of the other particles via the empirical measure of states of all the particles, hence the name mean-field interaction.

We now describe another Markov process associated with the particle system. Consider the mapping $$\begin{aligned}
\left( (X_n^N(t), 1 \leq n \leq N),  t \geq 0\right) \mapsto \left( \frac{1}{N} \sum_{n=1}^N \delta_{X^N_n(t)}, t \geq 0\right) = (\mu_N(t), t \geq 0)\end{aligned}$$ that maps the evolution of the states of all the particles to the empirical measure process $(\mu_N(t), t \geq 0)$. This process is also Markovian with state space $M_1^N(\mathcal{Z})$ – the elements of $M_1(\mathcal{Z})$ that can arise as empirical measures of $N$-particle configurations. Its generator $L^N$ acting on functions $f$ on $M_1^N(\mathcal{Z})$ is given by $$\begin{aligned}
L^N f(\xi) = N \sum_{(z,z^\prime) \in \mathcal{E}} \xi(z) \lambda_{z,z^\prime}(\xi) \left[f\left( \xi + \frac{\delta_{z^\prime}}{N} - \frac{\delta_z}{N}\right) - f(\xi)  \right].\end{aligned}$$

The mean-field limit
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It often turns out that we can study various properties of our particle system by examining its macroscopic behaviour encoded in the process $\mu_N$. Towards this, we often approximate the process $\mu_N$, for large $N$, using a deterministic dynamical system. This approximation is often referred to as the mean-field limit.

Under assumptions \[assm:a1\] and \[assm:a2\], one can show the following law of large numbers [@mckean-67; 