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

TL;DR

This paper develops continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs equations, providing insights into their long-term behavior and applications to ergodic problems and singular perturbations.

Contribution

It introduces new continuous dependence estimates for HJBI equations in unbounded domains and applies these to ergodic constants and singular perturbation problems.

Findings

Established continuous dependence estimates for parabolic HJBI equations.

Derived estimates for ergodic constants under periodicity and ellipticity.

Proved local uniform convergence for certain singular perturbation problems.

Abstract

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.