# Convergence of the solutions of discounted Hamilton--Jacobi systems

## TL;DR

This paper proves that solutions to a weakly coupled system of discounted Hamilton--Jacobi equations on a Riemannian manifold converge to a specific limit as the discount factor approaches zero, using generalized Mather measures and random representations.

## Contribution

It introduces a generalized Mather measure framework for Hamilton--Jacobi systems and establishes convergence results for discounted solutions on manifolds.

## Key findings

- Solutions converge to a specific limit as discount factor approaches zero
- Generalized Mather measures are used to analyze the system
- Random representation formulas facilitate the convergence proof

## Abstract

We consider a weakly coupled system of discounted Hamilton--Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to zero. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton--Jacobi systems and on suitable random representation formulae for the discounted solutions.