  We study stochastic Mean Field Games on networks with sticky transition
conditions. In this setting, the diffusion process governing the agent's
dynamics can spend finite time both in the interior of the edges and at the
vertices. The corresponding generator is subject to limitations concerning
second-order derivatives and the invariant measure breaks down into a
combination of an absolutely continuous measure within the edges and a sum of
Dirac measures positioned at the vertices. Additionally, the value function,
solution to the Hamilton-Jacobi-Bellman equation, satisfies generalized
Kirchhoff conditions at the vertices.
