Stability in determination of states for the mean field game equations

TL;DR

This paper establishes stability results for solutions to linearized mean field game equations with Neumann boundary conditions, demonstrating how solutions can be stably determined from data at the boundary or in subdomains.

Contribution

It provides new stability estimates for mean field game solutions, including Hölder and Lipschitz stability, under specific boundary and subdomain data conditions.

Findings

Hölder stability in time interval (ε, T) from boundary data at T

Lipschitz stability in space from subdomain data over (0,T)

Stability results for solutions with Neumann boundary conditions

Abstract

We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $\Omega \times (0,T)$, where $\Omega$ 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 H\"older stability in time interval $(\epsilon, T)$ with arbitrarily fixed $\epsilon>0$ by data of solutions in $\Omega \times \{T\}$. The second is the Lipschitz stability in $\Omega \times (\epsilon, T-\epsilon)$ by data of solutions in arbitrarily given subdomain of $\Omega$ over $(0,T)$.