Fluid-limit trajectories conjecture for stable mean field game equilibria

Let $N$ be the number of players, and let $n^N(t)$ denote the number of players in state $0$ at time $t$ under the unique symmetric Markov perfect equilibrium for the $N+1$-player game. A trajectory $\theta=(\theta_t:0\leq t\leq T)$ is a fluid limit Markov perfect trajectory (FLMP trajectory) if, for some sequence of initial states with $n^N(0)/N\to\theta_0$, the normalized state process converges to $\theta$ in probability uniformly over $0\leq t\leq T$. Let ${\cal T}$ denote the mean field best-response mapping. FLMP trajectory conjecture. The FLMP trajectories are the stable fixed points of the MFG mapping ${\cal T}$. The paper gives evidence for this conjecture after showing that FLMP trajectories are mean field game Nash equilibria when $\eta>0$; identifying which mean field equilibria arise as finite-population limits remains open.