HJB + Fokker-Planck system, Nash equilibrium, crowd congestion model (Lasry-Lions 2006)
HJB: −∂ₜu − σ²/2 · Δu + |∇u|²/2 = f(x,m) + r (backward in time)
FPK: ∂ₜm − σ²/2 · Δm − div(m∇u) = 0 (forward in time)
Mean Field Games (Lasry-Lions, Huang-Malhamé-Caines 2006): N→∞ players in Nash equilibrium. Each agent minimizes their cost accounting for the aggregate density m(x,t). Congestion: f(x,m)=g·m (crowded areas more costly). Fixed point: (u,m) solving HJB + FPK simultaneously. Uniqueness under monotonicity: ∫(f(x,m₁)−f(x,m₂))(m₁−m₂)≥0. Numerical: iterative MFG algorithm (Achdou-Capuzzo Dolcetta 2010).