Jeux à champ moyen. I – Le cas stationnaire

We introduce here a general approach to model games with a large number of players. More precisely, we consider N players Nash equilibria for long term stochastic problems and establish rigorously the ‘mean field’ type equations as N goes to infinity. We also prove general uniqueness results and determine the deterministic limit. To cite this article: J.-M. Lasry, P.-L. Lions, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Mean field limit of a dynamical model of polymer systems

This paper provides a mathematically rigorous foundation for self-consistent mean feld theory of the polymeric physics.We study a new model for dynamics of mono-polymer systems.Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces.Every two points on the same string or on two diferent strings also interact under a pairwise potential V.The dynamics of the system is described by a system of N coupled stochastic partial diferential equations(SPDEs).We show that the mean feld limit as N→∞of the system is a self-consistent McKean-Vlasov type equation,under suitable assumptions on the initial and boundary conditions and regularity of V.We also prove that both the SPDE system of the polymers and the mean feld limit equation are well-posed.     

Discrete time,finite state space mean field games

In this paper we study a mean field model for discrete time, finite number of states, dynamic games. These models arise in situations that involve a very large number of agents moving from state to state according to certain optimality criteria. The mean field approach for optimal control and differential games was introduced by Lasry and Lions (2006, 2007) [3], [4], [5]. The discrete time, finite state space setting is motivated both by its independent interest as well as by numerical analysis questions which appear in the discretization of the problems introduced by Lasry and Lions. The main contribution of this paper is the exponential convergence to equilibrium of the initial-terminal value problem.     

On the convergence of finite state mean-field games through Γ-convergence

In this study, we consider the long-term convergence (trend toward an equilibrium) of finite state mean-field games using Γ-convergence. Our techniques are based on the observation that an important class of mean-field games can be viewed as the Euler–Lagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a long-term convergence problem into a Γ-convergence problem. Our results generalize previous results related to long-term convergence for finite state problems.     

Asymptotically optimal strategies in repeated games with incomplete information

It is known that for repeated zero-sum games with incomplete information the limit of the values of theN-stage game exists asN tends to infinity. In this paper strategies are constructed that guarantee in theN-stage game the limit of values up to an error term \(\frac{K}{{\sqrt N }}.\)      

Asymptotic modelling of conductive thin sheets

We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H ¹-modelling error for an expansion with N terms is bounded by O(ε ^N+1) in the exterior of the sheet and by O(ε ^N+1/2) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results.     

Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms

We derive the defocusing cubic Gross–Pitaevskii (GP) hierarchy in dimension d = 3, from an N-body Schrödinger equation describing a gas of interacting bosons in the GP scaling, in the limit N → ∞. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies (Chen and Pavlovi? in Discr Contin Dyn Syst 27(2):715–739, 2010; http://arxiv.org/abs/0906.2984; Proc Am Math Soc 141:279–293, 2013), which are inspired by the solution spaces based on space-time norms introduced by Klainerman and Machedon (Comm Math Phys 279(1):169–185, 2008). We note that in d = 3, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schrödinger equation (NLS) in d = 3.     

Asymptotics of unitary and orthogonal matrix integrals

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large N limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the orthogonal group.     

A symmetry problem with applications to finance

In this paper we will consider an overdetermined problem in an unknown ring-shaped domain, and we prove that the domain has to be spherical ring. We will apply this problem to an \(n\) -dimensional toy model in the mean field game theory introduced by Lasry-Lions too. We show that if in Lasry-Lions model, the additional extra data ”the boundary is a level set” is assumed, then the region, where the solution is harmonic, has to have spherical symmetry.     

On the convergence of sequences of stationary jump Markov processes

This paper presents two main results: first, a Liapunov type criterion for the existence of a stationary probability distribution for a jump Markov process; second, a Liapunov type criterion for existence and tightness of stationary probability distributions for a sequence of jump Markov processes. If the corresponding semigroups T_N(t) converge, under suitable hypotheses on the limit semigroup, this last result yields the weak convergence of the sequence of stationary processes (T_N(t), π_N) to the stationary limit one.     

A convergence method for stochastic differential games with a small parameter

A stochastic differential game problem for wideband noise driven system is considered. Often in applications, one have a single realization, then expectation is not appropriate in the cost function. First we will consider the payoff structure in the pathwise but not necessarily in the expected value sense. For N-person noncooperative games, under very general conditions, it will be shown that the optimal equilibrium policies of the limit diffusion when applied to the physical processes, will be δ-equilibrium as the parameters ε > 0 and T→ ∞. A combination of direct averaging and perturbed test function techniques will be used in convergence analysis. Results are shown to hold when mathematical expectations are used in the payoff structure. Two person zero sum games can also be considered in this framework     

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.     

Time delay and finite differences for the non-stationary non-linear Navier–Stokes equations

In the present paper we use a time delay ? > 0 for an energy conserving approximation of the non-linear term of the non-stationary Navier–Stokes equations. We prove that the corresponding initial-value problem (N_?) in smoothly bounded domains G ? ?³ is well-posed. We study a semidiscretized difference scheme for (N_?) and prove convergence to optimal order in the Sobolev space H²(G). Passing to the limit ?→0 we show that the sequence of stabilized solutions has an accumulation point such that it solves the Navier–Stokes problem (N_o) in a weak sense (Hopf).     

A general characterization of the mean field limit for stochastic differential games

The mean field limit of large-population symmetric stochastic differential games is derived in a general setting, with and without common noise, on a finite time horizon. Minimal assumptions are imposed on equilibrium strategies, which may be asymmetric and based on full information. It is shown that approximate Nash equilibria in the n-player games admit certain weak limits as n tends to infinity, and every limit is a weak solution of the mean field game (MFG). Conversely, every weak MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria in the n-player games. Thus, the MFG precisely characterizes the possible limiting equilibrium behavior of the n-player games. Even in the setting without common noise, the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of MFG solution.     

Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions

We consider wave and Klein-Gordon equations in the whole space ?ⁿ with arbitraryn≥2. We assume initial data to be homogeneous random functions in ?ⁿ with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.     

The behavior of the best Sobolev trace constant and extremals in thin domains

In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W1,p(Ω)?Lq(∂Ω) in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of Ω, W1,p(P(Ω),α)?Lq(P(Ω),β).For the special case p=q, this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case.     

A generalized Stieltjes criterion for the complete indeterminacy of interpolation problems

The main result of this paper is a generalized Stieltjes criterion for the complete indeterminacy of interpolation problems in the Stieltjes class. This criterion is a generalization to limit interpolation problems of the classical Stieltjes criterion for the indeterminacy of moment problems. It is stated in terms of the Stieltjes parameters M _j and N _j . We obtain explicit formulas for the Stieltjes parameters. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class.     

A remark on the mean curvature of a graph-like hypersurface in hyperbolic space

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space Hn+1. We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the (n−1)-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.     

Averaging methods for finding periodic orbits via Brouwer degree

We consider the problem of finding T-periodic solutions for a differential system whose vector field depend on a small parameter ε. An answer to this problem can be given using the averaging method. Our main results are in this direction, but our approach is new. We use topological methods based on Brouwer degree theory to solve operator equations equivalent to this problem. The regularity assumptions are weaker then in the known results (up to second order in ε). A result for third order averaging method is also given.As an application we provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function. A concrete example is given.