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

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## 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 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 The proofs make use of Discrete Weak KAM theory. We also characterize \(u_0\) in terms of Peierls barrier and projected Mather measures.

*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$$\begin{aligned} \forall x\in M, \quad u_\lambda (x) = \min _{y\in M} \lambda u_\lambda (y) + c(y,x). \end{aligned}$$

$$\begin{aligned} \forall x\in X, \quad u_0(x) = \min _{y\in X}u_0(y) + c(y,x)+\alpha . \end{aligned}$$

## Keywords

Cost Function Comparison Principle Discrete Version Jacobi Equation Discrete Case
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## 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

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- 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
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**5**, 366–372 (1980)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Zavidovique, M.: Strict sub-solutions and Mañé potential in discrete weak KAM theory. Comment. Math. Helv.
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