31 research outputs found
Coupling, Attractiveness and Hydrodynamics for Conservative Particle Systems
Attractiveness is a fundamental tool to study interacting particle systems
and the basic coupling construction is a usual route to prove this property, as
for instance in simple exclusion. The derived Markovian coupled process
satisfies: (A) if
(coordinate-wise), then for all , a.s. In this
paper, we consider generalized misanthrope models which are conservative
particle systems on such that, in each transition, particles may
jump from a site to another site , with . These models include
simple exclusion for which , but, beyond that value, the basic coupling
construction is not possible and a more refined one is required. We give
necessary and sufficient conditions on the rates to insure attractiveness; we
construct a Markovian coupled process which both satisfies (A) and makes
discrepancies between its two marginals non-increasing. We determine the
extremal invariant and translation invariant probability measures under general
irreducibility conditions. We apply our results to examples including a
two-species asymmetric exclusion process with charge conservation (for which
) which arises from a Solid-on-Solid interface dynamics, and a stick
process (for which is unbounded) in correspondence with a generalized
discrete Hammersley-Aldous-Diaconis model. We derive the hydrodynamic limit of
these two one-dimensional models
Quadratic Mean Field Games
Mean field games were introduced independently by J-M. Lasry and P-L. Lions,
and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new
approach to optimization problems with a large number of interacting agents.
The description of such models split in two parts, one describing the evolution
of the density of players in some parameter space, the other the value of a
cost functional each player tries to minimize for himself, anticipating on the
rational behavior of the others.
Quadratic Mean Field Games form a particular class among these systems, in
which the dynamics of each player is governed by a controlled Langevin equation
with an associated cost functional quadratic in the control parameter. In such
cases, there exists a deep relationship with the non-linear Schr\"odinger
equation in imaginary time, connexion which lead to effective approximation
schemes as well as a better understanding of the behavior of Mean Field Games.
The aim of this paper is to serve as an introduction to Quadratic Mean Field
Games and their connexion with the non-linear Schr\"odinger equation, providing
to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure
'Phase diagram' of a mean field game
Mean field games were introduced by J-M.Lasry and P-L. Lions in the
mathematical community, and independently by M. Huang and co-workers in the
engineering community, to deal with optimization problems when the number of
agents becomes very large. In this article we study in detail a particular
example called the 'seminar problem' introduced by O.Gu\'eant, J-M Lasry, and
P-L. Lions in 2010. This model contains the main ingredients of any mean field
game but has the particular feature that all agent are coupled only through a
simple random event (the seminar starting time) that they all contribute to
form. In the mean field limit, this event becomes deterministic and its value
can be fixed through a self consistent procedure. This allows for a rather
thorough understanding of the solutions of the problem, through both exact
results and a detailed analysis of various limiting regimes. For a sensible
class of initial configurations, distinct behaviors can be associated to
different domains in the parameter space . For this reason, the 'seminar
problem' appears to be an interesting toy model on which both intuition and
technical approaches can be tested as a preliminary study toward more complex
mean field game models
Schrödinger apprach to Mean Field Games with negative coordination
Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled objects in interaction. Here we consider such systems when the interaction between controlled objects are negatively coordinated and analyze the behavior of their solutions using the correspondence which have been evidenciated with the non linear Schrödinger equation. When the system is conned, we rely on the existence of an ergodic state which notion has been shown previously to characterize most of the dynamics for long optimization times. In the case of an unbounded domain, such an ergodic state does not exist, and we show the existence of a scaling solution that can play a similar role in the analysis.
Schr\"odinger approach to Mean Field Games with negative coordination
Mean Field Games provide a powerful framework to analyze the dynamics of a
large number of controlled agents in interaction. Here we consider such systems
when the interactions between agents result in a negative coordination and
analyze the behavior of the associated system of coupled PDEs using the now
well established correspondence with the non linear Schr\"odinger equation. We
focus on the long optimization time limit and on configurations such that the
game we consider goes through different regimes in which the relative
importance of disorder, interactions between agents and external potential
varies, which makes possible to get insights on the role of the
forward-backward structure of the Mean Field Game equations in relation with
the way these various regimes are connected