  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.
