I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite state space $Z = \{0, 1, . . . , r − 1\}$. The transition rate for a particle from state $i$ to state $j$ is governed by mean field dynamics: the transition rate is $λ_{i,j}(\mu_N (t))$ where $\mu_N (t)$ is the empirical distribution of the states of particles at time t:
\begin{align}
\mu_N(t)=\sum_{i=1}^N\delta_{x_i}
\end{align}
The particles interact only through the dependence of their transition rates on the current empirical measure $\mu_N (t)$ and therefore each particle $X_n^N(t)$ is a continuous inhomogeneous-time Markov chain with state-space $Z$.
The authors of the paper claim, without proving that, that the family $(\mu_N , N \geq 1)$ satisfies the weak law of large numbers in the following sense: if $\mu_N(0)\rightarrow\nu$ weakly as $N\rightarrow\infty$ for some $\nu\in\mathcal{M}_1(Z)$, then $\mu_N\rightarrow\mu$ uniformly on compacts in probability, where $\mu$ solves the McKean-Vlasov equation \begin{align} \dot{\mu}(t) = A^* \mu(t) \end{align} with initial condition $\mu(0) =\nu$. Does anybody knows how to prove this claim or provide some references?
Thanks!
Reference: Vivek S. Borkar, Rajesh Sundaresan (2012) Asymptotics of the Invariant Measure in Mean Field Models with Jumps. Stochastic Systems 2(2):322-380. https://doi.org/10.1287/12-SSY064
