The graph limit for a pairwise competition model

This paper is aimed at extending the graph limit with time dependent weights obtained in [1] for the case of a pairwise competition model introduced in [10], in which the equation governing the weights involves a weak singularity at the origin. Well posedness for the graph limit equation associated with the ODE system of the pairwise competition model is also proved.


Introduction
General Background.In this work, we are concerned with analyzing the graph limit of the following system of (d + 1)N ODEs The notation is as follows: the unknowns are x N i ∈ R d and m N i ∈ R are referred to as the opinions and weights respectively.The evolution of the opinions is given in terms of the weights and a function a : R d → R d which is called the influence.The evolution of the weights is given by means of functions ψ N i : R dN × R N → R where we apply the notation x N (t) := (x N 1 (t), ..., x N N (t)), m N (t) := (m N 1 (t), ..., m N N (t)).This model has been proposed in [10], along with several other models which are meant to idealize social dynamics.We refer to [10,13] for more details of how these models originate from biology and social sciences.Mathematically, the system (1.1) is a weighted version of the first order N −body problem (simply by taking all the weights to be identically equal to 1).By now, the mean field limit of the N −body problem ẋi is fairly well understood even for influence functions with strong singularities at the origin [14].The mean field limit can be analysed in terms of the empirical measure defined by Thanks to the work of Dobrushin [4] it is possible to prove quantitative convergence of µ N (t) to the solution µ of the (velocity free) Vlasov equation with respect to the Wasserstein metric (provided this is true initially of course).The mean field limit with time dependent weights has been investigated in [1,5,6] for Lipschitz continuous interactions and ψ N i which are at least Lipschitz in each variable, and more recently in [2] for the case of the 1D attractive Coulomb interaction (but still with ψ N i regular enough).There is a different regime, the so called graph limit, closely related to the mean-field limit.In the graph limit, we pass from a discrete system of ODEs to a "continuous" system in the following sense: we associate to x N (t), m N (t) the following Riemman sums x N : [0, T ] × I → R, m N : [0, T ] × I → R defined by Using the equation for the trajectories of the opinions and weights, one easily finds that x N , m N are governed by the following equations Lebesgue differentiation theorem leads us formally to the following integro-differential equation    ∂ t x(t, s) = I m(t, s * )a(x(t, s * ) − x(t, s))ds * , x(0, s) = x 0 (s) ∂ t m(t, s) = Ψ(s, x(t, •), m(t, •)), m(0, s) = m 0 (s). ( Here Ψ : R is a functional whose relation to ψ N i is given by the formula (2.5) in the next section.The formula relating x 0 (s), m 0 (s) to x 0 N (s), m 0 N (s) will be given in the next section as well (formula (2.6)).Hence, one expects that the sums x N (t, s), m N (t, s) are an approximation of the solution (x(t, s), m(t, s)) of the Equation (1.4) .
Before going further, let us briefly comment on the origin of the terminology "graph limit".This name stems from the fact that the system (1.1) can be viewed as a nonlinear heat equation on a graph.For example, in the case where the weights are time independent and the a is taken to be the identity, then the system (1.1) can be rewritten as the linear heat equation with respect to the Laplacian associated to the underlying simple graph.This is the point of view which has been taken in [11].However, this underlying combinatorial structure seems to come into play mostly when the weights may vary from one opinion to another, in which case methods from graph theory prove as highly useful.We also refer to the more recent work [9] for a demonstration of the power of graph theory techniques in the context of the mean field limit, and [3] in the context of convergence to consensus for the graph limit equation.See also [12] for a proof of the graph limit for metric valued labels, alongside an extensive explanation of the relation between the graph limit and the hydrodynamic and mean field limits.In our settings, which are very similar to the framework in [1], this graph structure is not as relevant, and we shall therefore not dwell on this matter.It is instructive to view the system (1.4) as continuous version of (1.1), in the sense that it is obtained by replacing averaged sums by integrals on the unit interval and summation indices by variables in the unit interval.
Relevant Literature and Contribution of the Present work.It appears that the graph limit point of view has not received as much attention as the mean-field limit.The study of this problem was initiated in [11], which as already remarked, considers time independent weights which may depend on the index of the opinion as well.This result has been extended in [1] to cover time dependent weights (although in [1] the weights depend only on the summation index).The evolution in time of the weights renders difficult the problem both at the microscopic and graph limit level-since the corresponding ODE/integro-differential equation become coupled (compare for instance Equations (1.1) and (1.2)), and at the macroscopic level-since the mean field PDE includes a non-local source term (see Section 4 for more details).In both of these results, the functions ψ N i are assumed to be well behaved in terms of regularity.On the other hand, models corresponding to scenarios where the functions ψ N i exhibit singularities recently received attention in [10].For instance, the following ODE has been studied in [10]: where a : R d → R d is Lipschitz and takes the form a(x) = a(|x|)x for some radial a : R → R, and s : R d → S d−1 is the projection on the unit sphere, i.e.
Of course, inserting the equation for x i into the equation for m i transfers the system to the form (1.1).System (1.5) is referred to as a pairwise competition model in [10], and its well posedness can be proved provided opinions are separated initially (i = j =⇒ x 0 i = x 0 j ).It is the aim of this work to investigate how to overcome the challenges created due to the singularity in the weight function in the context of the graph limit.The problem of the graph limit for singularities in the influence function is also interesting.As already remarked, for the mean field limit this has been successfully achieved in [2] for the 1D attractive Coulomb case.However, it is not clear how to study the graph limit regime in this whole generality.The 1D repulsive Coulomb interaction however is manageable, and can be handled by similar methods to the one demonstrated in the present work.
A first contribution of the present work is reflected on two levels, both of which are considered in 1D: the well posedness of the graph limit equation (1.4), and the derivation of (1.4) from the opinion dynamics (1.1) in the limit as N → ∞.As for the first point, we note that in the case when a and ψ N i are well behaved then equation (1.4) can be viewed as a Banach valued ODE, and noting that at each time t our unknowns (x(t, •), m(t, •)) are functions of the variable s, and therefore there is a straightforward analogy between the well posedness of the discrete System (1.1) and equation (1.4).As already mentioned, the global well-posedness of the finite dimensional version of Equation (1.4), namely System (1.5) has been (among other things) proved in [10] using the theory of differential inclusions as developed by Fillipov [7].Originally, Fillipov formulated his theory for unknowns taking values in a finite dimensional space in contrast to Equation (1.4).We follow a slightly different route which is in fact more elementary and does not require any familiarity with convex analysis.A second contribution of the present work, is studying the graph limit in arbitrary dimensions d > 1.In higher dimensions, a natural assumption to impose on the initial datum x 0 is that it is bi-Lipschitz in s -an assumption of this type is strictly stronger from what is needed in 1D.This in turn leads to considering Riemann sums whose labeling variable s varies on the d-dimensional unit cube rather on the unit interval, because cubes of different dimensions cannot be diffeomorphic.This labelling procedure does not have any modelling interpretation since particles (opinions) are still exchangeable or indistinguishable.It would in fact be possible to still work on the unit interval through a change of variable on the labeling variable, since all cubes (and most measurable spaces that one may use) are isomorphic to the unit interval per the Borel isomorphism theorem.However the corresponding analysis would be far more convoluted, and instead having the labeling variable on the d-dimensional unit cube make the various technical steps more transparent.These considerations are therefore detailed separately in Section 5.For both points it is crucial to observe the lower bound |x(t, s 2 ) − x(t, s 1 )| x 0 (s 2 ) − x 0 (s 1 ) .In 1D the initial separation at the continuous level will be replaced by the assumption that x 0 is increasing, whereas in higher dimensions this assumption will be replaced by requiring that x 0 is bi-Lipschitz.Finally we remark that the method here extends the case of the main results in [1], in the sense that it simultaneously covers functions s which are either Lipschitz or have a jump discontinuity at the origin.This last observation is simple but not obvious-for example in the case where the singularity emerges from the influence part, as mentioned earlier, it is not clear how to unify both results.
We organize the paper as follows: Section 2 reviews the terminology introduced in [1] in the specific context of system (1.5).In particular, Section 2 includes preliminaries such as the existence and uniqueness of classical solutions to the system (1.1) in the present settings and other basic properties of solutions (of course, uniqueness is not strictly needed for the purpose of the graph or mean field limit).Section 3 is a continuous adaptation of section 2, namely well posedness for the 1D graph limit equation for which uniqueness is essential.Section 4 includes the main evolution estimate leading to the 1D graph limit, and clarifies the link between the mean field and the graph limit.In Section 5 we introduce multi-dimensional Riemann sums and study the graph limit for arbitrary d > 1.

Preliminaries
2.1 The ODE system.Recall that the system which will occupy us is where When d = 1, which is the case of main interest here, we note that s identifies with the sign function.
We start by reviewing the well-posedness theory which has been established for the System (2.1) in [10].As usual with ODEs with weakly singular right hand sides, the argument in [10] rests on the theory of differential inclusions as developed by Fillipov [7] and the fact that opinions remain separated for all times provided this is true initially.Unless necessary, we omit the super index N in the opinions and weights.
Proposition 2.1.([10, Proposition 3] Suppose a : R d → R d is Lipschitz with a(0) = 0 and x 0 i = x 0 j for all i = j.Then there exists a unique classical solution (x N (t), m N (t)) to the System (2.1) with x i (t) = x j (t) for all i = j and t ≥ 0.
We also recap the following basic properties of solutions, which already appear implicitly or explicitly in [1,10], and will appear in the course of the proof of the main theorems.
ii. (Uniform bound in time on opinions).If x 0 i ≤ X then for all t ∈ [0, T ] it holds that iii. (Uniform bound in time on weights).m i (t) > 0 for all t ∈ [0, T ] with the estimate iv. (Opinions are separated).There is a constant C = C(L, T ) > 1 such for all t ∈ [0, T ] the following bound holds Proof.For i. see Proposition 2 in [10].For ii., fix a time τ > 0 such that m j (τ ) ≥ 0, j = 1, ..., N for all t ∈ [0, τ ] (such a time exists by continuity).We utilize i. and the assumption a(0) = 0 to find that for each t ∈ [0, τ 0 ] which by Gronwall's Lemma implies We prove iii., from which we will conclude ii. for all t ∈ [0, T ].We start by explaining why m i (t) > 0. Indeed, if on the contrary m i (t) ≤ 0 for some 1 ≤ i ≤ N and t ∈ [0, T ] and let Then the bound from ii. and preservation of total mass of i. imply that for all t ∈ [0, τ ) we have d dt Integration in time yields that for all t ∈ [0, τ ] . Letting t ր τ yields a contradiction.Therefore m i (t) > 0 for all t ∈ [0, T ] which in turn implies that (2.3) holds for all t ∈ [0, T ].Remark also that the same estimate done on the interval [0, T ] yields the asserted bound on [0, T ].Point iv. is Proposition 7 in [10].

2.2
The graph limit equation.
In the graph limit we attach to the flow of System (1.1) the following "Riemman sums" (2.4) The functional Ψ : I × L ∞ (I) × L ∞ (I) → R and the functions x 0 : I → R d , m 0 : I → R are given and the functions ψ N i and the initial data x 0,N i , m 0,N i are defined in terms of these functions through the following formula If Ψ is given by then one readily checks that the ψ N i in Formula (2.2) are recovered via Formula (2.5).Notice that by Lebesgue's differentiation theorem x N (0, s), m N (0, s) well approximate x 0 (s), m 0 (s) because for a.e.s we have pointwise convergence Also, it is worthwhile remarking that unlike in the mean field limit regime, where the initial data realizing the initial convergence can be chosen from a set of full measure, here we use a very specific choice for the initial data, and in particular all initial data of the form specified by formula (2.6) constitute a set of measure 0, which means that the probabilistic methods that we have at our disposal in the mean field limit become useless in the graph limit.We will return to this point in Section 4.
The functions x N (t, s), m N (t, s) defined through Formula (5.2) are governed by the following equations, which should be compared with the graph limit Equation (1.4).
Proposition 2.2.Let the assumptions of Proposition 2.1 hold and let (x Proof.We start with the equation for x N (t, s).Fix s ∈ i0−1 N , i0 N , we get On the other hand, we have The equation for m N is obtained due to the following identities 3 Well Posedness for the Graph Limit Equation
The most restrictive assumption for the graph limit is H3' since S is not Lipschitz in problems of interest mentioned in the introduction, see [5,1].Furthermore, any solution to the graph limit equation (1.4) is expected to satisfy an estimate analogue to Inequality iv. in Lemma 2.1, namely which would imply Lipschitz continuity along the trajectories provided x 0 is one to one.This also leads us to remark that the initial separation in the microscopic system (1.5) can be replaced by the assumption that x 0 is one to one in the infinite dimensional case, which means that we need to be able to evaluate x 0 pointwise, and therefore a more natural assumption is x 0 ∈ C(I) rather than x 0 ∈ L ∞ .To summarize, in contrast to [1], we assume the hypotheses: ii. x 0 ∈ C(I) is one to one and x 0 ≤ X for some X > 0.
H3 i.The restrictions s| (0,∞) and s| (−∞,0) are Lipschitz, i.e. there is some S > 0 such that Clearly, the sign function is a particular example of hypothesis H3.In the following Lemma, which is a variant of [1, Lemma 3], the new considerations discussed above will be taken into account.
1. Suppose that Then, for all t ∈ [0, T ] it holds that Then, for all t ∈ [0, T ] it holds that Proof.
Step 1.For readability, we suppress the time variable (unless unavoidable).Set a(s, s * , s * * ) := a(x(s * * ) − x(s)) + a(x(s * * ) − x(s * ), we have Using the assumption that I m 1 (s)ds = I m 2 (s)ds = 1, the first integral in the right hand side of (3.3) can be estimated as Therefore, integrating (3.3) in s over I produces , we can also estimate as As a result, we obtain Lemma 3.2.Let hypotheses H1-H3 hold.Suppose also Then, there exists a unique solution (3.4) The solution x is such that s → x(t, s) is one to one and the solution m is non-negative such that I m(t, s)ds = 1.
Step 1. Existence and uniqueness for the equation for x.Fix 0 < T < . Let M x0 be the metric space of functions in C([0, T ] × I) with x(0, s) = x 0 (s).Define the operator We view M x0 as a complete metric space.We then have The choice of T ensures 2L m ∞,∞ T < 1, thereby making the Banach contraction principle available which implies there exist a unique solution x ∈ C([0, T ] × I) to the equation By a standard iteration argument we have existence and uniqueness on the whole interval [0, T ].
Evidently the map τ → I m(τ, s * )a(x(τ, s * ) − x(τ, s))ds * is continuous so that by the fundamental theorem of calculus we conclude x ∈ C 1 ([0, T ]; C(I)).Next we claim that this solution must be one to one.
Then for all t ∈ [0, T ] and all s 1 , s 2 ∈ I it hold that In particular, s → x(t, s) is increasing.
Proof.We start by showing that |x(t, s 2 ) − x(t, s 1 )| 2 > 0 for all t ∈ [0, T ].Assume to the contrary there is some t ∈ [0, T ] and s 2 > s 1 such that x(t, s 2 ) = x(t, s 1 ) and set Then for all t ∈ [0, τ 0 ) we have |x(t, s 2 ) − x(t, s 1 )| > 0 and as a result which in turn gives the inequality Taking t ր τ 0 gives a contradiction.Repeating now the estimate (3.5) shows that in fact for all By continuity and the assumption that x 0 is increasing it follows that s → x(t, s) is increasing.
Step 2. Existence and uniqueness for the equation for m.
We start by observing that K maps M m0 into itself.
Proof.To see why K m0 (m) is non-negative notice that because how T was chosen we have To show that K m0 (m) has unit integral we use that s is odd.Changing variables s ←→ s * and using that s is odd, the second integral in the right hand side is recast as We view M m0 as a complete metric space.Let m, n ∈ M m0 .Thanks to point 1. in Lemma 3.1 we have and thus The choice of T makes the Banach contraction theorem available thereby ensuring the existence of a unique solution m ∈ M m0 on [0, T ] to the equation Moreover, from the choice of T > 0 we evidently have

long time.
Let m(t, s) be the unique solution on [0, T ] to given by step 2.1.Then we obtain and as a result we deduce that Then we get a solution on 0, 2 × 1 16LXS∞τ .Iterating the process k > 16LT XS ∞ τ times we get existence and uniqueness of a solution on [0, T ].We claim now to have the upgrade m ∈ C 1 ([0, T ]; L ∞ (I)).Indeed, we have Taking into account (3.6), we finally conclude m ∈ C 1 ([0, T ]; L ∞ (I)).

The coupled equation
The well posedness for the decoupled equation serves as the main tool for proving well posedness of the original system.We prove Proof.
Step 1. Existence.We define recursively the following sequence of functions (x n , m n ): i.For all t ∈ [0, T ] and all s ∈ I we set x 0 (t, s) = x 0 (s) and for all t ∈ [0, T ] and a.e.s ∈ I we set m 0 (t, s) = m 0 (s).
ii.If (x n−1 , m n−1 ) have been defined we define (x n , m n ) to be the unique solution guaranteed by Lemma 3.2 to the equation Start by noting that x n is uniformly bounded (with respect to n) in the space C([0, T ] × I).We have This also implies a uniform bound in n for the weights since in view of Inequality (3.7) The proof of existence essentially boils down to proving that (x n , m n ) is a Cauchy sequence in the space C([0, T ]; C(I)) ⊕ C([0, T ]; L 1 (I)).

Estimate for sup
Integrating in s ∈ I gives Utilizing Lemma 3.1 shows that the first inner integral is whereas the second inner integral is As a result, we get Estimate for u n (t) := sup Collecting the Inequalities (3.9) and (3.10) we find which by easy induction implies It follows that sup x n (t, s) = x 0 (s) We explain how the passage to the limit as n → ∞ in the equation for m n is done, and the passage for the equation of x n is a standard verfication left to the reader.By Claim 3.1 we have for all Therefore, Lemma 3.1 is applicable and entails Hence, it follows that the right hand side in the equation for which by uniqueness of the limit implies that for all t ∈ [0, T ] and a.e.s ∈ I we have ) is exactly by the same reasoning of Lemma 3.2.
Step 2. Uniqueness.Suppose we are given 2 solutions (x 1 , m 1 ) and (x 2 , m 2 ) with the same initial data.We have In addition, Lemma 3.1 yields It follows that and this a fortiori forces m 1 = m 2 and 4 The Graph Limit and consequences.
This section is devoted to obtaining a Gronwall estimate on the time dependent quantity ξ N (t)+ ζ N (t), where , where x N , m N are given by Formula (5.2) and x, m are the corresponding solutions to Equation (3.8).We modify the argument demonstrated in Theorem 1 in [1] to our weakly singular settings.The estimate for ζ N (t) reflects the main novelty of this section.The estimates we obtain are locally uniform in time.The symbol stands for inequality up to a constant which may depend only on L, M, X, T, S, S ∞ .The main theorem is Theorem 4.1.Let the hypotheses H1-H3 hold.Let (x, m) ∈ C 1 ([0, T ]; C(I)) ⊕ C 1 ([0, T ]; L ∞ (I)) be the solution to Equation (3.8).Let (x N , m N ) ∈ C 1 [0, T ] ; R 2N be the solution to the system (2.1).Then x Proof.
Step 1.The time derivative of ζ N (t).The estimate for the time derivative of ζ N (t) reflects the main difference with the argument in [1].The time derivative of ζ N is computed as follows.
By Lebesgue differentiation theorem, for each t ∈ [0, T ] it holds that g N (t, s) → N →∞ 0 pointwise a.e.s ∈ I.In addition, x C([0,T ]×I) , m C([0,T ];L ∞ (I)) are bounded which implies that g N (t, s) is uniformly bounded (with respect to N ) so that by the dominated convergence theorem we find that the second integral in (4.1) is where for each t ∈ [0, T ] it holds that For the first integral, note Note that at this stage we cannot quite appeal to the Estimate (3.1) since it was formulated for x which are one to one in the variable s .The main difference is in the estimate of the second integral, which is now bounded by x N (t, •) − x(t, •)2 2 up to an error term which decays to 0 as N → ∞.Precisely put Lemma 4.1.It holds that i. ii.
Proof.Unless unavoidable, we supress the time variable.i. Thanks to Lemma 2.1 we have The estimate i. is almost identical to the estimate demonstrated in (3.1), the only minor difference being that here we take the The first integral is Therefore squaring and integrating in s over I, Inequality (4.4) produces ii.We have Let us concentrate on the estimate of I m 2 (t, s) |J 2 (t, s)| 2 ds.For each s ∈ I set We abbreviate We estimate the integral as follows By the assumption H3 we have Recall that by Lemma 2.1 and Claim 3.1 there exist a constant C > 1 such that By Lebesgue differentiation theorem, for a.e.s ∈ I it holds that a.e.s * .For all s ∈ I the set s * x 0 (s * ) = x 0 (s) (being an atom) is null due to H2, and therefore for a.e.s it holds that a.e.s * .By dominated convergence we obtain which shows The same reasoning also shows that The combination of (4.5), (4.6), (4.8) and (4.9) implies the announced claim.
Gathering i.,ii.and (4.2) gives Step 2. The time derivative of ξ N (t).The time derivative of ξ N (t) is mastered exactly as in [1].Following the argument in [1] one finds that Step 3. Conclusion.The combination of Inequalities (4.10) and (4.11) yields Applying Gronwall's lemma entails 2 is uniformly bounded, by (4.3) and dominated convergence which concludes the proof.
In the last part of this section, we recall how to obtain as a consequence a special version of the mean field limit for the empirical measure associated with the System (2.1).We start by pointing out that currently the existing literature does not cover the well posedness theory of the mean field equation, namely the non-local non-homogeneous transport equation where S(x, y, z)dµ(t, y)dµ(t, z), S(x, y, z) = 1 2 (a(z − y) + a(z − x)) s(x − y).

The case d > 1
In this section we explain how to extend the graph limit for arbitrary higher dimensions d > 1.In some places the proof requires only minor modifications and we therefore concentrate only on the parts which require special treatment.

The graph limit equation d > 1
The first notable difference in comparison to the case d = 1 (or the work [1]) is reflected in the definition of the Riemann sums.Instead of labeling the opinions along a multi-index of length d we label them along a multi-d-dimensional matrix of indices.This is a particular case of the metric valued labelling procedure introduced in [12] when the labelling space is [0, 1] d .At the level of the graph limit equation this choice corresponds to considering the equation posed on [0, T ] × I d rather than [0, T ] × I. Indeed, the fact that x(t, •) is a map from I d to itself enables to consider bi-Lipschitz initial data, which is crucial for the sake of properly analyzing the singularity in s as is clarified in Lemma 5.1.This labelling procedure does not have any modelling interpretation since particles (opinions) are still exchangeable or indistinguishable, it is solely needed for pure technical reasons.As we mentioned at the beginning of the paper, it would still be possible to go back to using [0, 1] as a labeling space through a change of variable, since [0, 1] and [0, 1] d are isomorphic as measurable spaces per the Borel isomorphism theorem.But, obviously, this would lead to painful technical assumptions to replace the bi-Lipschitz condition on x(t, .), while the analysis is otherwise much more transparent when considering [0, 1] d .Precisely put, we take the number of opinions to be perfect powers of d in which case the opinion dynamics system becomes the following system of (d + 1) We attach to the flow of System (5.1) the following "Riemman sums" like quantities, as in the one dimensional case, defined by Here the labeling variable s varies on the d-dimensional unit cube I d .Generalizing the constructions of Section 2.2, the functional Ψ : m 0 (s)ds. (5.5) If Ψ is given by then one readily checks that the ψ N i in Formula (2.5) are recovered via Formula (5.4).Notice that x N (0, s) and m N (0, s) well approximate x 0 (s), m 0 (s) because by Lebesgue's differentiation theorem for a.e.s ∈ I d we have pointwise convergence The functions x N (t, s), m N (t, s) defined through formulas (5.2), (5.3) are governed by the following equation, which is the obvious higher dimensional version of Equation (1.4).Proposition 5.1.Let the assumptions of Proposition 2.1 hold and let (x N (t), m N (t)) be the solution to System (5.1) on [0, T ].Let x N , m N be given by (5.2) and (5.3) respectively.Then (5.6) Proof.We start with the equation for x N (t, s).Fix s ∈ Q i0 .Then On the other hand The equation for m N is due to the following identities

Well posedness for d > 1
The point which requires most care for the proof of well posedness is point 2. in Lemma 3.1.Let us first state the assumptions we impose on the initial data and the other functions involved.
ii.There is some locally Remark 5.1.The assumption A2 that x 0 is bi-Lipschitz is strictly stronger than the assumption that it is 1- is a particular example of hypothesis A3 as can be seen from through the following elementary inequalities Furthermore, it is clear that the condition i. in A3 is more general than the assumption s ∈ Lip(R d ).
• The function x 2 is bi-Lipschitz in the labeling variable, i.e. there is some C > 1 such that for all (t, s, s Then where we define We start with the estimate on J 2 (t, s).Using assumption A3, we have that From the bi-Lipschitz assumption on The estimate for J 1 (t, s) follows in a similar way,  The symbol stands for inequality up to a constant which may depend only on L, L 0 , M, X, T, S, A2S ∞ .By Lebesgue differentiation theorem, for each t ∈ [0, T ] it holds that g N (t, s)→0 as N → ∞ pointwise a.e.s ∈ I d .In addition, x C([0,T ]×I d ) , m C([0,T ];L ∞ (I d )) are bounded which implies that g N (t, s) is uniformly bounded (with respect to N ) so that by the dominated convergence theorem we find that for each t ∈ [0, T ] it holds that g N (t, •) 1 → N →∞ 0. We now estimate the first integral as   for some compact set K ⊂ R d .
Step 2. The time derivative of ξ N (t).The time derivative of ξ N (t) is mastered exactly as in [1].Following the argument in [1] one finds that ξN (t) ξ N (t) + ζ N (t). (5.8) Step  Remark 5.3.Note that Theorem 5.1 proves convergence with respect to the L 1 norm, whereas Theorem 4.1 proves convergence with respect to the L 2 norm.This minor difference is because when d = 2 the L 2 norm of 1 |s| blows up, which prevents getting the inequality ii. in Lemma 5.2.Notice that for any d ≥ 3, the L 2 approach is perfectly valid.Remark 5.4.Essentially the same argument of Theorem 4.2 allows one to conclude the weak mean field limit from the graph limit in higher dimension.

( 1 ≤
and the functions x 0 : I d → R d , m 0 : I d → R are given and the functions ψ N i and the initial data x 0,N i , m 0,N i i ≤ N d ) are defined in terms of these functions through the following formulas m(s) m(s * ) (|s(x 1 (s) − x 1 (s * ))| + |s(x 2 (s) − x 2 (s * ))|) ds ds * ≤ 6 L M S L 1 (K) sup I d |x 1 (t, •) − x 2 (t, •)| .