the process $\mu_N$ from that of the ODE (\[eqn:MVE\]).

One can also show convergence in the stationary regime. Take for instance the situation when the McKean-Vlasov equation has a unique globally asymptotically stable equilibrium. Since $\mu_N$ is a Markov process on a finite set and is irreducible (by assumption \[assm:a1\]), there exists a unique invariant probability measure for the process $\mu_N$ which we denote by $\wp_N$. (Note that this is a probability measure on $M_1(\mathcal{Z})$.) Let $\xi^* \in M_1(\mathcal{Z})$ denote the unique globally asymptotically stable equilibrium of the McKean-Vlasov equation (\[eqn:MVE\]). One can then show that $\wp_N$ converges weakly to $\delta_{\xi^*}$ as $N \to \infty$, the point mass at $\xi^*$ (see [@benaim-leboudec-08; @bordenave-etal-12]). This also suggests that we can understand the stationary behaviour of the $N$-particle system by examining the limiting behaviour of the deterministic McKean-Vlasov equation, i.e. we can effectively “interchange" the limits $N \to \infty$ and $t \to \infty$ of $\mu_N(t)$ to understand its stationary behaviour, when the McKean-Vlasov equation has a unique globally asymptotically stable equilibrium. Settings when this is not the case are what we study in this paper.

Motivation, contributions, and outline of the paper
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We now describe the questions that we address in 