08-AUG-2024
We study a stationary discounted mean field game on the one-dimensional torus $\mathbb{T}=[0,1]$, in which a continuum of identical agents controls a Brownian drift subject to a local congestion cost and a sinusoidal external potential $f(x)=A\sin(2\pi x)$. The equilibrium pair $(V^*,m^*)$ satisfies a coupled Hamilton–Jacobi–Bellman and Kolmogorov–Fokker–Planck system. Lasry–Lions monotonicity guarantees a unique Nash mean field equilibrium. Numerically, Howard policy iteration solves the HJB at each Picard step, while the stationary KFP is handled by a direct adjoint-matrix solve; the outer loop converges in 33 iterations to an equilibrium featuring a bang-bang optimal policy and a unimodal density concentrated in the low-cost region.
