Ergodic control of random singular diffusions

We study an ergodic control problem of random singular diffusions. This is a typical hybrid system which arises in numerous applications such as fault tolerant control systems, flexible manufacturing systems etc. Under a certain Liapunov type stability condition we establish the existence of an optimal control. We then study the corresponding Hamilton-Jacobi-Bellman equation and establish the existence of a unique viscosity solution in a certain class