We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a unique absolutely minimal (AM) extension to X, i.e., a unique Lipschitz extension u : X → ℝ for which Lip U u = Lip ∂u u for all open U ⊂ X \ Y.

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Tug-of-war and the infinity Laplacian

The mean-field framework was developed to study systems with an infinite number of rational agents in competition, which arise naturally in many applications. The systematic study of these problems was started, in the mathematical community by Lasry and Lions, and independently around the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhame. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this paper we present a brief survey of mean-field models as well as recent results and techniques. In the first part of this paper, we study reduced mean-field games, that is, mean-field games, which are written as a system of a Hamilton–Jacobi equation and a transport or Fokker–Planck equation. We start by the derivation of the models and by describing some of the existence results available in the literature. Then we discuss the uniqueness of a solution and propose a definition of relaxed solution for mean-field games that allows to establish uniqueness under minimal regularity hypothesis. A special class of mean-field games that we discuss in some detail is equivalent to the Euler–Lagrange equation of suitable functionals. We present in detail various additional examples, including extensions to population dynamics models. This section ends with a brief overview of the random variables point of view as well as some applications to extended mean-field games models. These extended models arise in problems where the costs incurred by the agents depend not only on the distribution of the other agents, but also on their actions. The second part of the paper concerns mean-field games in master form. These mean-field games can be modeled as a partial differential equation in an infinite dimensional space. We discuss both deterministic models as well as problems where the agents are correlated. We end the paper with a mean-field model for price impact.

Mean Field Games Models—A Brief Survey

There is an abundancy of systems characterized by parabolic PDEs in science and engineering, especially in chemistry and physics. These systems have a scalar variable, we generally call time, defining the evolution of the system under consideration. The governing equation(s) involves the unknown(s) and their first order partial derivative(s) with respect to this variable. Time variant Schrodinger equations where the unknown is the wavefunction which is responsible for the probability density for the system and Liouville equations for the statistical mechanics where the unknown is somehow responsible for a density in the systems’ phase space (here we use the plurality since both case may differ from Hamiltonian to Hamiltonian). Certain PDE(s), depending on so-called spatial coordinates, govern the behavior of the system in these and similar cases even though the partial differential equation nature is not necessarily needed. Hence we give the following equation for more

MATHEMATICAL MODELS and METHODS in APPLIED SCIENCES

This article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations (PDEs), a relatively new subarea within numerical PDEs Due to their ever increasing importance in mathematics itself (eg, differential geometry and PDEs) and in many scientific and engineering fields (eg, astrophysics, geostrophic fluid dynamics, grid generation, image processing, optimal transport, meteorology, mathematical finance, and optimal control), numerical solutions to fully nonlinear second order PDEs have garnered a great deal of interest from the numerical PDE and scientific communities Significant progress has been made for this class of problems in the past few years, but many problems still remain open This article intends to introduce these current advancements and new results to the SIAM community and generate more interest in numerical methods for fully nonlinear PDEs

Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations

In this paper we consider extended stationary mean-field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean-field games in Gomes et al. (2012). In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish the existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in Lasry and Lions (2006), Gomes and Sanchez-Morgado (2011) or Gomes et al. (2012) this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where the existence of classical solutions can be proved.

On the existence of classical solutions for stationary extended mean field games

We collect a number of technical assertions and related counterexamples about viscosity solutions of the infinity-Laplacian PDE − Δ∞ u = 0 for .

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Various Properties of Solutions of the Infinity-Laplacian Equation

G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in "Periodic homogenization of G-equations and viscosity effects," Nonlinearity, to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of degree one. It is also coercive if we further assume that the flow is mean zero.

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Periodic homogenization of the inviscid G-equation for incompressible flows

Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension

We present a proof of qualitative stochastic homogenization for a nonconvex Hamilton–Jacobi equations. The new idea is to introduce a family of “sub-equations” and to control solutions of the original equation by the maximal subsolutions of the latter, which have deterministic limits by the subadditive ergodic theorem and maximality.

Stochastic homogenization of a nonconvex Hamilton-Jacobi equation

In this paper, we prove that viscosity solutions of Aronsson equations are absolute minimizers in certain L∞ variational problems.

Yifeng Yu

Papers

Various Properties of Solutions of the Infinity-Laplacian Equation

Periodic homogenization of the inviscid G-equation for incompressible flows

Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension

Stochastic homogenization of a nonconvex Hamilton-Jacobi equation

L∞ Variational Problems and Aronsson Equations