AS **MEAN**-**FIELD** **LIMITS**
MATHIEU LEWIN, PHAN TH ` ANH NAM, AND NICOLAS ROUGERIE
Abstract. We prove that Gibbs measures based on 1D defocusing nonlinear Schr¨ odinger functionals with sub-harmonic trapping can be obtained as the **mean**- **field**/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices.

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Abstract. These notes deal with the **mean**-**field** approximation for equilibrium states of N -body systems in classical and quantum statistical mechanics. A general strategy for the justification of effective models based on statistical independence assumptions is presented in details. The main tools are structure theorems ` a la de Finetti, describing the large N **limits** of admissible states for these systems. These rely on the symmetry under exchange of particles, due to their indiscernability. Emphasis is put on quantum aspects, in particular the **mean**-**field** approximation for the ground states of large bosonic systems, in relation with the Bose-Einstein condensation phenomenon. Topics covered in details include: the structure of reduced density matrices for large bosonic systems, Fock-space localization methods, derivation of effective energy functionals of Hartree or non-linear Schr¨ odinger type, starting from the many-body Schr¨ odinger Hamiltonian.

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We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in R d
and study the associated **mean** **field** **limits**. As the total number of neurons tends to infinity, we prove that the evolution of a typical neuron, attached to a given spatial position, can be described by a nonlinear limit differential equation driven by a Poisson random measure. The limit process is described by a neural **field** equation. As a consequence, we provide a rigorous derivation of the neural **field** equation based on a thorough **mean** **field** analysis.

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Here we investigate in what extend the **mean** **field** game problem is the limit of the N ´person differential games. This question is surprisingly difficult in general and is not completely un- derstood so far in full generality. When, in the N ´player game, players play in open-loop (i.e., observe only their own position but not the position of the other players), the **mean** **field** limit is a **mean** **field** game. The first result in that direction goes back to [24] in the ergodic setting (see also [1, 17] for statements in the same direction); extensions to the non Markovian setting can be found in Fischer [18] while Lacker gave a complete characterization of the limit [22] (see also [25] for an exit time problem). Note that these (often technically difficult) results are not entirely surprising since, in the N-player game as well as in the **mean** **field** game, the players do not observe the position of the other players: therefore there is no real change of nature between the N ´player problem and the **mean** **field** game.

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2017 ) generalize this result to a multi-population frame and show how oscillations emerge in the large population limit. Note again that the interactions in both papers are scaled in N −1 , which
leads to limit point processes with deterministic intensity.
The purpose of this paper is to study the large population limit (when N goes to innity) of the multivariate Hawkes processes (Z N,1 , . . . , Z N,N ) with **mean** eld interactions scaled in N −1/2 .

4.1 Indistinguishability and **mean**-**field** limit
In the context of the classical Hegselmann-Krause model without weights (5), the **Mean**-**Field** Limit process consists of representing the population by its density rather than by a collection of individual opinions. The limit density ν(t, x) represents the (normalized) quantity of agents with opinion x at time t and satisfies a non-local transport equation, where the transport vector **field** V is defined by the convolution of ν with the interaction function φ. The proof of the limit lies on the fact that the empirical measure

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The functional E is the one which is used to study in L1 the asymptotic behaviour of drift-di usion problems corresponding to a nonlinear diffusion see [CJMTU00, BDM99] for the case with[r]

- half-relaxed **limits**, strong maximum principle.
( Note: most of such arguments are not suitable for our system )
As far as m is concerned , if one proves that ∇u → ∇¯u, then m(t)
converges to the unique invariant probability measure ¯ m associated to the process

The paper is organized as follows. First, Section 2 discusses the main assumptions A, definitions of strong MFG solutions, and existence of discretized MFG solutions. Section 3 defines weak MFG solutions in detail, discusses some of their properties, and proves existence by refining the discretiza- tions of the previous section and taking **limits**. Section 4 discusses how to strengthen the notion of control, providing general existence results without relaxed controls under additional convexity hypotheses. The brief Section 5 discusses two counterexamples, which explain why we must work with weak solutions and why we cannot relax the growth assumptions placed on the coefficients. Uniqueness is studied in Section 6, discussing our analog of the Yamada-Watanabe theorem and its application to an existence and uniqueness result for strong MFG solutions.

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In a recent series of papers (J-M. Lasry and P-L. Lions [16, 17, 18, 19, 20, 21]), we introduce a general mathematical modelling approach to situations which involve a great number of “agents”. Roughly speaking, we derive these models from a “continuum limit” (in other words letting the number of agents go to infinity) which is somewhat reminiscent of the classical **mean** **field** approaches in Statistical Mechanics and Physics (as for instance, the derivation of Boltzmann or Vlasov equations in the kinetic theory of gases) or in Quantum Mechanics and Quantum Chemistry (density functional models, Hartree or Hartree-Fock type models. . . ). This general approach leads in various situations to new nonlinear equations which contain as particular examples many classical problems and are linked to several research fields of Analysis. We describe rapidly these equations in the next section. And we conclude this introduction by a brief overview of the economical and/or financial issues that we address through our “**mean**-**field**” approach.

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Abstract
In this paper we attack the figure-ground discrimination problem from a combinatorial optimization perspective. In general the solutions proposed in the past solved this problem only partially: either the mathematical model encoding the figure-ground problem was too simple or the optimization methods that were used were not efficient enough or they could not guarantee to find the global minimum of the cost function describing the figure-ground model. The method that we devised and which is described in this paper is tailored around the following contributions. First, we suggest a mathematical model encoding the figure- ground discrimination problem that makes explicit a definition of shape (or figure) based on cocircularity, smoothness, proximity, and contrast. This model consists of building a cost function on the basis of image element interactions. Moreover, this cost function fits the constraints of a interacting spin system which in turn is a well suited physical model to solve hard combinatorial optimization problems. Second, we suggest a combinatorial optimization method for solving the figure-ground problem, namely **mean** **field** annealing which combines **mean** **field** approximation and annealing. **Mean** **field** annealing may well be viewed as a deterministic approximation of stochastic methods such as simulated annealing. We describe in detail the theoretical bases of this method, derive a computational model, and provide a practical algorithm. Finally, some experimental results are shown both with synthetic and real images.

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3.1. Solvability of FBSDE for **Mean**-**Field** Games. In this subsection, we apply the abstract exis- tence result of Section 2 to the **Mean** **Field** Game (MFG) problem described in the introduction. As in [4], we assume that the volatility function is a constant matrix σ (of size d × m), but here we assume in addition that det(σσ † ) > 0. Since the stochastic optimization problem is solved after the flow of measures is frozen, for each fixed µ, the Hamiltonian of the system reads:

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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During their time evolution the particles interact one another. The nature of the interactions depends on the McKean interpretation model. Nevertheless, in all interesting cases the independence property and the adequacy of the laws of the particles with the desired distributions are broken. The propagation of chaos properties of **mean** **field** particle models presented in this section can be seen as a way to measure these properties. In the first subsection, we estimate the relative entropy of the laws of the first q paths of a complete genealogical tree model with respect to the q-tensor product of the McKean measure. In the second part of this section, we present an original Laurent type and algebraic tree-based integral representations of particle block distributions. In contrast to the first entropy approach, this technique applies to any genetic type particle model without any regularity condition on the mutation transition.

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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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First, we suggest a mathematical encoding of the figure-ground discrimination problem that consists of separating shape from noise using a combinatorial optimization m[r]

The summation on the m γ ’s in Eq. (11) can be restricted according to constraints of the
types given in expressions (5) and (6).
Note that, at orders more than zero, the corrective terms added to the original con- traction method Hamiltonian, correspond, here, to **mean** **field** effects of the approximate ground states of the spectator modes. As explained in [3], it is not suitable to use an excited state **mean** **field**, (and that holds too for an excited state generalized **mean** **field**), if one intends to perform further GMFCI steps. In contrast, at the last step of a sequence of GMFCI calculations, one can modify the equations above to use a generalized **mean** **field** corresponding to an arbitrary spectator state. This is straightforward if the specta- tor approximate, reference state is non degenerate, that is to say, if it can be represented by N

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In the present paper, we show how a dynamical **mean**-**field** theory can be rigorously derived for a simple model of flexible homopolymers in an implicit good solvent. The un- derlying microscopic model is a Rouse-Brownian dynamics for the polymer segments of each chain and for simplicity hydrodynamic interactions and externally imposed flows are not included. Our results show that indeed, at the **mean**-**field** level, there is an instantaneous relation between the **mean**-**field** and the density that coincides with that employed in the SCMF framework. Furthermore, we observe that in a numerical implementation, chains and fields must be moved simultaneously for exact evolution on the dynamical **mean**-path.

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existence of PDF at all times.
Once the problem is reformulated with the Fokker-Planck equation, it becomes a somewhat standard exercise to find the optimality necessary conditions by a calculus of variations. So the note begins likewise. Then a similar result is obtained by using dynamic programming and the connection with the previous approach and with stochastic dynamic programming is established, with the advantage that sufficient con- ditions for optimality are obtained. Finally we apply the method to two **mean**-**field** type control problems stated in [1] and [7].