Continuous dependence estimates for large time behavior for Bellman-Isaacs equations and applications to the ergodic problem

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators (briefly, HJBI). For the parabolic Cauchy problem, we establish such an estimate in the whole space $[0,+\infty)\times\Rn$. Moreover, under some periodicity and ellipticity assumptions, we obtain a similar estimate for the ergodic constant associated to the HJBI operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.