AA NOTE ON FOKKER-PLANCK EQUATIONS AND GRAPHONS
FABIO COPPINI
Abstract.
Fokker-Planck equations represent a suitable description of the finite-timebehavior for a large class of particle systems as the size of the population tends to infinity.Recently, the theory of graph limits have been introduced in the mean-field framework toaccount for heterogeneous interactions among particles. In many instances, such networkheterogeneity is preserved in the limit which turns from being a single Fokker-Planckequation (also known as McKean-Vlasov) to an infinite system of non-linear partialdifferential equations (PDE) coupled by means of a graphon. While appealing froman applied viewpoint, few rigorous results exist on the graphon particle system. Thisnote addresses such limit systems focusing on the relation between initial conditionsand interaction network: if the system initial datum and the graphon degrees satisfya suitable condition, a significantly simpler representation of the solution is available.This in turn implies that very different graphons can lead to exactly the same particlebehavior, shedding some light on the network influence on the dynamics.Examples of such representation are provided. In particular, step kernels representa class of graphons to which our result applies: this in turn opens the way to approxi-mate the graphon particle system with a finite system of Fokker-Planck equations. Asa byproduct, we show that when the initial condition is uniform, every graphon withconstant degree leads to a behavior indistinguishable from the well-known mean-fieldlimit.
Keywords:
Fokker-Planck equation, graphons, mean-field systems, Interacting particles,McKean-Vlasov, graphon particle system, step kernels. Introduction, model and literature
Introduction, aim of this note and organization.
In the last years, the study ofinteracting particle systems with a non-trivial dense network structure has been repeatedlyaddressed in the mathematical community: see, e.g., [4, 10, 21, 9] for interacting diffusions,[5, 6, 20] for applications in mean-field games and [11] in the context of dynamical systems.Depending on the setting, many results on interacting particle systems are nowadaysavailable [1, 3, 8, 7, 17] whenever the underlying graph sequence is converging, in somesense depending case by case, to a suitable object. More precisely, if the graph limit isa graphon, then, as the size of the system tends to infinity, the finite-time populationbehavior is suitably described by an infinite system of coupled non-linear Fokker-Planckequations, the coupling between equations being made by the graphon limit itself (seeequation (1.1) for a simple example).This note addresses the graphon particle system obtained in the limit and tries to makeclear how potentially different graphons can lead to the same particle behavior. Sucha perspective is important not only to better understand the limit system, but also incase one is interested in reconstructing the network structure by looking at the particledynamics: we prove that, depending on the initial conditions, the class of suitable graphons a r X i v : . [ m a t h . P R ] F e b FABIO COPPINI can be potentially very large. As it will be clear later on, the relation between the graphonand the initial datum plays the most important role.Graphons have been used as a model for many real-world networks, yet, to the author’sknowledge, known results on graphon particle systems are limited to existence and unique-ness of solutions; we refer to Subsection 1.3 for the current literature. A mathematicalstudy of the limit object may lead to a better understanding of the complex phenomenaat the heart of these models.A labeled graphon (we use the same notation of [15]) is a symmetric measurable function W defined on the unit-square W : [0 , → [0 , x, y ) (cid:55)→ W ( x, y ) . If the limit population is represented as a continuum of particles labeled by the unit interval[0 , W ( x, y ) stands for the connection strength between the particle labeled with x and the one labeled with y . The function W thus describes the connection networkunderlying a (possibly infinite) particle population.Fix a finite time horizon T >
0. For one-dimensional particles which are interactingthrough a (regular enough) function Γ, the graphon particle system is an infinite systemof partial differential equations (PDE) coupled by means of W , i.e., ∂ t µ t ( θ, x ) = 12 ∂ θ µ t ( θ, x ) − ∂ θ (cid:20)(cid:90) W ( x, y ) µ t ( θ, x ) (cid:90) R Γ( θ, θ (cid:48) ) µ t (d θ (cid:48) , y )d y (cid:21) , x ∈ [0 , , (1.1)for t ∈ [0 , T ] and where µ = { µ t ( · , x ) , t ∈ [0 , T ] } x ∈ [0 , is a collection of probability mea-sures. The initial datum is given by a probability measure µ ∈ P ( R × [0 , µ ( · , x ) ∈ P ( R ) for almost every x ∈ [0 , µ ( x ) := µ ( · , x ) = { µ t ( · , x ) , t ∈ [0 , T ] } ∈ P ( C ([0 , T ] , R ))represents the law of the trajectory associated to the x -labeled particle, this last onebeing connected to the others by means of W ( x, · ) : [0 , → [0 , , T ] to R , i.e., an element of the space C ([0 , T ] , R ).To the system of non-linear Fokker-Planck equations (1.1), it is associated a family ofcontinuous processes { θ x } x ∈ [0 , ⊂ C ([0 , T ] , R ), which solve (cid:40) θ xt = θ x + (cid:82) t (cid:82) W ( x, y ) (cid:82) R Γ( θ xs , θ ) µ s (d θ, y )d y d s + B xt , t ∈ [0 , T ] ,µ t ( · , y ) = L ( θ yt ) , for t ∈ [0 , T ] and y ∈ [0 , , (1.2)where the law of the initial condition θ x , denoted by L ( θ x ), is thus given by µ ( x ). Thefamily { B x } x ∈ [0 , is composed of independent and identically distributed (IID) Brownianmotions, independent of the initial conditions as well.The link between (1.1) and (1.2) is given by the fact that µ t ( x ) = L ( θ xt ) for every t ∈ [0 , T ] and x ∈ [0 , σ ≡ NOTE ON FOKKER-PLANCK EQUATIONS AND GRAPHONS 3 their mathematical properties. We thus aim at mathematically addressing (1.1), firstlyby showing the strong link with the graphon theory (including unlabeled graphons, seeProposition 1.5), secondly by proving that a simpler representation for both (1.1) and(1.2) exists, whenever the initial conditions and the underlying graphon satisfy a suitablecondition, see Theorem 2.1 and, in particular, Corollary 2.3 and Proposition 2.6.The note is organized as follows: the next subsection presents the general model asso-ciated to (1.2), together with known results from the existing literature. Related worksare discussed at the end of this first section.Section 2 provides the main result, Theorem 2.1, as well as two corollaries and a relevantapplication to step kernels, see Proposition 2.6. Notably, Corollary 2.4 addresses theclassical mean-field scenario showing that, if every particle has the same initial law, thenfor every graphon with constant degree, i.e., such that d ( · ) = (cid:82) W ( · , y )d y is constant, thedynamics is mean-field.Section 3 contains the mathematical set-up and the proof of the main result.1.2. The model and some known results.
We consider a class of models slightlymore general than (1.2): fix µ ∈ P ( R × [0 , { θ x } x ∈ [0 , be the family solving the ∞ -dimensional coupled system: (cid:40) θ xt = θ x + (cid:82) t F ( θ xs )d s + (cid:82) t (cid:82) W ( x, y ) (cid:82) R Γ( θ xs , θ ) µ s (d θ, y )d y d s + (cid:82) t σ ( θ xs )d B xs µ t ( · , y ) = L ( θ yt ) , for t ∈ [0 , T ] and y ∈ [0 , , (1.3)where F , Γ and σ are 1-Lipschitz functions bounded by 1, and { B x } x ∈ [0 , is a family ofIID Brownian motions on R . For every x ∈ [0 , θ x is a randomvariable with law given by L ( θ x ) = µ ( x ), independent of the other initial conditions andof the Brownian motions. The probability measure induced by the initial conditions andthe family of Brownian motions is denoted by P and the corresponding expectation by E .We work under the following assumptions. Hypothesis 1.1.
We assume that (1) (measurability) The map [0 , (cid:51) x (cid:55)→ µ ( x ) ∈ P ( R ) is measurable; (2) (moment condition) E [ | θ x (0) | ] < ∞ for every x ∈ [0 , . Remark 1.2.
Measurability with respect to x is somehow necessary since we are workingwith measurable functions W on the unit interval [0 , ; we will explicitly use it in thefollowing relation (2.1) .The finite second moment condition is related to the use of the 2-Wasserstein distancebetween probability measures (see the Section 3 and, in particular, (3.1) ). It is obviouslypossible to work with initial conditions with finite p -moment for every p (cid:62) , and thechoice p = 2 is purely arbitrary. We stick to the one-dimensional setting, i.e., θ x taking values in R ; however, all theproofs presented below are easily extendable to any finite dimension.Existence and uniqueness for system (1.3) are known. Proposition 1.3.
Under the measurability assumption and the moment condition in Hy-pothesis 1.1, there exists a unique pathwise solution to system (1.3) . FABIO COPPINI
Moreover, if we denote by µ ( x ) the law of θ x for each x ∈ [0 , , then the map [0 , (cid:51) x (cid:55)→ µ ( x ) ∈ P ( C ([0 , T ] , R )) is measurable and µ ( x ) weakly solves ∂ t µ t ( θ, x ) = 12 ∂ θ (cid:0) σ ( θ ) µ t ( θ, x ) (cid:1) − ∂ θ ( µ t ( θ, x ) F ( θ )) − ∂ θ (cid:20)(cid:90) W ( x, y ) µ t ( θ, x ) (cid:90) R Γ( θ, θ (cid:48) ) µ t (d θ (cid:48) , y )d y (cid:21) , (1.4) for every x ∈ [0 , . We refer to [1, Proposition 2.1] and [17, Proposition 2.4] for two standard similar proofs.We point out that W needs only to be bounded and not necessarily with values in [0 , W , as in the notation below.Let W := { W : [0 , → R bounded, symmetric and measurable } be the space ofkernels . The cut-norm of W ∈ W is defined as (cid:107) W (cid:107) (cid:3) := max S,T ⊂ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S × T W ( x, y )d x d y (cid:12)(cid:12)(cid:12)(cid:12) (1.5)where the maximum is taken over all measurable subsets S and T of I . Let W := { W ∈W : 0 (cid:54) W (cid:54) } be the space of labeled graphons: for W, V ∈ W their cut-distance isdefined by δ (cid:3) ( W, V ) := min ϕ ∈ S [0 , (cid:107) W − V ϕ (cid:107) (cid:3) , (1.6)where the minimum ranges over S [0 , the space of invertible measure preserving mapsfrom [0 ,
1] into itself and where V ϕ ( x, y ) := V ( ϕ ( x ) , ϕ ( y )) for x, y ∈ [0 , δ (cid:3) is a pseudometric on W since it can be zero between two differentlabeled graphons. If we identify all labeled graphons with cut-distance zero, we obtain thespace of graphons (cid:102) W := W /δ (cid:3) . A well-known result of graph limits theory says that( (cid:102) W , δ (cid:3) ) is a compact metric space [15, Theorem 9.23].Proposition 1.3 can thus be restated by saying that for every kernel W ∈ W , thereexists a unique solution µ W to (1.4). It is thus natural to ask whether the application W (cid:55)→ µ W is continuous (e.g., in the topology of the weak-convergence) with respect to (cid:107)·(cid:107) (cid:3) . This point is discussed in the next remark. Remark 1.4.
Under suitable assumptions on the coefficients, see [3, Proposition 3.3] butalso [1, Theorem 2.1] and [14, Lemma 2.7] , it is possible to show that for two solutions µ W and µ V associated to W and V in W respectively, it holds that D T ( µ W , µ V ) (cid:54) C (cid:107) W − V (cid:107) (cid:3) , for some C > , (1.7) where D T is some distance on probability measures metricizing the weak-convergence.Equation (1.7) proves that the application W (cid:55)→ µ W is (H¨older-)continuous and, thus,that similar graphons (in cut-norm) leads to similar particle behaviors. However, it doesnot say anything whether two different graphons (possibly in δ (cid:3) -distance) lead to similarbehaviors. As Theorem 2.1 and its corollaries show, there are relevant examples where thishappens and where it is possible to prove that the particle dynamics coincide. For every W ∈ W , Proposition 1.3 shows that the infinite system (1.4) admits a uniquesolution, yet it does not address the unlabeled class of W in (cid:102) W . We explicit such relationin the next proposition. we always consider two kernels to be equal if and only if they differ on a subset of Lebesgue measurezero. We follow closely the notation in [15]. NOTE ON FOKKER-PLANCK EQUATIONS AND GRAPHONS 5
Proposition 1.5.
Let U be a uniform random variable on [0 , and θ U the non-linearprocess in { θ x } x ∈ [0 , with random label U . Then, the law of θ U is given by (cid:101) µ = (cid:82) µ ( x )d x and it is independent of the class of W in (cid:102) W .Proof. This result was firstly stated in [3, Proposition 2.1] in the case of interacting oscil-lators. In case the particles are living in R the proof does not change. Indeed, let ϕ be ameasure preserving map from [0 ,
1] to itself, we observe that µ ( ϕ ( x )) solves: ∂ t µ t ( θ, ϕ ( x )) = 12 ∂ θ (cid:0) σ ( θ ) µ t ( θ, ϕ ( x )) (cid:1) − ∂ θ ( µ t ( θ, ϕ ( x )) F ( θ )) − ∂ θ (cid:20)(cid:90) W ( ϕ ( x ) , y ) µ t ( θ, ϕ ( x )) (cid:90) R Γ( θ, θ (cid:48) ) µ t (d θ (cid:48) , y )d y (cid:21) , (1.8)where the last term is equal to ∂ θ (cid:20)(cid:90) V ( x, y ) µ t ( θ, ϕ ( x )) (cid:90) R Γ( θ, θ (cid:48) ) µ t (d θ (cid:48) , ϕ ( y ))d y (cid:21) with V ( x, y ) = W ( ϕ ( x ) , ϕ ( y )). Thus { µ ( ϕ ( x )) } x ∈ [0 , solves the same system of { ν ( x ) } x ∈ [0 , solution to (1.4) with V ∈ W and initial condition { µ ( ϕ ( x )) } x ∈ [0 , . Clearly (cid:101) µ = (cid:82) µ ( x )d x = (cid:82) µ ( ϕ ( x ))d x = (cid:82) ν ( x )d x = (cid:101) ν . (cid:3) Proposition 1.5 makes clear that, as for a graphon the relevant information is indepen-dent of the labeling, the same holds true for the behavior of an interacting particle system:it does not change under relabeling of the particles. Surprisingly, this result has never beenstated in the current literature (with the exception of [3] for interacting oscillators).1.3.
Related works.
The graphon framework [16, 15] has been introduced in the the-ory of particle systems [7, 18, 19, 21] as an useful ingredient to construct inhomogeneousrandom graph sequences with nice statistical properties (edge independence, graph homo-morphism, etc.). Most of the known literature on particle systems has been focused onthe converging properties of particle systems on such random graph sequences, see, e.g.,[1, 3, 17, 21]. They show that some of the classical mean-field arguments (as propagationof chaos [22]) can be extended to deal with random graphs and to include labeled graphonsin the limit description. Nevertheless, only little attention has been put into the study ofthe limit object (1.4) which remains, to the author’s knowledge, rather unknown.The recent work [3] shows a direct connection between particle systems and the graphontheory: under suitable hypothesis on (1.2), there exists a H¨older-continuous mappingbetween the space of graphons ( (cid:102) W , δ (cid:3) ) and (cid:101) µ as in Proposition 1.5. This allows toconsider general graph sequences, as exchangeable random graphs [12], which leads toa (possibly) random graphon W in the limit. However, despite the rather understoodconvergence properties, only a few insights are available on the limit particle system, werefer to the examples in Subsection 2.3 of [3].Although interesting on itself, the study of the limit Fokker-Planck equation does notnecessarily provide a suitable description of particle systems on diverging time scales, asalready raised in [9, 8] for the mean-field case. With the exception of dissipative dynamics[2], a substantial understanding of the phase space of (1.4) is needed to study the long-time behavior of finite particle systems on graphs as shown in [8]. It is thus important tobetter understand system (1.4) in order to address the finite particle system counterpart. FABIO COPPINI Main result and discussion
We suppose that the initial condition µ ∈ P ( R × [0 , W ∈ W are fixed.Before stating the main result, we introduce an equivalence relation on the unit intervaland give the definition of degree in W . For x and y in [0 , ∼ by x ∼ y if and only if µ ( x ) = µ ( y ) , (2.1)i.e., we identifies two labels whenever the initial conditions of the corresponding particleshave the same law. Denote by J the quotient space [0 , / ∼ and, for ¯ x ∈ J , denote itsorbit by [¯ x ] := { x ∈ [0 ,
1] : x ∼ ¯ x } . For some x ∈ [0 , x by d ( x ) = (cid:82) W ( x, y )d y . More generally,we define the degree of x with respect to a (measurable) subset A ⊂ [0 ,
1] by d A ( x ) = (cid:90) A W ( x, y )d y. (2.2)Observe that under the measurability assumption (1.1), [¯ y ] is a measurable subset of [0 , y ∈ J ; with a slight abuse of notation, we denote d ¯ y ( · ) := d [¯ y ] ( · ).2.1. Main result, examples and applications.
We have the following theorem.
Theorem 2.1.
Assume Hypothesis 1.1. Suppose that, for every ¯ x and ¯ y in J it holds that d ¯ y ( x ) = (cid:90) [¯ y ] W ( x, y )d y = (cid:90) [¯ y ] W (¯ x, y )d y = d ¯ y (¯ x ) for all x ∈ [¯ x ] . (2.3) Then for every ¯ x ∈ J , µ ( x ) = µ (¯ x ) , for all x ∈ [¯ x ] . Condition (2.3) is requiring that, if we partition the interval [0 ,
1] in J subsets { [¯ x ] } ¯ x ∈ J ,then the density of neighbors with respect to any of these subsets, i.e., d ¯ y ( · ) for some¯ y ∈ J , is piecewise constant on [0 ,
1] and completely characterized by the values on J . SeeFigure 1 (B) and Figure 2 (A) for graphons that satisfies (2.3).Theorem 2.1 basically states that, if one can group the particles so that within eachgroup they have the same initial condition (in law) and are equi-connected to the othergroups in the sense of (2.3), then the solution µ to (1.4) is constant on these groups.We observe that this result does not depend on the class of W as unlabeled graphon. Asa consequence, one can relabel the particles such that µ is piecewise constant as a functionof x ∈ [0 , Remark 2.2.
It is possible to weaken (2.3) by considering a larger J based on verticeswhich have not only the same initial law, but also the same degree in W . We refrainfrom increasing the complexity in the construction of J to keep the main ideas as clear aspossible. This aspect is briefly discussed in the application to step kernels below. A direct consequence of Theorem 2.1 is that the information in { µ ( x ) } x ∈ [0 , is containedin the possibly much smaller object { µ (¯ x ) } ¯ x ∈ J . Corollary 2.3.
Under the assumptions of Theorem 2.1. The infinite system (1.4) issuitably described by the (possible finite) system of coupled partial differential equations ∂ t ν t ( θ, ¯ x ) = 12 ∂ θ (cid:0) σ ( θ ) ν t ( θ, ¯ x ) (cid:1) − ∂ θ ( ν t ( θ, ¯ x ) F ( θ )) − ∂ θ (cid:20)(cid:90) J W (¯ x, ¯ y ) ν t ( θ, ¯ x ) (cid:90) R Γ( θ, θ (cid:48) ) ν t (d θ (cid:48) , ¯ y )d¯ y (cid:21) , ¯ x ∈ J, (2.4) NOTE ON FOKKER-PLANCK EQUATIONS AND GRAPHONS 7 (a)
Constant graphon (b)
Disconnected graphon (c)
Cayley graphon
Figure 1.
Examples of graphons that satisfy (2.5) (constant degree). Ob-serve that the graphon in (B) is composed of two connected components,the smaller one being more densely connected (darker) with respect to theother one so to satisfy (2.5). Finally, observe that (A), (B) and (C) are dif-ferent both as labeled and unlabeled graphons, i.e., different in δ (cid:3) -distance. with ν = µ . Notably, µ ( x ) = ν (¯ x ) for every ¯ x ∈ J and x ∈ [¯ x ] .Proof. Existence and uniqueness for system (2.4) directly follow from Proposition 1.3.Moreover, system (2.4) is written in closed form and can be thus solved independently of(1.4). (cid:3)
Graphons have proven to be an important tool for establishing the convergence ofparticle systems on graph sequences. Notably, it is now possible to show that interactingparticle systems on apriori different graph sequences have the same asymptotic behaviorwhenever the limit of these sequences coincides as unlabeled graphon. However, differentunlabeled graphons can lead to the same particle behavior: in these cases, the limit (1.4)tends to be a formal object instead of giving a precise description. Theorem 2.1 andCorollary 2.3 precise that both network structure and initial conditions are necessary tounderstand the system evolution.We now turn to an interesting application when | J | = 1, i.e., when the law of the initialcondition is label independent. It turns out that condition (2.3) boils down to a constantdegree assumption on W . Corollary 2.4.
Under the assumptions of Theorem 2.1. Suppose that µ ( x ) = µ ∈ P ( R ) for every x ∈ [0 , . Then there exists p ∈ [0 , such that condition (2.3) is equivalent to p = (cid:90) W ( x, y )d y, for all x ∈ [0 , . (2.5) In particular, { µ ( x ) } x ∈ [0 , is label independent, i.e., µ ( · ) ≡ µ ∈ P ( C ([0 , T ] , R )) , and µ solves the classical McKean-Vlasov equation ∂ t ν t ( θ ) = 12 ∂ θ (cid:0) σ ( θ ) ν t ( θ ) (cid:1) − ∂ θ ( ν t ( θ ) F ( θ )) − ∂ θ (cid:20) ν t ( θ ) p (cid:90) R Γ( θ, θ (cid:48) ) ν t (d θ (cid:48) ) (cid:21) , (2.6) with initial condition µ .Proof. The hypothesis on µ forces | J | = 1 and thus condition (2.3) becomes (2.5). Corol-lary 2.3 yields the result. (cid:3) FABIO COPPINI
The constant degree assumption on W is satisfied by a large class of non-trivial graphonsas the examples shown in Figure 1. Remark 2.5.
Observe that Corollary 2.4 can be derived from the results in [10] com-bined with the ones on particle systems on graphons, e.g., [17, 1] . Indeed, in [10] it isshown that, for a particle system defined on a graph sequence and with IID initial condi-tions, a sufficient condition to obtain the mean-field limit (2.6) is that each vertex in the(renormalized) graph sequence has the same asymptotic degree density. If one chooses suchsequence to converge to a graphon with constant degree, then it satisfies both the hypothesisin [10, Theorem 1.1] and in, e.g., [1, Theorem 4.1] . Combining [10, Theorem 1.1] and [1,Theorem 4.1] , we obtain that the same finite particle system converges to (2.6) but alsoto (1.4) . As a consequence, the solutions of (1.4) and (2.6) must be the same, giving anundirect proof of Corollary 2.4.
We step to a representative example whenever the graphon is a piecewise constantfunction, as the one in Figure 2.
Finite representation for step-kernels.
For simplicity, we suppose that µ is label indepen-dent, i.e., µ ( x ) = µ ∈ P ( R ). However, observe that the strategy presented here can beapplied by suitably approximating the initial condition with a constant-wise initial datum:indeed, the continuity of the solution µ to (1.4) with respect to the initial condition is aclassical result [22].Step kernels ([15, § W ∈ W with constant-wise functions. A function W ∈ W is a step kernel if there is a partition P = { S i } i =1 ,...,k of [0 ,
1] into measurable setssuch that W is constant on every product set S i × S j . We use the following notation W P ( x, y ) = k (cid:88) i,j =1 w ij S i × S j ( x, y ) , for x, y ∈ [0 , , (2.7)where { w ij } i,j =1 ,...,k are bounded real numbers. See Figure 2 for an example with equidis-tant partition.For a step kernel, condition (2.3) is clearly satisfied by taking J = { x i } i =1 ,...,k with x i ∈ S i for every i = 1 , . . . , k , thus refining the strategy presented before Theorem 2.1, asanticipated in Remark 2.2.Applying Theorem 2.1 and Corollary 2.3, we have the following representation for thegraphon particle system (1.4) on the step graphon W P . Proposition 2.6.
Suppose that µ ∈ P ( R ) and let W be the step kernel in (2.7) . Thenthe graphon particle system (1.4) is suitably described by the finite collection of probabilitymeasures { µ i } i =1 ,...,k which solve ∂ t µ it ( θ ) = 12 ∂ θ (cid:0) σ ( θ ) µ it ( θ ) (cid:1) − ∂ θ (cid:0) µ it ( θ ) F ( θ ) (cid:1) − ∂ θ k (cid:88) j =1 w ij µ it ( θ ) (cid:90) R Γ( θ, θ (cid:48) ) µ jt (d θ (cid:48) ) , (2.8) and where µ i = µ , for i = 1 , . . . , k . As shown in Proposition 2.6, the representation given in Corollary 2.3 becomes naturalin the case of step kernels, and thus for graph limits arising from the stochastic blockmodel. While Proposition 2.6 could be derived with direct computations , observe that A step kernel can be seen as a Stochastic Block Model, for which the representation (2.8) is somehowknown.
NOTE ON FOKKER-PLANCK EQUATIONS AND GRAPHONS 9 (a)
Step kernel (b)
Scale free graphon
Figure 2.
Suppose that J = { x , x , x } and that [ x ] = [0 , / x ] = [1 / , /
3) and [ x ] = [2 / , , S i × S j in(2.7) by a (suitable scaled) graphon with constant degree, equation (2.8) does not change.We also observe that combining Proposition 2.6 with the continuity estimates as inRemark 1.4, allows to approximate the graphon particle system (1.4) by a finite numberof coupled Fokker-Planck equations. Since it is beyond the scope of this note, we do notpurse such analysis, yet we make this point a bit more precise in the next remark. Remark 2.7.
By using the Weak Regularity Lemma [15, Corollary 9.13] , one can approx-imate every graphon by a step kernel with an explicit control on the error. Assuming thecontinuity of (1.4) with respect to W , recall Remark 1.4, Proposition 2.6 opens the wayto approximate the infinite graphon particle system (1.4) by using a finite (thus apriorinumerically solvable) system of coupled Fokker-Planck equations, with precise bounds onthe error. Proof of the main result
Distance between probability measures.
For two probability measures ¯ µ, ¯ ν ∈P ( C ([0 , T ] , R )), define their 2-Wasserstein distance as D T (¯ µ, ¯ ν ) = inf X,Y (cid:40) E (cid:34) sup t ∈ [0 ,T ] | X t − Y t | (cid:35) : L ( X ) = ¯ µ, L ( Y ) = ¯ ν (cid:41) / (3.1)where the infimum is taken on all random variables X and Y with values in C ([0 , T ] , R )and law L equal to ¯ µ and ¯ ν respectively. From (3.1) we obtain that for every s ∈ [0 , T ]and for every bounded 1-Lipschitz function f (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R f ( θ ) ¯ µ s (d θ ) − (cid:90) R f ( θ ) ¯ ν s (d θ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R f ( θ ) [¯ µ s (d θ ) − ¯ ν s (d θ )] (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) D s (¯ µ, ¯ ν ) . (3.2)Observe that (3.1) also makes sense with T = 0 and C ([0 , T ] , R ) replaced by R . Proof of Theorem 2.1.
Recall the definition of the 2-Wasserstein distance D T in(3.1), we aim at showing that max ¯ x ∈ Jx ∈ [¯ x ] D T ( µ (¯ x ) , µ ( x )) = 0 . (3.3)We first assume F ≡ σ ≡ x ∈ J and x ∈ [¯ x ], without loss of generality, we can suppose that the associatedrealizations of the Brownian motion are the same. Using the fact that x ∈ [¯ x ], the initialconditions cancel and we have that θ xt − θ ¯ xt = (cid:90) t (cid:90) W ( x, y ) (cid:90) R Γ( θ xs , θ ) µ (d θ, y )d y d s − (cid:90) t (cid:90) W (¯ x, y ) (cid:90) R Γ( θ ¯ xs , θ ) µ (d θ, y )d y d s, (3.4)In particular, this can be rewritten as (cid:90) t (cid:90) W ( x, y ) (cid:90) R (cid:2) Γ( θ xs , θ ) − Γ( θ ¯ xs , θ ) (cid:3) µ (d θ, y )d y d s + (cid:90) t (cid:90) [ W ( x, y ) − W (¯ x, y )] (cid:90) R Γ( θ ¯ xs , θ ) µ (d θ, y )d y d s. (3.5)We now use hypothesis (2.3) and add the following term in the previous equation0 = (cid:90) J (cid:34)(cid:90) [¯ y ] [ W ( x, y ) − W (¯ x, y )] (cid:90) R Γ( θ ¯ xs , θ ) µ s (d θ, ¯ y )d y (cid:35) d¯ y, (3.6)where the integrals are sums whenever J or [¯ y ] are countable (recall that [¯ y ] is a measurablesubset of [0 ,
1] because of Hypothesis (1.1)).By taking the squares and using ( a + b ) (cid:54) a + b ), this leads to (cid:12)(cid:12) θ xt − θ ¯ xt (cid:12)(cid:12) (cid:54) T (cid:90) t (cid:12)(cid:12) θ xs − θ ¯ xs (cid:12)(cid:12) d s ++ 2 T (cid:90) t (cid:90) J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) [¯ y ] [ W ( x, y ) − W (¯ x, y )] (cid:90) R Γ( θ ¯ xs , θ )[ µ s (d θ, y ) − µ s (d θ, ¯ y )]d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d¯ y d s, (3.7)where we have used Cauchy-Schwartz inequality as well as the fact that Γ is 1-Lipschitz.Applying Cauchy-Schwartz again, and using (3.2), we are left with (cid:12)(cid:12) θ xt − θ ¯ xt (cid:12)(cid:12) (cid:54) T (cid:90) t (cid:12)(cid:12) θ xs − θ ¯ xs (cid:12)(cid:12) d s + 4 T (cid:90) t (cid:90) J (cid:34)(cid:90) [¯ y ] D s ( µ ( y ) , µ (¯ y ))d y (cid:35) d¯ y d s, (3.8)Thus, taking the supremum with respect to the time and the expected value E this leadsto E (cid:34) sup t ∈ [0 ,T ] (cid:12)(cid:12) θ xt − θ ¯ xt (cid:12)(cid:12) (cid:35) (cid:54) T (cid:90) T E (cid:34) sup u ∈ [0 ,s ] (cid:12)(cid:12) θ xu − θ ¯ xu (cid:12)(cid:12) (cid:35) d s ++ 4 T (cid:90) T (cid:90) J (cid:34)(cid:90) [¯ y ] D s ( µ ( y ) , µ (¯ y ))d y (cid:35) d¯ y d s. (3.9) NOTE ON FOKKER-PLANCK EQUATIONS AND GRAPHONS 11
Finally, we can take the maximum with respect to ¯ x ∈ J and x ∈ [¯ x ] and use the charac-terization of (3.1), to obtainmax ¯ x ∈ Jx ∈ [¯ x ] D T ( µ (¯ x ) , µ ( x )) (cid:54) T (cid:90) T max ¯ x ∈ Jx ∈ [¯ x ] D s ( µ (¯ x ) , µ ( x ))d s, (3.10)which implies (3.3).Whenever F is not zero, the proof is basically the same: using the Lipschitz propertiesof F , a term equal to | θ xs − θ ¯ xs | appears, thus adding the constant factor of 2 T to thefinal equation (3.9).If σ is not constant, then we may use the Burkholder-Davis-Gundy inequality (and theLipschitz property of σ ) to bound: E (cid:34) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T (cid:0) σ ( θ xs ) − σ ( θ ¯ xs ) (cid:1) d B s (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) (cid:54) CT (cid:90) T E (cid:34) sup u ∈ [0 ,s ] (cid:12)(cid:12) θ xu − θ ¯ xu (cid:12)(cid:12) (cid:35) d s, with C a universal positive constant. The rest of the proof remains unchanged. (cid:3) Acknowledgments
The author is thankful to Gianmarco Bet, Giambattista Giacomin and Francesca Nardifor discussions while writing this note.
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