| Abstract: |
| The problem I will present is motivated by the study of housing dynamics inside a city through a Mean Field Game model. Each agent jumps to move from one place to another and minimizes a cost composed of the number of jumps and an increasing function of the density of the population. A Nash equilibrium is of the form of a probability measure on the set of individual trajectories. This probability measure minimizes a problem which depends on the probability measure itself. By using tools from optimal transport, we will see that the latter problem can be expressed in an Eulerian point of view by minimizing the total variation on the set of curves of measures.
The solution to the Eulerian problem exists, is unique and is Lipschitz in time, despite the discontinuous trajectories taken by each agent. With additional hypothesis on the data, boundedness or continuity in space can be obtained with Dirichlet conditions in time.
Numerical simulations are carried out on the Eulerian problem by using a splitting method called Fast Dual Proximal Gradient method for which the convergence of the iterations is guaranteed by Beck and Teboulle in 2014.
The regularity results allows us to show the existence of a Nash equilibrium. |
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