We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton–Jacobi–Bellman equation and a forward in time Fokker–Planck equation. The value function u is continuous and satisfies general Kirchhoff conditions at the vertices. The density m of the distribution of states satisfies dual transmission conditions: in particular, m is generally discontinuous across the vertices, and the values of m on each side of the vertices satisfy some compatibility conditions. The stress is put on the case when the Hamiltonian is Lipschitz continuous. Existence, uniqueness and regularity results are proven.

我们考虑状态空间是网络的有限水平随机平均场博弈。它们由一个将时间倒向的汉密尔顿-雅各比-贝尔曼方程和时间向前的福克-普朗克方程耦合的系统来描述。值函数u是连续的，并且满足顶点处的一般Kirchhoff条件。状态分布的密度m满足双重传输条件：特别是，m在整个顶点上通常是不连续的，并且在顶点的每一侧上的m值都满足某些兼容性条件。当哈密顿量为Lipschitz连续时，就施加了应力。存在性，唯一性和规律性结果得到证明。