We study the relative value iteration for the ergodic control problem under a
near-monotone running cost structure for a nondegenerate diffusion controlled
through its drift. This algorithm takes the form of a quasilinear parabolic
Cauchy initial value problem in \RR^{d}. We show that this Cauchy problem
stabilizes, or in other words, that the solution of the quasilinear parabolic
equation converges for every bounded initial condition in \Cc^{2}(\RR^{d}) to
the solution of the Hamilton--Jacobi--Bellman (HJB) equation associated with
the ergodic control problem
