “most stable" equilibria, with nontrivial probability in the exponential scale (see Theorem \[thm:mixing\]). A similar result for the mean-field discrete-time setting but without the specification of the constant $\Lambda$ was established by Panageas and Vishnoi [@panageas-vishnoi-16]. Let us reemphasise that our setting is a continuous time setting. Further, our desire to identify the constant $\Lambda$ demands that we must study the large deviation asymptotics in some detail in the continuous time setting. The proof 