Stability in determination of states for the mean field game equations

We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $\mathrm{\Omega }\times (0,T)$, where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^d$ and $(0,T)$ is the time interval. We prove two kinds of stability results in determining the solutions. The first is Hölder stability in time interval $(\epsilon ,T)$ with arbitrarily fixed $\epsilon >0$ by data of solutions in $\mathrm{\Omega }\times \{T\}$. The second is the Lipschitz stability in $\mathrm{\Omega }\times (\epsilon ,T-\epsilon )$ by data of solutions in arbitrarily given subdomain of $\mathrm{\Omega }$ over $(0,T)$.