Renato Iturriaga
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Renato Iturriaga has a total of 29 co-authors. Here are the top co-authors by number of shared publications:
- Héctor Sánchez-Morgado
Universidad Nacional Autónoma de México
9 shared publications - Andrea Davini
Sapienza Università di Roma, Rome, Italy
6 shared publications - Albert Fathi
UMPA, ENS-Lyon & IUF, Lyon Cedex 7, France
5 shared publications - Maxime Zavidovique
IMJ-PRG (projet Analyse Algébrique) , UPMC , 4, place Jussieu, Case 247, 75252 Paris Cedex 5 , France
5 shared publications - Gonzalo Contreras
Centro de Investigación en Matemáticas, Guanajuato, México
4 shared publications - Konstantin Khanin
University of Toronto, Toronto, Canada
3 shared publications
Latest Publications
- 9 July 2026
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 65 (7) , 228
Discretization and vanishing discount problems for first-order mean field games
- 9 July 2026
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 65 (7) , 228
Discretization and vanishing discount problems for first-order mean field games
- Open Access
Discretization and Vanishing Discount Problems for First-order Mean Field Games
This article focuses two issues related to the first-order discounted mean field games system. The first is the time discretization problem. The time discretization approach enables us to prove the existence of solutions (u,m) of the system, where u is a viscosity solution of the discounted Hamilton-Jacobi equation and m is a projected minimizing measure satisfying the continuity equation in the sense of distributions. The second is the vanishing discount problems for both the discounted mean field games system and its discretized system. The methods we use primarily derive from weak KAM theory. Moreover, we provide an example demonstrating the non-uniqueness of solutions to the discounted mean field games system.
Open AccessDiscretization and Vanishing Discount Problems for First-order Mean Field Games
This article focuses two issues related to the first-order discounted mean field games system. The first is the time discretization problem. The time discretization approach enables us to prove the existence of solutions (u,m) of the system, where u is a viscosity solution of the discounted Hamilton-Jacobi equation and m is a projected minimizing measure satisfying the continuity equation in the sense of distributions. The second is the vanishing discount problems for both the discounted mean field games system and its discretized system. The methods we use primarily derive from weak KAM theory. Moreover, we provide an example demonstrating the non-uniqueness of solutions to the discounted mean field games system.
- Open Access
CONVERGENCE OF THE SOLUTIONS OF THE DISCOUNTED EQUATION: THE DISCRETE CASE
We derive a discrete version of the results of our previous work. 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∀x∈M,uλ(x)=miny∈Mλuλ(y)+c(y,x).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∀x∈X,u0(x)=miny∈Xu0(y)+c(y,x)+α.The proofs make use of Discrete Weak KAM theory. We also characterize u0 in terms of Peierls barrier and projected Mather measures.
Open AccessCONVERGENCE OF THE SOLUTIONS OF THE DISCOUNTED EQUATION: THE DISCRETE CASE
We derive a discrete version of the results of our previous work. 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∀x∈M,uλ(x)=miny∈Mλuλ(y)+c(y,x).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∀x∈X,u0(x)=miny∈Xu0(y)+c(y,x)+α.The proofs make use of Discrete Weak KAM theory. We also characterize u0 in terms of Peierls barrier and projected Mather measures.
- 29 March 2021
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- Vol. 9 (4) , 1081-1104
Discrete approximation of the viscous HJ equation
- 29 March 2021
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- Vol. 9 (4) , 1081-1104
Discrete approximation of the viscous HJ equation
- 28 October 2019
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- Vol. 8 (3) , 544-579
Exponential convergence of solutions for random Hamilton–Jacobi equations
- 28 October 2019
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- Vol. 8 (3) , 544-579
Exponential convergence of solutions for random Hamilton–Jacobi equations
- Open Access
- 1 January 2018
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 38 (10) , 4997-5010
Selection of calibrated subaction when temperature goes to zero in the discounted problem
Consider $T(x) = d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, zhongwenzy<\lambda0$ and $\lambda \in (0, 1)$, there exists a unique fixed point $u_{\lambda, \beta} :S^1\to \mathbb{R}$ for the equation $e^{u_{\lambda, \beta}(x)} = \sum_{T(y) = x}e^{\beta A(y) +\lambda u_{\lambda, \beta}(y)}$. It is known that as $\lambda \to 1$ the family $e^{[u_{\lambda, \beta}- \sup u_{\lambda, \beta}]}$ converges uniformly to the main eigenfuntion $\phi_\beta $ for the Ruelle operator associated to $\beta A$. We consider $\lambda = \lambda(\beta)$, $\beta(1-\lambda(\beta))\to+\infty$ and $\lambda(\beta) \to 1$, as $\beta \to\infty$. Under these hypotheses we will show that $\frac{1}{\beta}(u_{\lambda, \beta}-\frac{P(\beta A)}{1-\lambda})$ converges uniformly to the above $V$, as $\beta\to \infty$. The parameter $\beta$ represents the inverse of temperature in Statistical Mechanics and $\beta \to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $\beta \to \infty$.- 1 January 2018
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 38 (10) , 4997-5010
Open AccessSelection of calibrated subaction when temperature goes to zero in the discounted problem
Consider $T(x) = d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, zhongwenzy<\lambda0$ and $\lambda \in (0, 1)$, there exists a unique fixed point $u_{\lambda, \beta} :S^1\to \mathbb{R}$ for the equation $e^{u_{\lambda, \beta}(x)} = \sum_{T(y) = x}e^{\beta A(y) +\lambda u_{\lambda, \beta}(y)}$. It is known that as $\lambda \to 1$ the family $e^{[u_{\lambda, \beta}- \sup u_{\lambda, \beta}]}$ converges uniformly to the main eigenfuntion $\phi_\beta $ for the Ruelle operator associated to $\beta A$. We consider $\lambda = \lambda(\beta)$, $\beta(1-\lambda(\beta))\to+\infty$ and $\lambda(\beta) \to 1$, as $\beta \to\infty$. Under these hypotheses we will show that $\frac{1}{\beta}(u_{\lambda, \beta}-\frac{P(\beta A)}{1-\lambda})$ converges uniformly to the above $V$, as $\beta\to \infty$. The parameter $\beta$ represents the inverse of temperature in Statistical Mechanics and $\beta \to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $\beta \to \infty$. - Open Access
The Lax-Oleinik semi-group on graphs
We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. Weak KAM solutions are viscosity solutions, and in the case of Hamiltonians called of eikonal type in [CS], we prove that the converse holds.
Open AccessThe Lax-Oleinik semi-group on graphs
We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. Weak KAM solutions are viscosity solutions, and in the case of Hamiltonians called of eikonal type in [CS], we prove that the converse holds.
- Open Access
Exponential convergence of solutions for random Hamilton-Jacobi equations
We show that for a family of randomly kicked Hamiton-Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the results in \cite{IK03} and \cite{KZ12}, this completes the program started in \cite{EKMS00} for the multi-dimensional setting.
Open AccessExponential convergence of solutions for random Hamilton-Jacobi equations
We show that for a family of randomly kicked Hamiton-Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the results in \cite{IK03} and \cite{KZ12}, this completes the program started in \cite{EKMS00} for the multi-dimensional setting.
- Open Access
- 1 January 2017
- Published by American Institute of Mathematical Sciences (AIMS) in Networks and Heterogeneous Media
- Vol. 12 (4) , 643-662
The Lax-Oleinik semigroup on graphs
- 1 January 2017
- Published by American Institute of Mathematical Sciences (AIMS) in Networks and Heterogeneous Media
- Vol. 12 (4) , 643-662
Open AccessThe Lax-Oleinik semigroup on graphs
- Open Access
Convergence of the solutions of the discounted equation: the discrete case
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.Open AccessConvergence of the solutions of the discounted equation: the discrete case
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.