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Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 1021–1034 | Cite as

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

  • Andrea Davini
  • Albert FathiEmail author
  • Renato Iturriaga
  • Maxime Zavidovique
Article

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, \(c : M\times M \rightarrow \mathbb {R}\) a continuous cost function and \(\lambda \in (0,1)\), the unique solution to the discrete \(\lambda \)-discounted equation is the only function \(u_\lambda : M\rightarrow \mathbb {R}\) such that
$$\begin{aligned} \forall x\in M, \quad u_\lambda (x) = \min _{y\in M} \lambda u_\lambda (y) + c(y,x). \end{aligned}$$
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 \(\lambda \rightarrow 1\) and that for this \(\alpha \), the family uniformly converges to a function \(u_0 : M\rightarrow \mathbb {R}\) which then verifies
$$\begin{aligned} \forall x\in X, \quad u_0(x) = \min _{y\in X}u_0(y) + c(y,x)+\alpha . \end{aligned}$$
The proofs make use of Discrete Weak KAM theory. We also characterize \(u_0\) in terms of Peierls barrier and projected Mather measures.

Keywords

Cost Function Comparison Principle Discrete Version Jacobi Equation Discrete Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

Part of this work was done while the first (AD) and second (AF) authors were visiting CIMAT in Guanajuato, that they both wish to thank for its hospitality. The final version was done while the second author was visiting DPMMS, University of Cambridge. The authors thank the anonymous referee for useful comments that helped improve the presentation of the present paper.

References

  1. 1.
    Bernard, P., Buffoni, B.: Weak KAM pairs and Monge-Kantorovich duality. In: Asymptotic Analysis and Singularities–Elliptic and Parabolic PDEs and Related Problems, vol. 47 of Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, pp. 397–420 (2007)Google Scholar
  2. 2.
    Davini, A., Fathi, A., Iturriaga, R., Zavidovique, M.: Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent. Math. (2016). doi: 10.1007/s00222-016-0648-6
  3. 3.
    Kohlberg, E.: Invariant half-lines of nonexpansive piecewise-linear transformations. Math. Oper. Res. 5, 366–372 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zavidovique, M.: Strict sub-solutions and Mañé potential in discrete weak KAM theory. Comment. Math. Helv. 87, 1–39 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Andrea Davini
    • 1
  • Albert Fathi
    • 2
    Email author
  • Renato Iturriaga
    • 3
  • Maxime Zavidovique
    • 4
  1. 1.Dip. di MatematicaSapienza Università di RomaRomeItaly
  2. 2.UMPAENS-Lyon & IUFLyon Cedex 7France
  3. 3.CimatValencianaMexico
  4. 4.IMJ-PRG (Projet Analyse Algébrique)UPMCParis Cedex 5France

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