Corollary 5.6.15]]{}. Note that, since the space $M_1(\mathcal{Z})$ is compact, we may take $K = M_1(\mathcal{Z})$ in the above corollary.

Large time behaviour {#section:large_time_behaviour}
====================

In the study of large-time behaviour of $\mu_N$, an important role is played by the Freidlin-Wentzell quasipotential $V : M_1(\mathcal{Z}) \times M_1(\mathcal{Z}) \to [0, \infty)$ defined by $$\begin{aligned}
V(\nu,\xi) \coloneqq \inf \{S_{[0,T]}(\mu|\nu):\mu(T)= \xi, T > 0\},\end{aligned}$$ i.e., $V(\nu, \xi)$ denotes the minimum cost of transport from $\nu$ to $\xi$ in an arbitrary but finite time.

We say that $\nu \sim \xi$ ($\nu$ is equivalent to $\xi$) if $V(\nu, \xi) = 0$ and $V(\xi, \nu) = 0$. It is easy to see that $\sim$ defines an equivalence relation on $M_1(\mathcal{Z})$. To study the large time behaviour of the process $\mu_N$, we make the following assumptions on the McKean-Vlasov equation (\[eqn:MVE\]) (see [@freidlin-wentzell Chapter 6, Section 2, Condition A]):

1.  There exists a finite number of compact sets $K_1, K_2, \ldots, K_l$ such that

    -   For each $i = 1,2, \ldots l$, $\nu_1, \nu_2 \in K_i$ implies $\nu_1 \sim \nu_2$.

    -   For each $i \neq j$, $\nu_1 \in K_i$ and $\nu_2 \in K_j$ implies $\nu_1 \nsim \nu_2$.

    -   Every $\omega$-limit set of the dynamical system (\[eqn:MVE\]) lies 