Ergodic problems for viscous Hamilton-Jacobi equations with inward drift

Abstract : In this paper we study the ergodic problem for viscous Hamilton-Jacobi equations with superlinear Hamiltonian and inward drift. We investigate (i) existence and uniqueness of eigenfunctions associated with the generalized principal eigenvalue of the ergodic problem, (ii) relationships with the corresponding stochastic control problem of both finite and infinite time horizon, and (iii) the precise growth exponent of the generalized principal eigenvalue with respect to a perturbation of the potential function.