Some ergodic problems for differential games

We present a notion of ergodicity for deterministic zero-sum differential games that extends the one in classical ergodic control theory to systems with two conflicting controllers. We describe its connections with the existence of a constant and uniform long-time limit of the value function of finite horizon games, and characterize this property in terms of Hamilton-Jacobi-Isaacs equations. We also give several sufficient conditions for ergodicity and describe some extensions of the theory to stochastic differential games.