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

TL;DR

This paper develops continuous dependence estimates for the ergodic Bellman problem, providing explicit bounds that are used to analyze the convergence rate in homogenization of Bellman equations, matching known rates in regular cases.

Contribution

It introduces explicit continuous dependence estimates for the ergodic Bellman problem, facilitating the analysis of homogenization convergence rates.

Findings

Explicit bounds on ||v_1-v_2||_ with coefficient dependence

Characterization of constants in estimates under regularity conditions

Rate of convergence in homogenization matching known results in regular cases

Abstract

This paper is devoted to establish continuous dependence estimates for the ergodic problem for Bellman operators (namely, estimates of (v_1-v_2) where v_1 and v_2 solve two equations with different coefficients). We shall obtain an estimate of ||v_1-v_2||_\infty with an explicit dependence on the L^\infty-distance between the coefficients and an explicit characterization of the constants and also, under some regularity conditions, an estimate of ||v_1-v_2||_{C^2(\R^n)}. 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 [11,26] for regular linear problems.