Ergodic control of diffusions with random intervention times

We study an ergodic singular control problem with constraint of a regular one-dimensional linear diffusion. The constraint allows the agent to control the diffusion only at jump times of independent Poisson process. Under relatively weak assumptions, we characterize the optimal solution as impulse type control policy, where it is optimal to exert the exact amount of control needed to push the process to a unique threshold. Moreover, we discuss the connection of the present problem to ergodic singular control problems, and finally, illustrate the results with different well-known cost and diffusion structures.