  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 special
compatibility conditions. The stress is put on the case when the Hamiltonian is
Lipschitz continuous. Existence and uniqueness are proven.
