437 research outputs found
Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games
Mean field type models describing the limiting behavior of stochastic
differential games as the number of players tends to +, have been
recently introduced by J-M. Lasry and P-L. Lions. Under suitable assumptions,
they lead to a system of two coupled partial differential equations, a forward
Bellman equation and a backward Fokker-Planck equations. Finite difference
schemes for the approximation of such systems have been proposed in previous
works. Here, we prove the convergence of these schemes towards a weak solution
of the system of partial differential equations
Mean field type control with congestion
We analyze some systems of partial differential equations arising in the
theory of mean field type control with congestion effects. We look for weak
solutions. Our main result is the existence and uniqueness of suitably defined
weak solutions, which are characterized as the optima of two optimal control
problems in duality
On the system of partial differential equations arising in mean field type control
We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations
arising from the finite horizon control of McKean-Vlasov dynamics. We give
examples of existence and uniqueness results. Finally, we propose some simple
models for the motion of pedestrians and report about numerical simulations in
which we compare mean filed games and mean field type control
Mean Field Games models of segregation
This paper introduces and analyses some models in the framework of Mean Field
Games describing interactions between two populations motivated by the studies
on urban settlements and residential choice by Thomas Schelling. For static
games, a large population limit is proved. For differential games with noise,
the existence of solutions is established for the systems of partial
differential equations of Mean Field Game theory, in the stationary and in the
evolutive case. Numerical methods are proposed, with several simulations. In
the examples and in the numerical results, particular emphasis is put on the
phenomenon of segregation between the populations.Comment: 35 pages, 10 figure
Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface
We consider a family of optimal control problems in the plane with dynamics
and running costs possibly discontinuous across an oscillatory interface
. The oscillations of the interface have small period and
amplitude, both of the order of , and the interfaces
tend to a straight line . We study the asymptotic
behavior as . We prove that the value function tends to the
solution of Hamilton-Jacobi equations in the two half-planes limited by
, with an effective transmission condition on keeping track of
the oscillations of
Asymptotic behaviour of a rapidly rotating fluid with random stationary surface stress
The goal of this paper is to describe in mathematical terms the effect on the
ocean circulation of a random stationary wind stress at the surface of the
ocean. In order to avoid singular behaviour, non-resonance hypotheses are
introduced, which ensure that the time frequencies of the wind-stress are
different from that of the Earth rotation. We prove a convergence result for a
three-dimensional Navier-Stokes-Coriolis system in a bounded domain, in the
asymptotic of fast rotation and vanishing vertical viscosity, and we exhibit
some random and stationary boundary layer profiles. At last, an average
equation is derived for the limit system in the case of the non-resonant torus.Comment: 45 page
Mean field games: convergence of a finite difference method
Mean field type models describing the limiting behavior, as the number of
players tends to , of stochastic differential game problems, have been
recently introduced by J-M. Lasry and P-L. Lions. Numerical methods for the
approximation of the stationary and evolutive versions of such models have been
proposed by the authors in previous works . Convergence theorems for these
methods are proved under various assumption
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