M_1(\mathcal{Z}))$-valued martingale problem for $L^N$, i.e., the law of the empirical measure process $(\mu_N(t), 0 \leq t \leq T)$, and let $p_{\nu_N,T}^{(N)}$ denote the law of the terminal-time empirical measure $\mu_N(T) \in M_1(\mathcal{Z})$, with initial condition $ \mu_N(0) = \nu_N $. Let $\mathcal{AC}[0,T]$ denote the space of absolutely continuous $M_1(\mathcal{Z})$-valued paths on $[0,T]$ (in particular they are differentiable for almost all $t \in [0,T]$; see [[@leonard-95 Definition 3.1]]{}). Define $$\begin{aligned}
\tau^*(u) \coloneqq \left\{
\begin{array}{lll}
\infty & \text{ if } u < -1 \\
1 & \text{ if } u = -1 \\
(u+1) \log (u+1) - u& \text{ if } u > -1,
\end{array}
\right.\end{aligned}$$ which is the Fenchel-Legendre transform of $\tau(u) = e^u-u-1, u \in \mathbb{R}$. We recall the following large deviation principle (LDP) for the sequence $\{p_{\nu_N}^{(N)}\}_{N \geq 1}$ on $D([0,T], M_1(\mathcal{Z}))$ (see [[@leonard-95 Theorem 3.1]]{}, [[@borkar-sundaresan-12 Theorem 3.2]]{}). See [[@dembo-zeitouni Section 1.2]]{} for the definition of LDP and a good rate function.

Suppose that the initial conditions $\nu_N \to \nu$ in $M_1(\mathcal{Z})$. Then the sequence of probability measures $\{p_{\nu_N}^{(N)}, N\geq 1\}$ on the space $D([0,T], M_1(\mathcal{Z}))$ satisfies the LDP with a good rate function $S_{[0,T]}(\mu|\nu)$. Moreover, if $S_{[0,T]}(\mu|\nu) < \infty$, then $\mu \in \mathcal{AC}[0,T],\,  \mu(0) = \nu$ 