Before revision:  A stochastic control problem optimizing a long-run objective function is an ergodic control problem whose resolution can be achieved by solving an associated  Hamilton-Jacobi-Bellman (HJB) equation having an effective Hamiltonian.
After revision:  A stochastic control problem optimizing a long-run objective function is an ergodic control problem whose resolution can be achieved by solving an associated non-local Hamilton-Jacobi-Bellman (HJB) equation having an effective Hamiltonian.