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 quasi-linear parabolic Cauchy initial value problem in $\mathbb{R}^{d}$. We show that this Cauchy problem stabilizes or, in other words, that the solution of the quasi-linear parabolic equation converges for every bounded initial condition in $\mathcal{C}^{2}(\mathbb{R}^{d})$ to the solution of the Hamilton--Jacobi--Bellman equation associated with the ergodic control problem.
