Joana Mohr

Continuous Time Finite State Mean Field Games

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.

ON THE GENERAL ONE-DIMENSIONAL XY MODEL: POSITIVE AND ZERO TEMPERATURE, SELECTION AND NON-SELECTION

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.

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

We generalize several results of the classical theory of thermodynamic formalism by considering a compact metric space

THE MATHER MEASURE AND A LARGE DEVIATION PRINCIPLE FOR THE ENTROPY PENALIZED METHOD

M

Entropy, pressure and duality for Gibbs plans in Ergodic transport

as the state space. We analyze the shift acting on

Large deviations for quantum spin probabilities at temperature zero

M^{\mathbb{N}}

Thermodynamic Formalism for the General One-Dimensional XY Model: Positive and Zero Temperature

and consider a general a priori probability for defining the transfer (Ruelle) operator. We study potentials

Discrete time, finite state space mean field games

A

Negative Entropy, Zero temperature and stationary Markov chains on the interval.

which can depend on the infinite set of coordinates in

On the general XY Model: positive and zero temperature, selection and non-selection

M^{\mathbb{N}}