Convergence of the solutions of the discounted equation: the discrete case

TL;DR

This paper studies the convergence behavior of solutions to a discrete discounted equation on compact metric spaces, establishing conditions for uniform convergence to a limit function as the discount factor approaches one.

Contribution

It extends previous continuous results to the discrete setting, proving convergence and characterizing the limit using Discrete Weak KAM theory, Peierls barrier, and Mather measures.

Findings

Existence of a unique constant  for convergence.

Uniform convergence of shifted solutions to a limit function.

Characterization of the limit function via Discrete Weak KAM theory.

Abstract

We derive a discrete version of the results of our previous work. If $M$ is a compact metric space, $c : M\times M \to \mathbb R$ a continuous cost function and $\lambda \in (0,1)$, the unique solution to the discrete $\lambda$-discounted equation is the only function $u_\lambda : M\to \mathbb R$ such that $$\forall x\in M, \quad u_\lambda(x) = \min_{y\in M} \lambda u_\lambda (y) + c(y,x).$$ We prove that there exists a unique constant $\alpha\in \mathbb R$ such that the family of $u_\lambda+\alpha/(1-\lambda)$ is bounded as $\lambda \to 1$ and that for this $\alpha$, the family uniformly converges to a function $u_0 : M\to \mathbb R$ which then verifies $$\forall x\in X, \quad u_0(x) = \min_{y\in X}u_0(y) + c(y,x)+\alpha.$$ The proofs make use of Discrete Weak KAM theory. We also characterize $u_0$ in terms of Peierls barrier and projected Mather measures.