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

Abstract

We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent Math, 2016). If M is a compact metric space, c: M× M→ R a continuous cost function and λ∈ (0 , 1) , the unique solution to the discrete λ-discounted equation is the only function uλ: M→ R such that (Formula presented.). We prove that there exists a unique constant α∈ R such that the family of uλ+ α/ (1 - λ) is bounded as λ→ 1 and that for this α, the family uniformly converges to a function u0: M→ R which then verifies (Formula presented.).The proofs make use of Discrete Weak KAM theory. We also characterize u0 in terms of Peierls barrier and projected Mather measures.