The generalized principal eigenvalue for Hamilton–Jacobi–Bellman equations of ergodic type

This paper is concerned with the generalized principal eigenvalue for Hamilton–Jacobi–Bellman (HJB) equations arising in a class of stochastic ergodic control. We give a necessary and sufficient condition so that the generalized principal eigenvalue of an HJB equation coincides with the optimal value of the corresponding ergodic control problem. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function.