and there exists a family of rate matrices $L(t), 0 \leq t \leq T$ such that $\mu$ is the unique solution to $$\begin{aligned}
\dot{\mu}(t) = L(t)^* \mu(t), \, 0 \leq t \leq T, \, \mu(0) = \nu,\end{aligned}$$ and $$\begin{aligned}
& S_{[0,T]}(\mu|\nu) =  \int_{[0,T]} \sum_{(i,j)\in \mathcal{E}} \mu(t)(i) \lambda_{i,j}(\mu(t)) \tau^*\left( \frac{l_{i,j}(t)}{\lambda_{i,j}(\mu(t))} -1 \right) dt.\end{aligned}$$ \[thm:finite-duration-ldp\]

We can interpret the rate function $S_{[0,T]}$ as follows. Starting at $\nu_N$, the process $\mu_N$ is likely to be in the neighbourhood of the solution to the McKean-Vlasov equation (\[eqn:MVE\]) with initial condition $\nu$ (with very high probability). In order for the process $\mu_N$ to be in the neighbourhood of some other path, we need to apply a control given by the rate matrix $L$; $S_{[0,T]}(\mu|\nu)$ is the cost of this control. In particular, since the solution to the McKean-Vlasov equation starting at $\nu$ has zero-cost (i.e. $S_{[0,T]}(\mu_{\nu}|\nu) = 0$ where $\mu_\nu$ denotes the solution to (\[eqn:MVE\]) starting at $\nu$), the limiting behaviour that $\mu_N(\cdot) \xrightarrow{P} \mu_\nu(\cdot)$ in $D([0,T],M_1(\mathcal{Z}))$ as $N \to  \infty$ follows.

Here is an outline of the proof of Theorem \[thm:finite-duration-ldp\]: one looks at a system of non-interacting particles where the transition rates of a particle do not depend on the empirical measure, and 