Stabilization of a quasi-linear parabolic Cauchy problem associated with ergodic control of diffusions

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 ℝ^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 C²(ℝ^d) to the solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the ergodic control problem.