completely in one of the compact sets $K_i$.

    \[assm:b1\]

It can be shown that $V(\nu_1, \nu_2) = 0$ whenever $\nu_1,\nu_2 \in K_i$ for any $1 \leq i \leq l$. We can therefore define $$\begin{aligned}
V(K_i, K_j) \coloneqq \inf\{S_{[0,T]}(\mu|\nu):\nu \in K_i, \mu(T) \in K_j\},\end{aligned}$$ which is interpreted as the minimum cost of going from $K_i$ to $K_j$. We also define the minimum cost of going from $K_i$ to $K_j$ without touching the other compact sets $K_k, k \neq i,j$ by $$\begin{aligned}
\tilde{V}(K_i, K_j) \coloneqq \inf & \{S_{[0,T]}(\mu|\nu):\nu \in K_i, \mu(t) \notin \cup_{k\neq i, j} K_k \\
&  \text{ for all } 0 \leq t \leq T, \mu(T) \in K_j, T>0\}.\end{aligned}$$

Preliminary results
-------------------

It turns out that, under assumption \[assm:b1\], the large time behaviour of the process $\mu_N$ can be studied via a discrete time Markov chain whose state space is the union of small neighbourhoods of the compact sets $K_i, 1 \leq i \leq l$. To study this chain, we introduce some notation. Let $L = \{1,2,\ldots, l\}$. Given $0 < \rho_1 < \rho_0$, let $\gamma_i$ (resp. $\Gamma_i$) denote the $\rho_1$-open neighbourhood (resp. $\rho_0$-open neighbourhood) of $K_i$. Let $\gamma = \cup_{i=1}^l \gamma_i$, $\Gamma = \cup_{i=1}^l \Gamma_i$, and $C =M_1(\mathcal{Z}) \setminus 