$$\begin{aligned}
S_T(\xi|\nu) \coloneqq \inf \{S_{[0,T]}(\mu|\nu): &\,  \mu(0) = \nu, \mu(T) = \xi,  \mu \in \mathcal{AC}[0,T]\}.\end{aligned}$$ Moreover, the infimum is attained, i.e., there exists a path $\hat{\mu} \in \mathcal{AC}[0,T] $, and a rate matrix $L (t) = (l_{ij}(t), (i,j) \in \mathcal{E})$ such that $$\begin{aligned}
\frac{d\hat{\mu}(t)}{dt} =L^*(t) \hat{\mu}(t),\,  \hat{\mu} (0)=\nu,\,  \hat{\mu} (T) = \xi,\end{aligned}$$ and $S_{[0,T]}(\hat{\mu}|\nu) = S_T(\xi|\nu)$. \[thm:ldp\_terminal\_time\]

Here, $S_T(\xi|\nu)$ can be interpreted as the minimum cost of passage from the profile $\nu$ to the profile $\xi$ in time $T$, among all paths from $\nu$ to $\xi$ in time $T$.

We also have the following *uniform* LDP for the sequence $\{p_{\nu_N}^{(N)}\}_{ N \geq 1}$ (see [[@borkar-sundaresan-12 Corollary 3.1]]{}) when the initial condition is allowed to lie in a compact set.

For any compact set $K \subset M_1(\mathcal{Z})$, any closed set $F \subset D([0,T], M_1(\mathcal{Z}))$, and any open set $G \subset D([0,T], M_1(\mathcal{Z}))$, we have $$\begin{aligned}
\limsup_{N \to \infty} \frac{1}{N} \log \sup_{\nu \in K} p_\nu^{(N)} \{\mu_N \in F\} \leq - \inf_{\nu \in K } \inf_{\mu \in F} S_{[0,T]} (\mu|\nu),\end{aligned}$$ and $$\begin{aligned}
\liminf_{N \to \infty} \frac{1}{N} \log \sup_{\nu \in K} p_\nu^{(N)} \{\mu_N \in G\} \geq - \sup_{\nu \in K} \inf_{\mu \in G} S_{[0,T]} (\mu|\nu).\end{aligned}$$ \[cor:uniform\_ldp\]

For a proof of the above, see [[@dembo-zeitouni 