The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

A Starter: The First-Order Master Equation

2019 ◽  
pp. 48-84
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Preliminary Analysis ◽  
Initial Conditions ◽  
Mean Field ◽  
Mean Field Games ◽  
Lipschitz Continuous ◽  
First Order ◽  
Common Noise ◽  
The Mean ◽  
Uniformly Lipschitz

This chapter contains a preliminary analysis of the master equation in the simpler case when there is no common noise. Some of the proofs given in this chapter consist of a sketch only. One of the reasons is that some of the arguments used to investigate the mean field games (MFGs) system have been already developed in the literature. Another reason is that the chapter constitutes a starter only, specifically devoted to the simpler case without common noise. It provides details of the global Lipschitz continuity of H. The solutions of the MFG system are uniformly Lipschitz continuous, which are independently of initial conditions.

Presentation of the Main Results

2019 ◽  
pp. 28-47
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Classical Solution ◽  
Mean Field ◽  
Approximate Solutions ◽  
Unique Classical Solution ◽  
Mean Field Game ◽  
The Mean ◽  
Derivatives Of

This chapter collects the main results of the master equation for the convergence of the Nash system. It explains the notation used, specifies the notion of derivatives in the space of measures, and describes the assumptions on the data. One of the striking features of the master equation is that it involves derivatives of the unknown with respect to the measure. This chapter also discusses the link between the two notions of derivatives, which have been used in the mean field game (MFG) theory. The main result states that the master equation has a unique classical solution under the regularity and monotonicity assumptions on H, F, and G. Once the master equation has a solution, this solution can be used to build approximate solutions for the Nash system with N-players.

Re-Examination of A1 → L10 Ordering: Generalized Bragg-Williams Model with Elastic Relaxation

2011 ◽  
Vol 172-174 ◽  
pp. 608-617 ◽  
Author(s):  
William A. Soffa ◽  
David E. Laughlin ◽  
Nitin Singh
Keyword(s):  
Nearest Neighbor ◽  
Binary Solution ◽  
Mean Field ◽  
Lattice Relaxation ◽  
Two Phase ◽  
First Order ◽  
Tetragonal Lattice ◽  
Wide Range ◽  
Third Law Of Thermodynamics ◽  
The Mean

The tetragonal lattice relaxation has been included in the thermodynamics of the fcc→L10ordering to produce a first-order character of the transition within the mean field description of the binary solution energetics. In view of growing interest in such systems e.g. Fe-Pd and Co-Pt alloys, which display a wide range of applications relevant to current and futuristic technologies, the fcc→L10two-phase field is re-examined utilizing a generalized Bragg-Williams approach including first and second nearest neighbor interactions. The thermodynamic behavior is examined in the limit of T→0K and discussed in terms of the implications of the Third Law of Thermodynamics.

scholarly journals Viability analysis of the first-order mean field games

2020 ◽  
Vol 26 ◽  
pp. 33
Author(s):  
Yurii Averboukh
Keyword(s):  
Mean Field ◽  
Initial Distribution ◽  
Initial Time ◽  
Mean Field Games ◽  
Sufficient Condition ◽  
Mean Field Game ◽  
First Order ◽  
Viability Analysis ◽  
A Value ◽  
Object Of Study

In the paper, we examine the dependence of the solution of the deterministic mean field game on the initial distribution of players. The main object of study is the mapping which assigns to the initial time and the initial distribution of players the set of expected rewards of the representative player corresponding to solutions of mean field game. This mapping can be regarded as a value multifunction. We obtain the sufficient condition for a multifunction to be a value multifunction. It states that if a multifunction is viable with respect to the dynamics generated by the original mean field game, then it is a value multifunction. Furthermore, the infinitesimal variant of this condition is derived.

Microstructure Evaluation for Time Dependent Nucleation Protocols in KJMA Kinetics

1999 ◽  
Vol 580 ◽  
Author(s):  
Eloi Pineda ◽  
Daniel Crespo
Keyword(s):  
Mean Field ◽  
Quantitative Agreement ◽  
Nucleation And Growth ◽  
Time Dependent ◽  
First Order ◽  
Microstructure Evaluation ◽  
The Mean ◽  
Dependent Growth ◽  
First Order Phase

AbstractThe microstructure developed in a first order phase transformation is obtained using a populational extension of the Kolmogorov, Johnson-Mehl and Avrami (KJMA) model, PKJMA. PKJMA allows one to determine the grain size distribution resulting from nucleation and growth kinetics. PKJMA is grounded on the mean field hypothesis that the free space around the growing grains is randomly distributed, independent of the grain radius. Although this approach is perfectly valid for the case of constant nucleation, a detailed analysis of the model shows that this hypothesis does not hold in the case of time dependent nucleation protocols or pre-existing nuclei. In this work, the PKJMA model has been improved by estimating the average free surface around the grains as a function of their radius and the time elapsed since nucleation. The resulting model gives quantitative determination of the microstructure developed under a variety of nucleation and growth processes: pre-existing nuclei, constant nucleation, and the combination of both mechanisms, constant and radius dependent growth rates. Comparison with Monte Carlo simulations, showing a quantitative agreement will be presented.

scholarly journals Schrödinger approach to Mean Field Games with negative coordination

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Thibault Bonnemain ◽  
Thierry Gobron ◽  
Denis Ullmo
Keyword(s):  
Mean Field ◽  
Time Limit ◽  
External Potential ◽  
Mean Field Games ◽  
Relative Importance ◽  
Mean Field Game ◽  
Non Linear ◽  
The Mean ◽  
The Way

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 Schrödinger 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 vary, 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.

scholarly journals Convergence to the mean field game limit: A case study

2020 ◽  
Vol 30 (1) ◽  
pp. 259-286 ◽  
Author(s):  
Marcel Nutz ◽  
Jaime San Martin ◽  
Xiaowei Tan
Keyword(s):  
Mean Field ◽  
Mean Field Game ◽  
The Mean
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