Rafael Rigão Souza

Bio: Rafael Rigão Souza is an academic researcher from Universidade Federal do Rio Grande do Sul. The author has contributed to research in topics: Ergodic theory & Compact space. The author has an hindex of 13, co-authored 36 publications receiving 809 citations. Previous affiliations of Rafael Rigão Souza include World Maritime University.

Continuous time finite state mean field games

Abstract: In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study the $N+1$-player problem, which the mean field model attempts to approximate. Our main result is the convergence as $N\to \infty$ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.

Continuous Time Finite State Mean Field Games

Abstract: In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.

Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: positive and zero temperature

Abstract: We generalize several results of the classical theory of thermodynamic formalism by considering a compact metric space by considering a certain class of smooth potentials and we show some properties of the corresponding main eigenfunctions.

On the general one-dimensional xy model: positive and zero temperature, selection and non-selection

Abstract: We consider (M, d) a connected and compact manifold and we denote by the Bernoulli space Mℤ. The analogous problem on the half-line ℕ is also considered. Let be an observable. Given a temperature T, we analyze the main properties of the Gibbs state . In order to do our analysis, we consider the Ruelle operator associated to , and we get in this procedure the main eigenfunction . Later, we analyze selection problems when the temperature goes to zero: (a) existence, or not, of the limit , a question about selection of subactions, and, (b) existence, or not, of the limit , a question about selection of measures. The existence of subactions and other properties of Ergodic Optimization are also considered. The case where the potential depends just on the coordinates (x0, x1) is carefully analyzed. We show, in this case, and under suitable hypotheses, a Large Deviation Principle, when T → 0, graph properties, etc. Finally, we will present in detail a result due to van Enter and Ruszel, where the authors show, for a particular example of potential A, that the selection of measure in this case, does not happen.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Abstract: Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Mean Field Games and Applications

Abstract: This text is inspired from a "Cours Bachelier" held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials developed by the authors. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class. The content of this text is therefore far more important than the actual "Cours Bachelier" conferences, though the guiding principle is the same and consists in a progressive introduction of the concepts, methodologies and mathematical tools of mean field games theory.

Mean Field Games and Applications

Abstract: This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.