2. Constitutive laws for **mean** **field** **modeling** of DRX
Initial microstructures in **mean** **field** models are represented by spherical grains,
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each having a dislocation density ρ i and a grain size R i . In several existing models [20, 17], the concept of grain classes is used to reduce computational cost. This con- sists in gathering several similar grains in one single entity called ”class”, that is defined by a grain radius R i , a dislocation density ρ i and a number of grains N i in the considered class. The main drawback to this reduction is that all the grains belonging to the same

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Abstract
We present a full **field** framework based on the level-set (LS) approach, which enables to simulate grain growth in a multiphase material. Our formalism permits to take into account different types of second phases, which can be static or dynamic (i.e. evolving also by grain growth) and reproduce both transient (evolving relative grain sizes) and steady-state structures. We use previously published annealing experiments of porous olivine or olivine and enstatite mixtures to constrain the parameters of the full **field** model, and then analyse the results of a peridotite-like annealing simulation. The experimental grain growth kinetics is very well reproduced while the simulated microstructure morphologies show some differences with experimental ones. We then propose a **mean** **field** model calibrated thanks to the full **field** simulations, which allow us to predict the **mean** grain size evolution depending on the simplified peridotite composition (e.g. second phase **mean** grain sizes, fractions).

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1.2.2 **Mean** **field** models
Several **mean** **field** models can be found in the literature to model DRX and PDRX. The structure of these models is always the same: the microstructure is implicitly described by considering spherical grains with a given dislocation density (uniform per grain). To further decrease computational costs, grains with same characteristics (in terms of radius and dislocation density) can be gathered into grain classes. This concept was already used in few **mean** **field** models but can be easily extended to any other ones. The main drawback of this concept is related to the fact that all grains belonging to the same class follow the same behavior during the simulation. The individual physical mechanisms taking place during DRX, i.e. strain hardening, dynamic recovery, dynamic nucleation and grain boundary migration and those taking place during PDRX, i.e. static recovery, static nucleation and grain boundary migration, are independently described by analyt- ical laws. The main advantage of **mean** **field** models as compared to phenomenological laws is the **modeling** of each underlying physical mechanism, which makes them more versatile. In addition, not only the **mean** grain size but also the grain size distribution can be predicted along the hot deformation process. The first **mean** **field** model was proposed by Hillert and Abbruzzese [47, 67–69] for **modeling** GG. This model was ex- tended to DRX in 2009 by Montheillet et al. [48]. In Montheillet’s model, each grain is considered inside a HEM composed of all grains of the microstructure. Also in 2009, Cram & Zurob [70] proposed another kind of **mean** **field** model for DRX based on an- other assumption: during deformation, each grain undergoes the same mechanical work meaning that softer grains deform more than harder ones. Following Montheillet’s ap- proach, Bernard et al. [49] proposed two years later another physically-based **mean** **field** model, for both DRX and PDRX mechanisms, whose main novelty lies in the fact that the HEM is subdivided into two different media composed of RX and NR grains re- spectively. In 2015, Beltran et al. [50] proposed an improvement of Bernard’s model to handle multi-pass deformation routes. Very recently, Smagghe [71] developed in his PhD work a new approach for DRX and PDRX **modeling** in which each grain interacts with only one randomly-selected grain all along the process. All the quoted **mean** **field** models are detailed below.

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2
1 − λ 0
= N ψ 1 (r) 2 (C4)
In other words, increasing the chain length increases the **mean** density uniformly, as if the floppy chain is simply winding around more under the same confinement. This uninteresting behavior is refined by two interactions that have been ignored in getting (C4). First is the stiffness of successive pair orientations, equivalent to the monomers having coupled orientational degrees of freedom, a topic that has been addressed to some extent in this format in the past and will be attended to more forcefully in the future. Second is the effect of non-next neighbor interactions, which we have here studied in a preliminary fashion. The **mean** **field** v(r) that has been enountered will also have the effect of correlating pair orientations, and will of course alter the nature of the long chain resonant state. Let us see how this works. To keep the extrapolation from (C1-C4) transparent,

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Our second main result concerns the long time average of the solution of the **mean** **field** game system. Following standard arguments in control theory, one expects that, as horizon T tends to infinity, the value function φ converges to the value of an ergodic control problem, while the measure m stabilizes to an invariant measure. The resulting system should be therefore an ergodic MFG system, as introduced by Lasry and Lions in [LL06b]:

2.6 Some remarks
We conclude this section with some observations about the **modeling** with the "tool" MFG. The MFG seem to be particularly adapted to describe a situation that combines two economical ideas, positive externality and scale effect. In our model, we introduced a positive externality (if one insulates better her home, her neighbor has a better insulation of her apartment). We saw that there is a clear incentive on any agent to choose some insulation level.

5 An Academic Example: Production of an Exhaustible Resource
Following [24], we consider a continuum of producers exploiting an oil **field**. Each producer’s goal is to maximize his profit, knowing the price of oil; however, this price is influenced by the quantity of oil available on the market, which is the sum of all that the producers have decided to extract at a given time. Hence, while each producer does not affect the price of oil, because each producer solves the same optimization problem, in the end the global problem must take into account the market price as a function of oil availability. For a better understanding of the relation between the individual game and the global game, the reader is referred to [10].

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Finally, in section 2, we consider a general class of stochastic differential games for a large number N of players. The limit behavior, as N goes to infinity, is intimately connected to the modelling of economical equilibrium with rational anticipations. One indeed postulates in such a context that each agent is rational (and assumes the other agents to be rational as well . . . ) but also has a “tiny” (infinitesimal) influence on the equilibrium. A fundamental contribution to this issue has been given by R. Aumann [1], and, since then, many works have investigated it (see the recent work by G. Carmona [9] and the references therein). We propose a different approach based upon stochastic control. Roughly speaking, each player maximizes the expectation of a criterion by choosing a strategy on the parameters of a stochastic evolution. This criterion depends on one hand on individual parameters as is customary in stochastic control and on the other hand on the (spatial) density of the other players. The consistency of this equilibrium, that we call a “**mean** **field** equilibrium”, in the sense of rational anticipations is insured by the fact that the dynamics of the density of players results from the individual optimal strategies. Let us mention that the stochastic set-up we use for this dynamical equilibrium allows us to circumvent the use of approximations proposed in the abstract static set-up of the works mentioned above. We also emphasize the fact that the deduction of these **mean**-**field** equilibria is justified rigorously from the limit, as N goes to infinity, of Nash equilibria. And, as it is to be expected, we observe a significant simplification of the complexity of such N players equilibria thanks to that limit.

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probl`emes de jeux `a champs moyen par P.L. Lions (2007). Les r´esultats sont appliqu´es `a un probl`eme d’optimisation de portefeuille et `a un probl`eme de risque syst´emique.
1. Introduction
Stochastic control is an old topic [5, 11, 13, 14] which has a renewed interest in economy and finance due to **mean**-**field** games [8, 7, 12]. They lead, among other things, to stochastic control problems which involve statistics of the Markov process like means and variance. Optimality conditions for these are derived either by stochastic calculus of variation [1] or by stochastic dynamic programming in the quadratic case [2, 3], but not in the general case for the fundamental reason that Bellman’s principle does not apply in its original form on the stochastic trajectories of say X t if those depend upon statistics of X t like its **mean** value. As

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The study of the link between the ( PPS ) system and a **mean**-**field** interacting neu- ral network (modelled by point processes) was left as an open question in [ 11 ]. The heuristic of this **mean**-**field** interpretation comes from the specific structure of the vari- able X(t) which brings out a non-linearity of the McKean-Vlasov type. One of the main purpose of the present paper is to answer that left open question. To be precise, this kind of study is performed in a preliminary work [ 48 ] for a firing rate p that is continuous and non-decreasing in both variables and under Markovian assumptions. Transposed to the Hawkes framework, this last point corresponds to interaction functions of the form h j→i (t) = e −β(t−τ j ) 1 [τ j ,+∞) (t) where β is a constant and the τ j ’s are i.i.d. random vari-

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The **mean** **field** annealing algorithm is outlined in the Appendix.
VI Examples
We tested these algorithms over a wide variety of images. Figure 5 and Figure 6 show two such im- ages. These images are preprocessed as follows. Edges are first extracted using the Canny/Deriche operator [4]. A small set of connected edges are grouped to form an edgel as follows: The tangent direction associated with such an edgel is computed by fitting a straight line in the least-square sense to the small set of connected edges. Then this small set of edges is replaced by an edgel, i.e., the fitted line. The position of the edgel is given by its midpoint, its direction is given by the direction of the line, and its contrast is given by the average contrast of the edges forming the edgel. Figure 7 and Figure 8 show the input data of our experiments.

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The purpose of this note is to provide an existence result for the solution of Forward Backward Stochastic Differential Equations (FBSDEs) of the McKean-Vlasov type. Following the wave of interest created by the pathbreaking work of Lasry and Lions on **mean** **field** games [13, 14, 15], simple forms of Backward Stochastic Differential Equations (BSDEs) of McKean Vlasov type have been introduced and called of **mean** **field** type. Fully coupled FBSDEs are typically more involved and more difficult to solve than BSDEs. FBSDEs of **mean** **field** type occur naturally in the probabilistic analysis of **mean** **field** games and the optimal control of dynamics of the McKean Vlasov type as considered in [4, 3]. See also [1, 5, 17] for the particular case of Linear Quadratic (LQ) models. Detailed explanations on how these FBSDEs occur in these contexts and the particular models which were solved are given in Section 3 below.

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Discretization of **mean** **field** games: In Parts 1 to 3 of this paper, we study a lagrangian approximation of the solutions to the **mean** **field** game described above. In practice, this means reformulating it as a minimization over P(C 0 ([0; T ], R d )), then looking for minimizers of a similar problem, over discrete probability measures this time. For such measures, the term R 0 T F (µ(t))dt could be ill-defined and congestion must be penalized in a different manner. The main goal of this article is to show that replacing the functional F with a regularized version, in the form of a Moreau envelope in the Wasserstein space:

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√
2ν is a function of the state variable x can also be dealt with by finite difference schemes.
Remark 2 Deterministic **mean** **field** games (i.e. for ν = 0) are also quite meaningful. One may also consider volatilities as functions of the state variable that may vanish. When the Hamiltonian depends separately on ∇u and m and the coupling cost is a smoothing map, see Remark 1, the Hamilton-Jacobi equation (respectively the Fokker-Planck equation) should be understood in viscosity sense (respectively in the sense of of distributions), see the notes of P. Cardaliaguet [31]. When the coupling costs depends locally on m, then under some assumptions on the growth at infinity of the coupling cost as a function of m, it is possible to propose a notion of weak solutions to the system of PDEs, see [56, 34, 59]. The numerical schemes discussed below can also be applied to these situations, even if the convergence results in the available literature are obtained under the hypothesis that ν is bounded from below by a positive constant. The results and methods presented in Section 3 below for variational MFGs, i.e. when the system of PDEs can be seen as the optimality conditions of an optimal control problem driven by a PDE, hold if ν vanishes. In Section 8 below, we present some applications and numerical simulations for which the viscosity is zero.

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the **mean**-**field** model.
The **mean**-**field** model is not a physical model since there is no reason to believe that the interaction would be multiplied by 1/N in a real sys- tem. However it can be used for dilute gases with long-range tunable in- teractions [84], as well as for some other systems that can be recast in the form (2.1) in an appropriate regime. This includes for instance bosonic atoms [13, 97, 10, 11, 54] when the number of electrons is proportional to the nuclear charge, and stars [81]. The Hamiltonian H N is an interesting

t (ds)µ t (ds 0 ), as N → ∞, for all t ∈ T ,
which can be interpreted as the average intensity of interactions converges to a limit as the size of the graph tends to infinity.
We can now interpret the second order **mean** **field** operator in the context of an inho- mogeneous random graph as the average intensity of interactions in the network. Unlike classical **mean**-**field** models in which nodes’ states depend on the average state of all other nodes, here what matters is the average intensity of interactions. If nodes do not interact with each other, there is little structural reason why the nodes’ states depend on others. This suggests the following operator

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PGMO - COPI’14
A **mean** **Field** Game approach to technological transition
Imen Ben Tahar · Ren´ e A¨ ıd
Abstract We develop a model to assess the diffusion of a new-technology among a population of poten- tial adopters. Our main objective is to analyze the effect of the strategic interaction of the firms which supply this new technology in a context where production costs decline with cumulated production (learning by doing effect), and where a firm’s learning or experience benefits its rivals (learning spillover effect). To produce a tractable model in a dynamic setting, we adopt a **mean**-**field**-game approach. Keywords diffusion of a new technology · spill over effect · strategic interaction · **mean** **field** game approach

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1.4 A first class of **mean** **field** games
From a mathematical point of view, a first class of **mean** **field** games, and this class is the purpose of this paper, appeared in a stochastic control form. With this representation, each agent can control - with a cost - the drift and/or the volatility of a diffusion process and maximizes (in expectancy) a utility criterium that depends on this dynamical process and on the **mean** **field** of the problem. This type of framework is really common in finance, in economics or in engineering and corresponds, in the deterministic case to variations calculus. Noticeably, even in the stochas- tic case, the problem, as far as the players are concerned as a whole, stays deterministic because of the continuum of players and the law of large numbers. In what follows, we are going to see that the equations of this first class of **mean** **field** games have a forward/backward structure: a back- ward PDE (Hamilton-Jacobi-Bellman) to model the individual backward induction process that explains each agent’s choices ; a forward PDE (Kol- mogorov) to model the evolution of the players as a whole, the evolution of the community.

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and the cluster approach with strong intracluster
couplings and weak intercluster couplings.
In this note, we wish to study a class of spin systems,
with purely local disorder. For such systems, we perform in section 2, the quenched average and calculate the free energy (without replicas). A quench- ed partition function is defined (section 3) whose leading term is a **mean** density quenched free energy. **Mean** **field** theory follows from a saddle point treat- ment of the leading term only. In section 4, we consider

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2 B ASICS OF EEG **MODELING** IN ANESTHESIA
To our knowledge, researches conducted by Steyn-Ross et al were the first attempts to explain, using an MF model, why gradual increase of anesthetic concentration produces a sudden transition between awareness and unconsciousness [6-8]. In their model, anesthetic agents prolong the decay time of GABAA receptors. Equilibrium solutions of the coupled differential equations in various drug concentrations make an S-bend with one unstable and two stable branches. According to Steyn-Ross et al, when anesthetic concentration is gradually increased or decreased, the equilibrium solution of the model suddenly jumps from one stable branch to another and this can cause sudden transition between awareness and unconsciousness. Since phase transitions make a hysteresis path, emergence and induction phases of anesthesia take place in different drug concentrations. Steyn-Ross et al indicate that their model may simulate the biphasic response. Biphasic response is a kind of transient activation-depression of the EEG signal that occurs in induction and recovery phases of anesthesia [15]. Later, Bojak & Liley [9] modify the Steyn-Ross et al model. They develop better formulations to describe inhibitory and excitatory post synaptic potentials (IPSP/EPSP) in different anesthetic drug concentrations. Bojak & Liley also argue that anesthetic drugs reduce the firing rate of spontaneous action potentials in a relatively smooth dose-dependent manner and as a result, **mean** membrane potentials of inhibitory and excitatory populations do not change abruptly by increasing or decreasing anesthetic drug concentration. According to this statement, Bojak & Liley generate a large set of spectra and compare them to empirical EEG recordings using some classical features such as SEF90 (Spectral Edge Frequency defined as the frequency below which 90% of the power in the electroencephalogram resides). The biphasic response of the Bojak & Liley model is produced differently compared to that of the Steyn-Ross et al model. It takes place in the same drug concentration in induction and recovery phases (i.e. non-hysteresis path).

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