On the Solutions of the Problem for a Singular Ergodic Control

This paper discusses an eigenvalue problem for a singular ergodic control. The eigenvalue has a probabilistic interpretation which can be regarded as the least, long‐time averaged (ergodic) cost for a singular control problem. The existence and uniqueness of positive radial solutions of an equation with constraints involving gradient which is related to a stochastic optimal control problem under certain conditions on the nonlinearity of the equation are examined.