Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem

This paper is devoted to establish continuous dependence estimates for the ergodic problem for Bellman operators (namely, estimates of (v1 −v2 ) where v1 and v2 solve two equations with different coefficients). We shall obtain an estimate of ||v1 − v2||_∞ with an explicit dependence on the L^∞ -distance between the coefficients and an explicit characterization of the constants and also, under some regularity conditions, an estimate of (v1 − v2) in C 2(Rn )-norm . Afterwards, the former result will be crucial in the estimate of the rate of convergence for the homogenization of Bellman equations. In some regular cases, we shall obtain the same rate of convergence established in the monographs by Bensoussan et al. (Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978) and by Jikov et al. (Homogenization of differential operators and integral functionals, Springer, Berlin, 1994) for regular linear problems.