  The ergodic control problem for a non-degenerate controlled diffusion
controlled through its drift is considered under a uniform stability condition
that ensures the well-posedness of the associated Hamilton-Jacobi-Bellman (HJB)
equation. A nonlinear parabolic evolution equation is then proposed as a
continuous time continuous state space analog of White's `relative value
iteration' algorithm for solving the ergodic dynamic programming equation for
the finite state finite action case. Its convergence to the solution of the HJB
equation is established using the theory of monotone dynamical systems and
also, alternatively, by using the theory of reverse martingales.
