Long time dynamics for interacting oscillators 

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Inhomogeneous behavior versus mean-eld behavior.

In Subsection 3.2, we have seen that a description for n in the case of an inhomogeneous graph sequence is available. Namely, it is given by extending the classical non-linear process (1.5) to a collection of processes indexed by [0; 1] and coupled by means of a labeled graphon W, recall equation (1.15). We have also pointed out that the limit measure dened by (1.16) exists for every W 2 W0 and is independent of the particular labeling of fxgx2[0;1]. In other words, although existing proofs seem to require regular labeled graphons, the limit description depends on the equivalent class of W in fW0 only, i.e., on an unlabeled graphon. Furthermore, the graph convergence considered in [6, 27, 77, 82, 90] is always required to be in stronger topologies than the one in fW0.
Consider a sequence (n) that converges in fW0 to some graphon W. We want to understand the following issues:
(1) Is it possible to prove that the empirical measure n converges to ?.
(2) For which W, is the limit of n mean-eld for n which tends to innity? More precisely, under which hypothesis on the sequence f(n)gn2N, n is approximately described by the mean-eld limit solving equation (1.4) as n diverges?.
(3) What can be said if (n) is a random sequence which converges, e.g., in probability, to a random W 2 fW0?.

A n-dependent equation for the empirical measure.

Consider equation (1.1) with 0, i.e., a deterministic system of weakly interacting particles. It is well known [87, 20, 41] that, with the only hypothesis of the weak convergence of n0 , the empirical measure n weakly converges to the solution of the Vlasov equation, i.e., equation (1.4) without the diusive term. To the author’s knowledge, this result is missing in the stochastic framework where a nite moment condition on 0, or chaotic initial data in (1.1), is always required to show this same convergence.
In collaboration with Florian Bechtold [8], we have investigated whether a similar result holds in the case 1. Under the only hypothesis of the convergence of n0 , we show that the weak convergence of n to the solution of equation (1.4), can be established in a suitable class of Hilbert spaces. This is possible by giving a meaning to the stochastic term in the n-dependent equation satised by the empirical measure, recall equation (1.3), and by using the properties of the heat semigroup. Indeed, equation (1.3) can be rewritten by means of Duamel’s formula as hnt ; hi = hStn0 ; hi + h Z t  St􀀀s@x [(F + 􀀀 ns ) ns ] ds ; hi + hwn t ; hi.

Mean-eld behavior and two explanatory examples.

Theorem 2.3 allows for a better understanding of the relationship between random graph sequences and the behavior of the empirical measure. More precisely:
(1) It highlights the dierence between the randomness present in the graph (n) for every n 2 N and the one left in the limit W.
(2) It presents a new class of random Fokker-Planck equations as possible limit descriptions for the empirical measure n.
As a byproduct, it allows to derive a precise characterization of the graph sequences for which the empirical measure limit is mean-eld. Let us recall what we mean by mean-eld limit and rst discuss this last issue; we then address (1) and (2) with the help of two examples.

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Synchronization of mean eld systems on graphs.

In recent years, synchronization of complex networks has become a very important topic for explaining real world phenomena. While in the physics literature the analysis has been pushed quite far and several extended reviews are available (e.g. [43, 94]), from a mathematical point of view these studies and the associated numerical simulations, can be regarded more as heuristic arguments than conclusive proofs.
The mathematical community has started working on particle systems on (random) graphs from the statistical mechanics point of view in the equilibrium regime and, with respect to the graph setting, assuming a locally tree-like structure (e.g [37]). Only in the last few years the attention has been focused on the dynamics of weakly interacting particles, tackling mean eld systems on graphs, and their relationship with the corresponding thermodynamical limit (e.g. [13, 36]). These results, and the one presented here, are obtained for graphs in an intermediate regime between the sparse and the dense case, i.e. if Gn has n vertices and npn represents the average number of edges, then 1 npn n. In the case of sparse graphs, i.e. npn = O(1), the limiting system seems to show a dierent phenomenology ([70, 91]).
Today, many results on the behavior of the empirical measure of such systems are available ([13, 31, 36, 77, 90]), but there is no agreement on the weakest hypothesis the class of graphs should satisfy in order to obtain the classical mean eld limit. It turns out that, depending on the setting one is considering, i.e. the normalization chosen in the interaction and/or the hypothesis on the initial data, dierent requirements on the graph may be asked. To the author’s knowledge, there exists no result on the longtime dynamics of a system de- ned on a sequence of graphs and the question whether the network is inuencing the dynamics on long time scales, is still open and very much awaited with regards to applications.

Table of contents :

Chapter 1. Introduction 
1. Weakly interacting particle systems
2. Graph sequences and graphons
3. Interacting particles on graphs
4. Motivations
5. Overview of the results and general organization
Chapter 2. Interacting diusions on Erd®s-Rényi random graphs 
1. Introduction
2. Main results
3. Proof: The Law of Large Numbers
4. Proof: Large Deviation Principle
Chapter 3. Weakly interacting oscillators on dense random graphs
1. Introduction, organization and set-up
2. The models and main results
3. The non-linear process
4. Proof of Theorem 2.3
3.A. Graph convergence and random graphons
Chapter 4. Long time dynamics for interacting oscillators 
1. Introduction
2. Main results
3. Longtime dynamics close to M
4. Longtime behavior around 1=2
5. Finite time behavior
4.A. Graphs
4.B. H􀀀1 and Semigroups
Chapter 5. A Law of Large Numbers via a mild formulation 
1. Introduction
2. Main result
3. Proofs
5.A. Hilbert spaces and semigroups
5.B. Rough integration associated to semigroup functionals
Chapter 6. Perspectives 
1. From dense graph sequences to the sparse regime
2. Irregular graphons and random graphons
3. Unbounded graphons
4. Long-time dynamics and uniform propagation of chaos


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