64 research outputs found
On the existence of oscillating solutions in non-monotone Mean-Field Games
For non-monotone single and two-populations time-dependent Mean-Field Game
systems we obtain the existence of an infinite number of branches of
non-trivial solutions. These non-trivial solutions are in particular shown to
exhibit an oscillatory behaviour when they are close to the trivial (constant)
one. The existence of such branches is derived using local and global
bifurcation methods, that rely on the analysis of eigenfunction expansions of
solutions to the associated linearized problem. Numerical analysis is performed
on two different models to observe the oscillatory behaviour of solutions
predicted by bifurcation theory, and to study further properties of branches
far away from bifurcation points.Comment: 24 pages, 10 figure
A generalization of the Hopf-Cole transformation for stationary Mean Field Games systems
In this note we propose a transformation which decouples stationary Mean
Field Games systems with superlinear Hamiltonians of the form |p|^r, and turns
the Hamilton-Jacobi-Bellman equation into a quasi-linear equation involving the
r-Laplace operator. Such a transformation requires an assumption on solutions
of the system, which is satisfied for example in space dimension one or if
solutions are radial
On the existence and uniqueness of solutions to time-dependent fractional MFG
We establish existence and uniqueness of solutions to evolutive fractional
Mean Field Game systems with regularizing coupling, for any order of the
fractional Laplacian . The existence is addressed via the vanishing
viscosity method. In particular, we prove that in the subcritical regime
the solution of the system is classical, while if we find a
distributional energy solution. To this aim, we develop an appropriate
functional setting based on parabolic Bessel potential spaces. We show
uniqueness of solutions both under monotonicity conditions and for short time
horizons.Comment: 42 page
On the problem of maximal -regularity for viscous Hamilton-Jacobi equations
For , we prove that maximal regularity of type holds
for periodic solutions to in ,
under the (sharp) assumption .Comment: 11 page
Lipschitz regularity for viscous Hamilton-Jacobi equations with terms
We provide Lipschitz regularity for solutions to viscous time-dependent
Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces.
Our approach is based on a duality method, and relies on the analysis of the
regularity of the gradient of solutions to a dual (Fokker-Planck) equation.
Here, the regularizing effect is due to the non-degenerate diffusion and
coercivity of the Hamiltonian in the gradient variable.Comment: 31 page
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
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