A McKean-Vlasov SDE and particle system with interaction from reflecting boundaries
Michele Coghi, Wolfgang Dreyer, Paul Gajewski, Clemens Guhlke, Peter Friz, Mario Maurelli
aa r X i v : . [ m a t h . P R ] F e b A McKean-Vlasov SDE and particle system with interactionfrom reflecting boundaries
Michele Coghi ∗ , Wolfgang Dreyer † , Paul Gajewski ‡ , Clemens Guhlke § , PeterFriz ¶ , and Mario Maurelli ‖ Institut f¨ur Mathematik, Technische Universit¨at Berlin, Straße des 17. Juni 136,10623 Berlin, Germany Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39,10117 Berlin, Germany Dipartimento di Matematica, Universit`a degli Studi di Milano, via Saldini 50,20133 Milano, ItalyFebruary 25, 2021
Abstract
We consider a one-dimensional McKean-Vlasov SDE on a domain and the associated mean-field interacting particle system. The peculiarity of this system is the combination of theinteraction, which keeps the average position prescribed, and the reflection at the boundaries;these two factors make the effect of reflection non local. We show pathwise well-posedness forthe McKean-Vlasov SDE and convergence for the particle system in the limit of large particlenumber.
Contents ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ‖ [email protected] The setting and the main results 12 BV estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3.2 H¨older estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Convergence of the particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5 Pathwise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 In this paper, we consider a system of N interacting one-dimensional diffusions, with the followingtwo main features: a) they are confined in a bounded domain with reflecting boundaries; b) theirempirical average is prescribed. We show that, as the number of particles N goes to infinity, thesystem converges to the unique solution to a suitable McKean-Vlasov SDEs on the domain. The model and the results . The propotypical example is the following one:d X i,Nt = d W it + d K Nt − d k i,Nt ,X i,Nt ∈ [0 , , d k i,Nt = n ( X i,Nt )d | k i,N | t , d | k i,N | t = 1 X i,Nt ∈{ , } d | k i,N | t , N N X i =1 X i,Nt = q. (1)Here W i are independent real Brownian motions, q is the given average in (0 , n is the outer nor-mal on ∂ [0 ,
1] = { , } and the solution is a triple X ( N ) = ( X i,N ) i =1 ,...N , k ( N ) = ( k i,N ) i =1 ,...N , K N satisfying the above system; | k i,N | denotes the total variation process of k i,N . We will sometimesomit the superscript N from the notation. The term − k i represents the reflection of the process X i at the boundary of [0 ,
1] and the term K (independent of i ) represents the interaction between theparticles, which keeps the average equal to q . More generally, for modelling purpose, we consideralso the case with a given drift µ : [0 , → R , a given time-dependent average q : [0 , T ] → (0 , σ ∈ R , namely we take the systemd X i,Nt = − µ ( X i,Nt )d t + σ d W it + d K Nt − d k i,Nt ,X i,Nt ∈ [0 , , d k i,Nt = n ( X i,Nt )d | k i,N | t , d | k i,N | t = 1 X i,Nt ∈{ , } d | k i,N | t , N N X i =1 X i,Nt = q ( t ) . (2)2he last line of the above system can be easily converted into an expression for K N in terms of X ( N ) and k ( N ) , namelyd K Nt = ( 1 N N X i =1 µ ( X it ) + ˙ q ( t ))d t − σ N N X i =1 d W it + 1 N N X i =1 d k it . (3)The main novelty of this work is the peculiar combination of the reflecting boundary and thecondition on the average of the particles. This combination is reflected in formula (3), where theinteraction d K depends also on the empirical average of d k i . To guess the limiting behaviour (as N → ∞ ) of the system (2), we can replace the average over particles N − P Ni =1 with the averageover the probability space E . In this way, we get the following McKean-Vlasov SDE on the domain[0 , X t = − µ ( ¯ X t )d t + σ d W t + d ¯ K t − d¯ k t , ¯ X t ∈ [0 , , d¯ k t = n ( ¯ X t )d | ¯ k | t , d | ¯ k | t = 1 ¯ X t ∈{ , } d | ¯ k | t , E ¯ X t = q ( t ) , (4)where W is a real Brownian motion and the solution is a triple ¯ X, ¯ k, ¯ K satisfying the above SDE.As in the particle system, the last line of (4) can be converted into an expression for ¯ K :d ¯ K t = ( E µ ( ¯ X t ) + ˙ q ( t ))d t + E d¯ k t . (5)Our main results Theorems 9, 10 and Proposition 12 state roughly speaking that the McKean-Vlasov SDE (4) is well-posed in the pathwise sense and the empirical measure N P Ni =1 δ X i,N fromthe system (2) converges in probability, as N → ∞ , to the law of the unique solution to (4), withits time marginals converging in L . Motivation . Our motivation to study this system comes from a specific model for chargingand discharging in a lithium-ion battery, introduced in [DGH11] and further studied and expandedfor example in [DHM +
15, GGM + Y it represents the filling degree of the i -thiron phosphate particle at time t (for example, Y i = 1, = 0 resp., stands for the i -th particlefully filled with lithium atoms, fully empty resp.); by this definition of Y i , Y i has to stay in [0 , q ( t ) of Y it represents the current in the battery, which is given and isproportional to the percentage of lithium atoms inside the ensemble of particles. The reason toconsider reflecting boundaries Y i ∈ { , } comes from the boundary conditions in the Fokker-Planckequation in [DHM + +
15] shows global well-posedness for the nonlinear nonlocal Fokker-Planck equation associated with the McKean-VlasovSDE (4), namely ∂ t u ( t, x ) + ∂ x [( − µ ( x ) + ˙¯ K t ) u ( t, x )] = σ ∂ x u ( t, x ) , t > , x ∈ (0 , ,σ ∂ x u ( t, x ) + ( µ ( x ) − ˙¯ K t ) u ( t, x ) = 0 , t > , x ∈ ∂ (0 , , Z xu ( t, x )d x = q ( t ) , t ≥ . (6)3he paper [GGM +
18] considers the particle system (6), associated with (2), even in a more generalversion (to take into account variations in the radius of iron phosphate particles), but removesthe boundaries: without boundaries, the particle system (2) is reduced to a classical system ofmean field interacting diffusions, for which convergence to the corresponding McKean-Vlasov SDEis well-known. Hence the current paper arises from the natural (from a mathematical viewpoint)question whether convergence of the particle system for the model (6) holds. We also point outthat interacting diffusions with constraints both on the domain and on the empirical measure ofthe diffusions appear in several contexts, see e.g. [BCdRGL20, Jab17, Bar20] below.
Background . McKean-Vlasov SDEs are SDEs where the drift depends also on the law of thesolution, namely SDEs of the formd ¯ X t = b ( ¯ X t , Law( ¯ X t ))d t + d W t , (7)where W is a given Brownian motion (we do not consider here the case of general diffusion coeffi-cients). McKean-Vlasov SDEs are related to the mean field interacting diffusions, namely systemsof the form d X it = b ( X it , N N X i =1 δ X it )d t + d W it , i = 1 , . . . N, where W i are independent Brownian motions. By classical resuls, e.g. [Szn91, M´96, Tan84], if b isbounded and smooth (smoothness with respect to the measure variable is understood in the senseof Wasserstein distance), then the McKean-Vlasov SDE (7) is pathwise well-posed and, as N → ∞ ,the empirical measure N P Ni =1 δ X i converges to the law of the solution ¯ X to the McKean-VlasovSDE. This convergence result is a law of large numbers type result and is related to the asymptoticindependence of the particles, the so-called propagation of chaos, see e.g. [Szn84]. The Fokker-Planck equation associated with the McKean-Vlasov SDE (7), namely the equation for Law( ¯ X t ),is nonlinear, see Section 2.3.SDEs on a domain ¯ D ⊆ R m with reflecting boundaries take the formd X t = b ( X t )d t + d W t − d k t , (8) X t ∈ ¯ D, d k t = n ( X t )d | k | t , d | k | t = 1 X t ∈ ∂D d | k | t , (9)where W is a Brownian motion (we do not consider general diffusion coefficients) and n ( x ) is theouter normal to D at x ; | k | represents the total variation process associated with k . The solution is acouple ( X, k ) and − d k represents a “kick”, in the inward normal direction − n ( X ), that the diffusion X receives anytime it reaches the boundary, and that makes X stay in the domain ¯ D . Pathwise well-posedness for the SDE (8) has been proved under quite general conditions, see e.g. [LS84, Tan79].The Fokker-Planck equation associated with the SDE (8) has Neumann-type boundary conditions,see Section 2.2.To our knowledge, the first work to deal with both McKean-Vlasov SDEs and reflecting bound-aries is [Szn84]: there pathwise well-posedness is proved for the SDEd ¯ X t = b ( ¯ X t , Law( ¯ X t ))d t + d W t − d¯ k t , ¯ X t ∈ ¯ D, d¯ k t = n ( ¯ X t )d | ¯ k | t , d | ¯ k | t = 1 ¯ X t ∈ ∂D d | ¯ k | t , X it = b ( X it , N N X i =1 δ X it )d t + d W it , i = 1 , . . . N,X it ∈ ¯ D, d k it = n ( X it )d | k i | t , d | k i | t = 1 X it ∈ ∂D d | k i | t , i = 1 , . . . N. Other works have studied McKean-Vlasov SDEs with reflecting boundaries, in more general con-texts, especially in the context of backward SDEs, see e.g. [Li14]. However, in [Szn84] and in manyof these works, the reflection is local, that is: at the level of the McKean-Vlasov SDE, Law(d¯ k t )does not appear in the SDE; at the level of the particle system, d k i acts only on the i -th particle X i .Closer to our work are the mean reflected (possibly backward) SDEs and related particle systems,introduced in [BEH18, BCdRGL20], and their generalization, namely the SDEs with a constrainton the law and related particle systems, introduced in [BCCdRH20]. Roughly speaking, in theseSDEs a reflecting boundary is imposed on the law of the process. The typical example of this typeof SDEs is the following:d ¯ X t = b ( ¯ X t , Law( ¯ X t ))d t + d W t − d ¯ K t Law( ¯ X t ) ∈ ¯ D , d ¯ K t deterministic, “non-zero only when ¯ X t is on the boundary of ¯ D ” . (10)For such systems, under suitable conditions, [BCCdRH20] proves well-posedness and particle ap-proximation. As a particular case, taking ¯ D = { ρ | R xρ (d x ) ≥ q } for a given q ∈ R , the constraintbecomes E [ ¯ X t ] ≥ q , which is morally comparable to our constraint E [ ¯ X t ] = q ( t ) in the last line of (4).Due to the assumptions on ¯ D (which must have a non-empty “interior”), the condition E [ ¯ X t ] = q ( t )is not covered by [BCCdRH20], but this is not a big limitation: the condition E [ ¯ X t ] ≥ q is actuallymore difficult to take into account than E [ ¯ X t ] = q ( t ), which gives an explicit form for ¯ K and makesthe SDE a classical McKean-Vlasov SDE. However, compared to our equation (4), the restrictionto deterministic ¯ K t in [BCCdRH20] does not allow to consider reflecting boundaries for the process¯ X (for reflecting boundaries on ¯ X , the reflection d¯ k is not deterministic).Probably the closest work to ours is [Jab17]. This paper considers a more general case than[BCCdRH20], in particular removing from (10) the requirement that ¯ K is deterministic. In partic-ular, taking ¯ D = { ρ | Z xρ (d x ) ≥ q, supp( ρ ) ⊆ [0 , } (11)the constraint on ¯ X in (10) becomes E ¯ X t ≥ q, ¯ X t ∈ [0 , , which is morally similar to our constraints E ¯ X t = q ( t ), ¯ X t ∈ [0 ,
1] in (4). The work [Jab17] con-structs a weak solution to the SDE (10) (without the requirement of deterministic ¯ K , in particularincluding condition (11)) by a penalization approach. However it does not show uniqueness, nor itconsiders the related particle system.The work [Bar20] studies a system of N Brownian particles X i hitting a Newtonian movingbarrier Y . For this system the paper proves the convergence, as N → ∞ , to a McKean-Vlasovtype SDE, whose associated Fokker-Planck equation solves a free-boundary problem. Now, in the5rame of the moving barrier, that is taking Z it = X it − Y t , the system in [Bar20] is similar to ourmodel (2), without the drift µ , but with one important difference: in the expression (3) for d K N ,the term N P Ni =1 d k i is replaced in [Bar20] by N P Ni =1 k i d t (times some constant). In particular,unlike here, the term d K N becomes of bounded variation in time in [Bar20].The paper [DEH19] studies the case of backward SDEs with reflecting boundaries dependingboth on the diffusion process ¯ X and on the law of ¯ X , showing well-posedness for this type of SDEsand convergence for the corresponding penalization scheme. However, by the precise assumptionsin [DEH19], a condition of the form E [ ¯ X t ] ≥ q or = q cannot be taken in [DEH19].Finally, we mention the works [CCP11, DIRT15] and [HLSj19]: they deal with systems ofinteracting diffusions, which arise respectively in neuroscience and in finance, and include also anonlocal effect of boundaries, though the boundaries are not reflecting. More precisely, when oneor more particles hit the boundary, the other particles make a jump proportional to the number ofparticles hitting the boundary. Novelty of our work . The main feature of our model is the combination of reflecting bound-aries and nonlocal interaction. At the level of the McKean-Vlasov SDE (4), this combinationappears in the formula (5) for the interacting term d ¯ K , which contains the term d¯ k . At the level ofthe particle system (2), this fact corresponds to an oblique reflection for X ( N ) = ( X ,N , . . . X N,N ),where the direction of reflection depends on the empirical measure N P Ni =1 δ X i,Nt . As explainedbefore, to our knowledge, this kind of systems is studied only in [Jab17] (which shows an existenceresult for the McKean-Vlasov SDE).More specifically, in our model (2), the nonlocal interaction comes from the condition N P Ni =1 X i,Nt = q ( t ) and keeps the direction of reflection for X ( N ) on the iperplane { x | N P Ni =1 x i = 0 } . Intuitively,when a particle X i hits the boundary and receive a “kick” − d k i , then the other particles receive akick N d k j in the opposite direction, so that the average of the particles remains q ( t ).As we will explain below, our proof relies strongly on the constraint N P Ni =1 X i,Nt = q ( t ) and soon the specific form of the interaction. The question of well-posedness and particle approximationfor a more general dependence of d ¯ K on d¯ k , or equivalently, of the direction of reflection of X ( N ) on the empirical measure, remains open. Method of proof . The main idea of the proof is that both d¯ k and d ¯ K in (4) act as projectors.Precisely, d¯ k acts as projector on the set of paths staying in [0 , X, ¯ k ¯ X , ¯ K ¯ X ) and( ¯ X, ¯ k ¯ X , ¯ K ¯ X ) are two solutions to (4), then ( ¯ X − ¯ Y ) · d¯ k ¯ X ≤
0. The term d ¯ K acts as projector in L (Ω) (where (Ω , A , P ) is the underlying probability space) on the space of processes Z with average E [ Z t ] = q ( t ): indeed, if ( ¯ X, ¯ k ¯ X , ¯ K ¯ X ), ( ¯ X, ¯ k ¯ X , ¯ K ¯ X ) are two solutions, then E [( ¯ X − ¯ Y )d ¯ K ¯ X ] = 0.This idea of projectors allows to show uniqueness for the McKean-Vlasov SDE (4) easily. This ideais also behind the proof of convergence of the particle system (2).Concerning the convergence of the particle system (2), we use a pathwise approach introducedby Tanaka in [Tan84] and revisited in [CDFM20] (and further developed in [CL15, BCD20] in therough path context): for any fixed ω ∈ Ω, the particle system (2) can also be viewed as a McKean-Vlasov equation (4), where however the law on the driving signal ¯ W is not the Wiener measure,but the random empirical measure L W,N ( ω ) = N P Ni =1 δ W i ( ω ) . Under this viewpoint, the termd K N is also a projector in L on the space of processes with average q ( t ), but where the underlyingmeasure is the empirical measure L W,N ( ω ) instead of the Wiener measure. Intuitively then, since L W,N ( ω ) converges P -a.s. to the Wiener measure, we expect that d K N should converge to theprojector under the Wiener measure, that is d ¯ K ; this should imply the convergence of the particlesystem to the McKean-Vlasov SDE (4). 6owever, a direct proof based only on this pathwise approach seems not easy. Indeed, to makethis argument work, one needs to create an optimal coupling (in the sense of Wasserstein distance)between the Wiener measure and the random empirical measure L W,N ( ω ), and such coupling doesnot have a Gaussian structure. Having a Gaussian measure on the driving signal allows to useclassical stochastic analysis tools like Itˆo formula; such tools give in turn uniform BV estimates ond K N , which are also needed in the proof. Moreover, the pathwise argument gives a convergenceonly of the one-time marginals, that is, convergence of N P Ni =1 δ X i,Nt ( ω ) (as a random measure on[0 , X t for every fixed ¯ X t , and not convergence of N P Ni =1 δ X i,N · ( ω ) as a measure onthe path space C ([0 , T ]; [0 , N ) BV andH¨older type bounds on the solution to the particle system (2), which give a tightness result; we showthen that any limit point of (2) satisfies the McKean-Vlasov SDE (4), obtaining at once convergenceof the particle system and existence for the McKean-Vlasov SDE itself. Once we have these uniform BV bounds and the existence for the McKean-Vlasov SDE, we can then use the pathwise argumentexplained before. From this pathwise argument, we also get the rate of convergence O (1 / p log( N ))for N P Ni =1 δ X i,Nt ( ω ) .The fact that d¯ k acts like a projector is classical and used since at least [LS84]. However,combining this fact with standard fixed-point arguments for McKean-Vlasov SDEs, as in [Szn91,Szn84], seems not easy in presence of nonlocal effects of boundaries as here. The reason in ourmodel is that E [d¯ k ] is just a BV and continuous term in time (not Lipschitz-continuous) withouta distinguished sign. Even the use of the Lipschitz bounds from [DI91], on the sup norm of thereflecting term ¯ k in terms of the driving signal, seems not too helpful. For our model, the fact thatd ¯ K acts also as projector allows to overcome this difficulty.We remark that the action of d ¯ K as projector on the processes with prescribed average, aswell as some tricks used in the proof of uniform H¨older bounds for the particle system, are quitespecific to our model. For extension to more general nonlocal effects of the reflecting boundary,other methods may be useful, which we do not explore here, for example the penalization method(e.g. [Men83], used in [Jab17]), the approach based on Lions’ derivative (e.g. [Lio08, CD18], usedin [BCCdRH20]), pathwise approaches (e.g. [DGHT19, Aid16, FR13]), PDE-based or singularinteraction methods (see the paragraph below). The PDE and singular interaction viewpoint . The Fokker-Planck equation (6) associatedwith the McKean-Vlasov SDE (4), that is the equation for the probability density function (pdf) u ( t, · ) of ¯ X t , is a nonlinear nonlocal PDE. From this PDE (6), we can get another expression ford ¯ K in terms of u : ˙¯ K t = ˙ q ( t ) + Z µ ( x ) u ( t, x )d x + σ u (1) − u (0)) . The nonlocal effects of the reflecting boundary appear as an additional drift term (here u (1) − u (0))depending on the values of the probability density function u at the boundary. Hence our modelcan be interpreted as a McKean-Vlasov SDE with reflecting boundaries and singular interaction atthe boundary, and one may try to use PDE methods or an approach to singular interaction to dealwith our model.The literature about McKean-Vlasov SDEs with singular interaction and about PDE-basedapproach to McKean-Vlasov SDEs is large, though we are not aware of a work which can cover easilyour model. We only mention some references related to our model. We mention [Szn91, Chapter II],7hich deals with viscous Burgers equation as a McKean-Vlasov SDE with Dirac delta interaction(without boundaries), in particular the McKean-Vlasov SDE is also driven by the probability densityfunction of the solution. We also mention [BJ11, BJ15, BJ18] which study systems of interactingsecond-order diffusions (that is, SDEs for the acceleration of the particles) in a domain. Suchsystems contain a form of singular interaction and a form of reflection at the boundaries, thoughboth interaction and reflection are of different type than in our model. Finally we mention [Kol07]for an approach based on semigroup theory to McKean-Vlasov SDEs and particle approximation.In this paper we do not explore the PDE viewpoint and we give in Section 2.2 a formal argument,without any rigorous proof, to show that (6) is indeed the Fokker-Planck equation for the SDE (4). Organization of the paper . The paper is organized as follows: In Section 2 we show theformal link, without rigorous proofs, between SDEs and Fokker-Planck equations, in presence ofboundaries and mean-field interaction. In Section 3 we give the precise setting and the main results.The proofs of these results are given in Section 4. Finally, in the Appendix 5 we show well-posednessfor the particle systems (2).
The authors acknowledge support from Einstein Center for Mathematics Berlin through MATHEONprojects C-SE8 and C-SE17 ‘Stochastic methods for the analysis of lithium-ion batteries’. M.C.and M.M. acknowledge support from Hausdorff Research Institute for Mathematics in Bonn underthe Junior Trimester Program ‘Randomness, PDEs and Nonlinear Fluctuations’. This project hasreceived funding from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No. 683164, PI P.K.Friz.)We thank Boualem Djehiche, Jean-Francois Jabir, John Schoenmakers and Andreas Sojmarkfor pointing out relevant references on McKean-Vlasov SDEs on bounded domains and suggestingpossible alternative approaches to our problem.
In this section we revisit the link between second-order PDEs and associated SDEs, both in presenceof boundary and with nonlinearity, as the PDE (6) we consider here; we took inspiration from[Son07]. Our aim here is not to give rigorous results but to provide an easy, yet clear “translator”between SDEs and PDEs, which applies, but is not restricted, to our case and shows in particularwhy (4) is the SDE corresponding to (6). For this reason, we keep all the computations at a formallevel, without any rigorous proof.In the following we focus our attention on the 1D case, mostly for simplicity. We take b : I → R a given vector field on R or on an interval I of R when specified, and σ > W is a real Brownian motion. For two functions f, g : I → R , we call h f, g i = Z I f ( x ) g ( x )d x their L scalar product. 8 .1 Diffusion, forward and backward PDEs It is well-known that the (forward) PDE ∂ t p = ( σ / ∂ y p − ∂ y ( bp ) for t > s ,with time- s initial data δ x , models the evolution of the transition density function p = p ( s, x ; t, y )of the diffusion process X , given byd X t = σ d W t + b ( t, X t )d t, X s = x . More generally, if u is the solution to this forward PDE with time- s initial data u s , which is assumedto be a probability density function, then u is the pdf of the solution of the same SDE but initiallaw u s ( x )d x .The dual viewpoint will be important. Consider the (backward) PDE − ∂ s v = ( σ / ∂ x v + b∂ x v with terminal data v ( t, . ) = Ψ. Assuming v to be regular enough, Itˆo’s formula gives a representationof the backward PDE solution v as follows: v ( s, x ) = E [Ψ( X t ) | X s = x ] . (12)By a simple formal computation, one shows thatdd r h p ( s, x, r, · ) , v ( r, · ) i = 0 , which implies v ( s, x ) = Z Ψ( y ) p ( s, x, t, y )d y. (13)At last, comparing (12) and (13) shows that p ( s, x, t, y )d y is indeed the law of X t , started at X s = x ,as claimed in the beginning of this paragraph. We now discuss the case of a spatial domain, with focus on the simple case I = [0 , ,
1] is an SDE of the formd X ◦ t = σ d W t + b ( X ◦ t )d t − d k t , X ◦ s = x,X ◦ t ∈ ¯ I ∀ t ≥ s, d | k | = 1 X ◦ t ∈{ , } d | k | , d k = n ( X ◦ t )d | k | . Here, n (0) = − , n (1) = +1 are the outer normals of our domain [0 , X, k ) satisfying the above condition (it is implicitly assumed that k has BV paths). The lastcondition means that k acts only when X is on the boundary, giving a small “kick” so that X doesnot leave the domain I . 9rom the PDE viewpoint, the transition density function p ◦ ( s, x ; t, y ) associated with X is asolution to the following forward forward equation ∂ t p ◦ = ( σ / ∂ y p ◦ − ∂ y ( bp ◦ ) in I ◦ for t > s ,( σ / ∂ y p ◦ − bp ◦ = 0 at ∂I for t > sp ◦ ( s, x, s, · ) = δ x , More generally, if u is the solution to this forward PDE with time- s initial data u s , assumed to bea probability density function, then u is the pdf of the solution of the same SDE but initial law u s ( x )d x . We show this fact formally using the dual viewpoint.First step: Let v ◦ be a regular solution to the dual backward equation, i.e. time- t terminal valueprobelm with Neumann (no-flux) boundary data, − ∂ s v ◦ = ( σ / ∂ x v ◦ + b∂ x v ◦ in I ◦ , for s < t , ∂ x v ◦ ( s,
0) = ∂ x v ◦ ( s,
1) = 0 at ∂I , for all s < t , v ( t, . ) = Ψwith some (regular) time- t terminal data Ψ. Then necessarily v ◦ has the representation v ◦ ( s, x ) = E [Ψ( X ◦ t ) | X ◦ s = x ] . (14)Indeed, Itˆo formula givesd[ v ( s, X s )] = ∂ s v ( X )d s + ∂ x v ( X )d X + σ ∂ x v ( X )d s = [( ∂ s + b∂ x + σ ∂ x ) v ]( X )d s + σ∂ x v ( X )d W − ∂ x v ( X )d k = σ∂ x v ( X )d W, where we have used the equation for v to kill the first addend in the second line and the boundaryconditions on v to kill the term with d k . Taking expectation, we get that E [ v ( s, X s )] is constantin s , which implies (14).Second step: Again with simple formal computation, one shows,dd r h p ◦ ( s, x, r, · ) , v ◦ ( r, · ) i = 0 , which implies v ◦ ( s, x ) = h p t , Ψ i . (15)From (14) and (15) we conclude h p ◦ ( s, x, t, · ) , Ψ i = E [Ψ( X ◦ t ) | X ◦ s = x ] . Since this is true for every regular Ψ, then p ◦ is the law of X ◦ conditional to X ◦ s = x .If u ◦ is a solution to the same forward equation as above, but with generic initial condition u s ,integrating p in u s ( x ) shows that u is the pdf of the law of X ◦ ,u , the process satisfying the sameSDE but with initial law u s ( x )d x . 10 .3 McKean-Vlasov diffusion, nonlinear mean-field PDEs Here we introduce the McKean-Vlasov setting. For a given drift ¯ b : R × P ( R ) → R , where P ( R ) isthe space of probability measures on R , we considerd ¯ X t = σ d W t + ¯ b ( ¯ X t , Law( ¯ X t ))d t, Law( ¯ X s ) given . Existence and uniqueness for regular bounded ¯ b (regularity with respect to the measure variableis usually in terms of the Wasserstein distance) are proved via a fixed-point argument on the map( m t ) t ≥ s (Law( X mt )) t ≥ s , where X m is the solution to the (classical) SDEd X mt = σ d W t + ¯ b [ X mt , m t ]d t, Law( X ms ) = m s . See Sznitman [Szn91, Theorem 1.1] for the classical case of ¯ b linear in the measure argument or,for instance, [CDFM20] for a more general ¯ b .The corresponding forward Fokker-Planck equation now takes the form ∂ t ¯ u = ( σ / ∂ y ¯ u − ∂ y [¯ b ( y, ¯ u t ( · ))¯ u ] (16)with time- s initial data ¯ u s = Law( ¯ X s ) (for notational sake, we blur the difference between u ( t, . )and u ( t, y ) dy ). Note that this is a nonlinear nonlocal PDE.We again outline why ¯ u ( t, . ) is indeed the law of ¯ X t . Fix the family ¯ u := { ¯ u ( t, . ) : t ≥ s } andconsider the (linear!) backward PDE − ∂ s v = ( σ / ∂ x v + ¯ b [ x, ¯ u ( s, . )] ∂ x v with terminal data v t = Ψ. As in Section 2.1, we see that the corresponding forward PDE (which byconstruction coincides with (16)) yields the law of a diffusion process X ¯ u , that is Law( X ¯ ut ) = ¯ u ( t, · ).But then X ¯ u solves the McKean-Vlasov SDE, therefore (by uniqueness for the McKean-Vlasov SDE) X ¯ u = ¯ X and so ¯ u t is the pdf of the marginal ¯ X t at time t . We can consider the simplest case of reflected McKean-Vlasov SDE on I , as in [Szn84], namelyd ¯ X t = σ d W t + ¯ b [ ¯ X t , Law( ¯ X t )]d t − d¯ k t , Law( ¯ X s ) given , ¯ X t ∈ ¯ I ∀ t ≥ s, d | ¯ k | = 1 ¯ X t ∈{ , } d | ¯ k | , d¯ k = n ( ¯ X t )d | ¯ k | . Adapting the arguments in the two previous sections, one shows that the probability densityfunction of ¯ X t is given via following non-linear nonlocal forward PDE ∂ t ¯ u t = ( σ / ∂ y ¯ u − ∂ y [¯ b [ y, ¯ u t ( . )]¯ u ] in I ◦ for t > s ,( σ / ∂ y ¯ u − ¯ b [ · , ¯ u t ( . )]¯ u = 0 at ∂I for t > s, with time- s initial data ¯ u s = Law( ¯ X s ). 11 .5 Interaction coming from the boundaries As explained in the introduction, the SDE (4) does not fall in the previous class. Indeed, the driftdepends not only on the law of ¯ X t but also on the law of d¯ k t . We will not treat here the generalcase of drifts depending on the law of d¯ k , but we focus our attention on our model.We claim that the forward Fokker-Planck equation associated with (4) is (6): we show formallythat, if u satisfies the PDE (6), with initial condition u , then u ( t, x ) is the probability densityfunction of the random variable ¯ X t solving (4), with initial law u . The formal proof puts togetherthe arguments for boundary problems and McKean-Vlasov SDEs with the additional difficulty ofinteraction coming from the boundary, for which we will use the constraint on the average in (6).Take ˙¯ K as in the PDE (6), call ( X ¯ K , k ¯ K ) the solution of the reflecting SDEd X ¯ K = σ d W t + ( − µ ( X ¯ K ) + ˙¯ K ( t ))d t − d k ¯ K , Law( X ¯ K ) = u ,X ¯ Kt ∈ ¯ I ∀ t ≥ , d | k ¯ K | = 1 X ¯ Kt ∈{ , } d | k ¯ K | , d k ¯ K = n ( X ¯ Kt )d | k ¯ K | . As a consequence of Section 2.2 (applied with given ˙¯ K ), the law of X ¯ Kt must be given by u ( t, · ),i.e. the PDE solution to (6). It remains to show that X Λ = ¯ X , the solution to the McKean-Vlasov(4). To this end, note that E [ X ¯ Kt ] = R xu ( t, x )d x = q ( t ), using the basic constraint in (6). Thisshows that X ¯ K is a solution to (4) and, by uniqueness of this equation (formally, see also Theorem9), we conclude that X ¯ K = ¯ X . In the following, we consider a probability space (Ω , A , P ) and independent Brownian motions W i , i = 1 . . . N , on a filtration ( F t ) t (satisfying the standard assumption). We are given a function ofspace µ : [0 , → R and a function of time q : [0 , T ] → [0 , µ and q will begiven later. The noise intensity σ is assumed to be constant (possibly 0).We consider the system of N interacting particles:d X i,Nt = ( − µ ( X it ) + 1 N N X j =1 µ ( X j,Nt ))d t + ˙ q ( t )d t + σ d W i − N N X j =1 σ d W jt − d k i,Nt + 1 N N X j =1 d k j,Nt , i = 1 , . . . N,X i,N ∈ C ([0 , T ]; [0 , , k i ∈ C ([0 , T ]; R ) a.s. , i = 1 , . . . N, d | k i,Nt | = 1 X i,Nt ∈{ , } d | k i,Nt | , d k i,Nt = n ( X i,Nt )d | k i,Nt | , i = 1 , . . . N. (17)Here n is the outer normal vector of the domain ]0 ,
1[ and | k i,N | is the total variation pro-cess associated with k i,N (mind that it is not the modulus of k i,N ). A solution is a couple( X ( N ) , k ( N ) ) = ( X i,N , k i,N ) i =1 ,...N of a ( F t ) t -progressively measurable continuous semimartingale X ( N ) and a ( F t ) t -progressively measurable BV process k ( N ) , satysfying the above system. We willoften omit the second superscript N (which denotes the number of particles) when not needed.12his system is the exact formulation of the interacting particle system (2), with the term d K N in (2) given by d K N = ˙ q ( t )d t + 1 N N X j =1 µ ( X jt )d t − N σ N X j =1 d W j + 1 N N X j =1 d k j . The term − d k i represents the reflection at the boundary of [0 , K N is independentof i and ensures that the empirical average N − P Ni =1 X it stays equal to q ( t ).The system (17) can be interpreted as an SDE for X on [0 , N with oblique reflecting boundaryconditions, where the direction of reflection keeps X ( N ) in the moving hyperplane H t = { x ∈ R N | N P Ni =1 x i = q ( t ) } . Another interpretation of this system is as an SDE on H t ∩ [0 , N with normalboundary condition, where the domain, in the frame of H t , is a moving convex polygon and thenormal reflection is in the frame of H t . This interpretation is used in the proof of well-posednessof the system (17) (Proposition 5).We work under the following assumptions on µ and q (and X ( N )0 ): Condition 1. i) The function − µ is C on ]0 , and one-side Lipschitz-continuous, namely:there exists c ≥ such that, for every x , y in ]0 , , − ( µ ( x ) − µ ( y ))( x − y ) ≤ c | x − y | , ∀ x, y ∈ ]0 , . (18) ii) The function µ satisfies sup x ∈ ]0 , / | x || µ ( x ) | + sup x ∈ ]1 / , | − x || µ ( x ) | < + ∞ . iii) There exists < ρ < / such thatsign ( x − / µ ( x ) ≥ , ∀ x ∈ ]0 , ρ [ ∪ ]1 − ρ, ρ [; (19) moreover µ (0) = µ (1) = 0 . Condition 2.
The map q is a Lipschitz-continuous function of time (in particular ˙ q ( t ) exists fora.e. t ) and there exists < ξ < such that ξ ≤ q ( t ) ≤ − ξ for every t . Condition 3.
The (possibly random) initial datum X ( N )0 is F -measurable and verifies ≤ X i,N ≤ for i = 1 , . . . N and N P Ni =1 X i,N = q (0) P -a.s.. The typical example we have in mind for µ is the derivative of a double-well potential, withlogarithmic divergence at the boundary, and the typical example for q is a piecewise linear contin-uous function which does not touch 0 nor 1. These examples are used in the battery model from[GGM + µ and Condition 2 on q are structural assumptions of our model,Condition 1-(ii) seems not really necessary: if for example µ diverges like 1 /x α for some α >
1, wewould expect that the system does not even touch the boundary, hence classical McKean-Vlasovapproach should apply, but for technical reasons our proof does not apply to this situation, seeRemark 22. Condition 1-(iii) is also technical and we expect that it can be removed without toomuch effort, see Remark 35. 13ctually we do not work directly with the system (17) but, to avoid possible singularity of µ atthe boundary, we take a regularization µ ǫ , C on the closed domain [0 , µ ǫ = µ on [ ǫ, − ǫ ]and | µ ǫ | ≤ | µ | on ]0 ,
1[ and verifying both the one-side Lipschitz condition (18) and the condition(19) uniformly in ǫ . We then consider the system:d X it = ( − µ ǫ ( X it ) + 1 N N X j =1 µ ǫ ( X jt ))d t + ˙ q ( t )d t + σ d W it − N N X j =1 σ d W jt − d k it + 1 N N X j =1 d k it , i = 1 , . . . N,X i ∈ C ([0 , T ]; [0 , , k i ∈ C ([0 , T ]; R ) a.s. , i = 1 , . . . N, d | k it | = 1 X it ∈{ , } d | k it | , d k it = n ( X it )d | k it | , i = 1 , . . . N. (20)When we want to stress the dependence on N and ǫ , we write X i,N,ǫ and X ( N,ǫ ) = ( X , . . . X N )and similarly for k i,N,ǫ , k ( N,ǫ ) . Remark 4.
Here and in the following, when we talk about pathwise uniqueness, resp. uniquenessin law, we refer to pathwise uniqueness of X ( N,ǫ ) , resp. of the law of X ( N,ǫ ) . Uniqueness of X ( N,ǫ ) implies in turn uniqueness of k ( N,ǫ ) − E k ( N,ǫ ) , but we do not make any uniqueness statement on k ( N,ǫ ) itself. Proposition 5.
Assume Conditions 2 and 3 and assume that µ ǫ is Lipschitz-continuous on [0 , .Then there exists a solution to the particle system (20) and this solution is pathwise unique in X ( N,ǫ ) . The basic idea of the proof is simple: namely the SDE above is an SDE on the moving domain H t ∩ [0 , N with normal boundary conditions. However the proof is slightly technical and postponedto Appendix 5. Remark 6.
Similarly to (17) , any solution to (20) satisfies N P Ni =1 X it = q ( t ) for every t . In the following, we consider again a probability space (Ω , A , P ) and a Brownian motion W on afiltration ( F t ) t (satisfying the standard assumption); E denotes the expectation with respect to P .The functions µ , q and σ are as in the previous subsection.We consider the McKean-Vlasov SDEd ¯ X = − µ ( ¯ X )d t + σ d W + d ¯ K − d¯ k, Z T E [ | µ ( ¯ X ) | ]d r < ∞ , E Z T | d¯ k | < ∞ , d ¯ K = ( E [ µ ( ¯ X )] + ˙ q )d t + E [d¯ k ] , ¯ X ∈ C ([0 , T ]; [0 , , ¯ k ∈ C ([0 , T ]; R ) a.s. , d | ¯ k | = 1 ¯ X t ∈{ , } d | ¯ k | , d¯ k = n ( ¯ X t )d | ¯ k | . (21)Here again n is the outer normal vector of the domain ]0 ,
1[ and | ¯ k | is the total variation processassociated with ¯ k (not the modulus of ¯ k ). A solution is a couple ( ¯ X, ¯ k ) of a ( F t ) t -progressivelymeasurable continuous semimartingale ¯ X and a ( F t ) t -progressively measurable BV process ¯ k , sat-isfying the above equation. We sometimes say that ¯ X is a solution if there exists a process ¯ k suchthat ( ¯ X, ¯ k ) is a solution. 14he assumptions on µ and q remain unchanged with respect to the particle system. In theassumption on ¯ X , here the empirical average is replaced by the average with respect to P . Condition 7.
The (possibly random) initial datum ¯ X is F -measurable and verifies ≤ ¯ X ≤ and E ¯ X = q (0) . Remark 8.
As for the particle system, it is easy to see that (under Condition 7) any solution to (21) satisfies E ¯ X t = q ( t ) for every t . Our main results are well-posedness of the McKean-Vlasov SDE (21) and convergence of the particlesystem (20) to the McKean-Vlasov SDE as N → ∞ and ǫ → Theorem 9.
Take a probability space (Ω , A , P ) , a Brownian motion W on a filtration ( F t ) t (satisty-ing the standard assumption) and an initial condition ¯ X , assume Conditions 1, 2, 7. Then thereexists a unique solution ( ¯ X, ¯ k ) to the McKean-Vlasov SDE (21) . Let ¯ X be the solution to the McKean-Vlasov SDE (21) with initial datum ¯ X and, for ǫ > N in N , let ( X ,N,ǫ , . . . X N,N,ǫ ) be the solution to the particle system (20) with initial datum( X , . . . X N ). For E Polish space and Y i E -valued random variables, we consider the empiricalmeasures N P Ni =1 Y i as P ( E )-valued random variable, where P ( E ) is the space of probabilitymeasures on E , endowed with the Borel σ -algebra with respect to the weak convergence (convergenceagainst C b ( E ) test functions). Theorem 10.
Assume Conditions 1, 2, Condition 7 on ¯ X and Condition 3 on ( X , . . . X N ) .Assume also that N P Ni =1 δ X i,N converges in probability to Law ( ¯ X ) as N → ∞ . Then the sequenceof empirical measures N P Ni =1 δ X i,N,ǫ on P ( C ([0 , T ])) converges in probability to Law ( ¯ X ) , as ǫ → and N → ∞ . Note that the convergence result of the particle system (20) holds as N → ∞ , ǫ → ǫ to 0 and then N to ∞ to show theconvergence of the original particle system (17). Remark 11.
The assumptions on the initial conditions may sound a bit rigid, in particular theycannot be satisfied taking ( X , . . . X N ) i.i.d. copies of ¯ X (the empirical average is not q (0) for a.e. ω ). However: • An easy example of ( X , . . . X N ) satisfying this constraint is given by taking Y i i.i.d. copiesof a variable ¯ X with mean q (0) and X i = Y i − N P Nj =1 Y j + q (0) : by the law of large number N P Nj =1 Y j tends to q (0) = E ¯ X and so the empirical measure of X i tends to the law of ¯ X in probability. • The assumptions can easily be relaxed allowing q (0) = q N (0) to be random and dependenton N , but keeping deterministic increments q N ( t ) − q N (0) , with q N (0) tending to q (0) inprobability as N → ∞ . This allows to include the case of ( X , . . . X N ) i.i.d. copies of ¯ X . Finally, we give another convergence result and exhibit a rate of convergence for the timemarginals. We denote by W , [0 , the 2-Wasserstein distance on [0 , roposition 12. Assume that µ is C on [0 , so that we can take µ ǫ = µ for every ǫ > . Assumethe conditions of Theorem 10 and assume also that X i,N = Y i + P Nj =1 Y j + q (0) , where ( Y i ) i ∈ N is a sequence of independent and identically distributed random variables with law Law ( ¯ X ) (seeRemark 11). Then we have the following rate of convergence: E " sup t ∈ [0 ,T ] W , [0 , ( Law ( ¯ X t ) , N N X i =1 δ X i,Nt ) = O (1 / p log( N )) as N → ∞ . The strategy of the proof is as follows: • We first prove uniqueness for the McKean-Vlasov SDE. For later use, we prove uniquenessamong a larger class of solutions, namely possibly non-adapted processes. We also give astability result with respect to the drift µ . • For convergence of the particle system and existence of the McKean-Vlasov SDE, we proveuniform (in N and ǫ ) BV and H¨older estimates for k i,N,ǫ and uniform H¨older estimates for X i,N,ǫ . These estimates in turn imply tightness for the empirical measures N P Ni =1 δ X i,N,ǫ and more generally for N P Ni =1 δ ( W i ,X i,N,ǫ , − R · µ ǫ ( X i,N,ǫr )d r − k i,N,ǫ ) . • We then prove that any limit point of the empirical measures N P Ni =1 δ X i,N,ǫ is the law ofa (possibly non-adapted) solution to the McKean-Vlasov SDE. Uniqueness of the McKean-Vlasov SDE implies that the whole sequence of empirical measure converges to the law of theunique solution and that this solution is actually adapted. • Finally, we prove the rate of convergence using a pathwise approach. We first show that par-ticle system (20) can be interpreted as the McKean-Vlasov equation with a different measureon the inputs. The core of the proof is then a stability result of the McKean-Vlasov equationwith respect to the inputs.In the following subsections, we will use the letter C to denote a positive constant, whose valuemay change from line to line; we will sometimes use C p to stress the dependence on p . In this Subsection we establish uniqueness and stability results for the McKean-Vlasov SDE (21).The following result proves the uniqueness part of Theorem 9.
Proposition 13.
Assume Condition 1-(i) on µ and that q is measurable bounded (Conditions 1and 2 in particular are enough). Assume also Condition 7 on ¯ X . Strong uniqueness holds for theMcKean-Vlasov SDE (21) . Moreover, if ¯ X and ¯ Y are two solution to (21) starting from ¯ X , ¯ Y ,with E [ ¯ X ] = E [ ¯ Y ] = q (0) , it holds for some C > (independent of ¯ X and ¯ Y ), for every t , E | ¯ X t − ¯ Y t | ≤ e Ct E | ¯ X − ¯ Y | . roof. It is enough to prove stability. We will use the superscripts ¯ X , ¯ Y for the quantities ¯ K , ¯ k , ...associated with ¯ X , ¯ Y . By Itˆo formula for continuous semimartingales [RY99] we haved | ¯ X − ¯ Y | = 2( ¯ X − ¯ Y )( − µ ( ¯ X ) + µ ( ¯ Y ))d t + 2( ¯ X − ¯ Y )d ¯ K ¯ X −
2( ¯ X − ¯ Y )d ¯ K ¯ Y −
2( ¯ X − ¯ Y )d¯ k ¯ X + 2( ¯ X − ¯ Y )d¯ k ¯ Y . For the first addend, the one-side Lipschitz condition of µ implies( ¯ X − ¯ Y )( − µ ( ¯ X ) + µ ( ¯ Y )) ≤ c | ¯ X − ¯ Y | . For the addends with ¯ k , the orientation of ¯ k (as the outward normal) implies − Z t ( ¯ X − ¯ Y )d¯ k ¯ X ≤ X − ¯ Y )d¯ k ¯ Y . For the addends with K , we take the expectation and use that K is deterministic and that E [ ¯ X t ] = E [ ¯ Y t ] = q ( t ) (see Remark 8): we obtain E Z t ( ¯ X − ¯ Y )d K ¯ X = Z t ( E [ ¯ X ] − E [ ¯ Y ])d K ¯ X = 0 . Putting all together, we get E | ¯ X t − ¯ Y t | ≤ E | ¯ X − ¯ Y | + C Z t E | ¯ X r − ¯ Y r | d r. We conclude by Gronwall inequality.
Proposition 14.
Assume Condition 1-(i) and that q is measurable bounded. Let µ n a sequence offunctions, with uniformly bounded one-side Lipschitz constant, converging uniformly to µ on everycompact subset of ]0 , , such that | µ n | ≤ C | µ | on ]0 , . Call ¯ X n , ¯ X the solutions to the SDE (21) resp. with µ n , µ and with the same initial condition. Then it holds, as n → ∞ , sup t ∈ [0 ,T ] E | ¯ X t − ¯ X nt | → . Proof.
By Itˆo formula we have, proceeding as in the previous proof, we obtaind | ¯ X − ¯ X n | = 2( ¯ X − ¯ X n )( − µ ( ¯ X ) + µ n ( ¯ X ))d t + 2( ¯ X − ¯ X n )( − µ n ( ¯ X ) + µ n ( ¯ X n ))d t + d(other terms) , where the other terms have non-positive expectation. For the second addend, the uniform one-sideLipschitz condition implies, for some c > n ,( ¯ X − ¯ X n )( − µ n ( ¯ X ) + µ n ( ¯ X n )) ≤ c | ¯ X − ¯ X n | . For the first addend, the integrability condition on µ in (21) implies that, for every ǫ >
0, thereexists δ > E Z T ¯ X / ∈ [ δ, − δ ] | µ ( ¯ X ) | d r < ǫ µ n since | µ n | ≤ C | µ | . By the uniform convergence of µ n to µ on [ δ, − δ ], thereexists n such that, for every n ≥ n , | µ n − µ | < ǫ on [ δ, − δ ]. Therefore we have E Z t ( ¯ X − ¯ X n )( − µ ( ¯ X ) + µ n ( ¯ X ))d r ≤ E Z t ¯ X ∈ [ δ, − δ ] | µ ( ¯ X ) − µ n ( ¯ X ) | d r + E Z t ¯ X / ∈ [ δ, − δ ] ( | µ ( ¯ X ) | + | µ n ( ¯ X ) | )d r ≤ Cǫ.
Finally we obtain, for every n ≥ n , E | ¯ X t − ¯ X nt | ≤ Cǫ + C Z t E | ¯ X r − ¯ X nr | d r. We conclude again by Gronwall lemma.For the proof of convergence of the particle system, it is actually useful a slightly strongeruniqueness result, among a generalized class of solutions. Given a probability space (Ω , A , P ) anda Brownian motion W on it (with respect to its natural filtration), we call generalized solutiona couple ( ¯ X, ¯ k ) of A ⊗ B ([0 , T ])-measurable maps, satisfying the system (21) P -a.s., without anyadaptedness condition; we also do not require A to be complete with respect to P . We also callweak generalized solution the object (Ω , A , W, ¯ X, ¯ k, P ). Note that the system makes sense evenwithout adaptedness, since the noise is additive. The difference with the usual concept of solutionlies exactly in the lack of adaptability (and lack of completeness of the σ -algebra A ). We say that¯ X is a generalized solution if there exists a A ⊗ B ([0 , T ])-measurable map ¯ k such that ( ¯ X, ¯ k ) is ageneralized solution. Lemma 15.
Assume Condition 1-(i) and that q is measurable bounded. Assume also Condition 7on ¯ X . Given (Ω , A , P ) and W , uniqueness holds among generalized solutions.Proof. Let ( ¯ X, ¯ k ¯ X ) and ( ¯ Y , ¯ k ¯ Y ) be two solutions. Then ¯ X − ¯ Y is a BV and continuous processsatisfying P -a.e. d( ¯ X − ¯ Y ) = ( − µ ( ¯ X ) + µ ( ¯ Y ))d t + d( ¯ K ¯ X − ¯ K ¯ Y ) + d(¯ k ¯ X − ¯ k ¯ Y ) . Each of the addends in the right-hand side is BV and continuous, in particular we can fix ω (outsidea P -null set in A ) and apply the chain rule to get the expression for the differential of | ¯ X − ¯ Y | .The rest of the proof goes as in the proof of Proposition 13.Another useful tool in view of particle convergence is Yamada-Watanabe principle, which,roughly speaking, states that strong uniqueness and weak existence imply uniqueness in law andstrong existence. Since we are working in a slightly non-standard context, with McKean-VlasovSDEs and with generalized solutions, we repeat the statements and the proofs for our case: Proposition 16 (Yamada-Watanabe, uniqueness in law) . Let (Ω i , A i , P i ) , i = 1 , , be two prob-ability spaces, with associated Brownian motions W i and generalized solutions ( ¯ X i , ¯ k i ) , i = 1 , ,such that Law ( ¯ X ) = Law ( ¯ X ) . Then the laws of ( W , ¯ X ) and ( W , ¯ X ) coincide.Proof. We take ˆΩ = ( C ([0 , T ]) × R ) × C ([0 , T ]) × C ([0 , T ]) , endowed with the Borel σ -algebraˆ A = B ( ˆΩ) (with respect to the uniform topology). We call ˆ ω = (( w, x ) , ( γ , κ ) , ( γ , κ )) a generic18lement of Ω. Let P i,W i , ¯ X i be the conditional law of ( ¯ X i , ¯ k i ) with respect to W i and ¯ X i , i = 1 ,
2. Wetake on B ( ˆΩ) the probability measure ˆ P = P W, ¯ X ⊗ P ,w,x ⊗ P ,w,x , where P W, ¯ X is the product ofthe Wiener measure and the law of ¯ X . We define ˆ W (ˆ ω ) = w , ˆ X (ˆ ω ) = x , ( ˆ X i (ˆ ω ) , ˆ k i (ˆ ω )) = ( γ i , κ i ), i = 1 ,
2, the canonical projections. Now, for each i = 1 ,
2, the law of ( ˆ
W , ˆ X i , ˆ k i ) is the law of( W i , ¯ X i , ¯ k i ), in particular ¯ K ˆ X i = ¯ K ¯ X i . Therefore ( ˆ X i , ˆ k i ), i = 1 ,
2, are two generalized solutionsto (21), defined on the same probability space ( ˆΩ , ˆ A , ˆ P ) with respect to the same Brownian motionˆ W and with the same initial datum ˆ X = ˆ X = ˆ X ˆ P -a.s.. By the uniqueness result, ˆ X and ˆ X must coincide ˆ P -a.s.. Hence (calling γ the push-forward of the projection on the γ component), γ P ,w,x and γ P ,w,x , the conditional laws of ˆ X and ˆ X given ( ˆ W , ˆ X ) = ( w, x ), coincide for P W, ¯ X -a.e. ( w, x ). Therefore Law( W , ¯ X , ¯ X ) = P W, ¯ X ⊗ γ P ,w,x and Law( W , ¯ X , ¯ X ) = P W, ¯ X ⊗ γ P ,w,x coincide. The proof is complete. Proposition 17 (Yamada-Watanabe, strong existence) . The generalized solution ( ¯ X , ¯ k ) is actu-ally a strong solution to (21) , that is, it is progressively measurable with respect to ( F W , ¯ X t ) t , thefiltration generated by W , ¯ X and the P -null sets (and similarly for ( ¯ X , ¯ k ) ).Proof. We continue using the notation of the previous proof. Call ( ˆ F ˆ W , ˆ X t ) t the filtration generatedby ˆ W , ˆ X and the P W, ¯ X -null sets on C ([0 , T ]) × R . Note that the conditional law of ( ˆ X , ˆ k , ˆ X , ˆ k given ( ˆ W , ˆ X ) = ( w, x ) is P ,w,x ⊗ P ,w,x . Hence, for P W, ¯ X -a.e. ( w, x ), conditioning to( ˆ W , ˆ X ) = ( w, x ), ˆ X and ˆ X coincide a.s. and are independent. Hence, for P W, ¯ X -a.e. ( w, x )given, conditioning to ( ˆ W , ˆ X ) = ( w, x ), ˆ X must coincide with an element Y T ( w, x ) a.s.. Therandom element Y T , extended on a P W, ¯ X -null set, defines a solution map Y T : C ([0 , T ]) × R → C ([0 , T ]) which is ˆ F ˆ W , ˆ X T -measurable: indeed, for every Borel subset B of C ([0 , T ]), { Y T ∈ B } coincides P W, ¯ X -a.s. with { ( w, x ) | γ P ,w,x ( B ) = 1 } , which belongs to ˆ F ˆ W , ˆ X T (since P ,w,x ( B )is ˆ F ˆ W , ˆ X T -measurable). From the previous proof, we haveLaw( W , ¯ X , ¯ X ) = P W, ¯ X ⊗ γ P ,w,x = Law( W, ¯ X ) ⊗ δ Y T ( w,x ) , therefore ¯ X = Y T ( W , ¯ X ) P -a.s.. Since ( W, ¯ X ) is measurable from F W , ¯ X T to ˆ F ˆ W , ˆ X T , weconclude that ¯ X is F W , ¯ X T -measurable.Concerning progressive measurability, we can restrict W , ¯ X and ¯ k on [0 , t ] and repeat theprevious argument: calling π t : C ([0 , T ]) → C ([0 , t ]) the restriction operator, we get that π t ( ¯ X ) = Y t ( π t ( W ) , ¯ X ) P -a.s. and π t ( ¯ X ) is F W , ¯ X t -measurable. Hence ¯ X is adapted and thereforeprogressively measurable, by continuity of its paths. Progressive measurability of ¯ k follows because, P -a.s., d¯ k = − d ¯ X − µ ( ¯ X )d t + d W + E P [ µ ( ¯ X )]d t + E P [d¯ k ] . The proof is complete.
Here we consider the particle system (20) and we give estimates which are uniform in N and ǫ . Wewill often omit the superscripts N and ǫ for notational simplicity.19 .3.1 BV estimates We start estimating the BV norm of the average of the drift over the particles. Throughout thissubsection, we will assume Conditions 1-(i) on µ , 2 on q and 3 on X . Lemma 18.
For every ≤ p < ∞ , it holds sup N,ǫ E N N X i =1 Z T | µ ǫ ( X i,N,ǫr ) | d r ! p + sup N,ǫ E N N X i =1 Z T | d k i,N,ǫr | ! p < + ∞ . The proofs of this lemma and of the next one use mainly two facts: • the one-side Lipschitz property of − µ and the reflection condition on k i , that is k i has thesame sign of n ( X i ); • the property N P Ni =1 X it − q ( t ) = 0.Let us explain briefly how these two facts yield the BV estimates. We will focus only on the boundson k i , the bounds on µ ( X i ) being similar. As in the standard argument for boundary terms, wetake the differential of | X it − q ( t ) | :d | X i − q | = − X i − q )d k i + 2( X i − q ) 1 N N X j =1 d k j + . . . (22)Disgarding the interaction terms, using the reflection condition, we would get an inequality of theform d | X i − q | = − X i − q )d k i ≤ − | X i − q | d | k i | This inequality would give a bound on | X i − q ( t ) | d | k i | and so on d | k i | (since | X i − q ( t ) | is boundedfrom below when X i is on the boundary). However, the interaction term 2( X i − q ) N P Nj =1 d k j in(22) cannot be bounded as before, due to the + sign instead of − sign. To deal with it, firstly weaverage over i : thanks to the property N P Ni =1 X it − q ( t ) = 0, the average of the interaction termsdisappears: 1 N N X i =1 ( X i − q ) 1 N N X j =1 d k j = 0 , hence we get a bound on the average of d | k i | (Lemma 18). Secondly, this bound allows to controlthe interaction term 2( X i − q ) N P Nj =1 d k j . Using this control in (22), we get a bound on d | k i | forany i (Lemma 20). Remark 19.
The one-side Lipschitz condition on − µ and the regularity of µ in the interior ]0 , imply that, for any < c < / , sup ǫ,x ∈ ]0 , ( µ ǫ ( x ) sign ( x − / − < + ∞ , sup ǫ,x ∈ [ c, − c ] | µ ǫ ( x ) | < + ∞ . he condition < ξ ≤ q ( t ) ≤ − ξ < implies that ( x − q ( t )) sign ( x − / x/ ∈ [ ξ/ , − ξ/ ≥ ξ x/ ∈ [ ξ/ , − ξ/ . Putting together the above bounds, we get, for some C ≥ independent of x and ǫ , for every x in ]0 , and every t , ( x − q ( t )) µ ǫ ( x ) = ( x − q ( t )) sign ( x − / µ ǫ ( x ) sign ( x − / + x/ ∈ [ ξ/ , − ξ/ − ( x − q ( t )) sign ( x − / µ ǫ ( x ) sign ( x − / − x/ ∈ [ ξ/ , − ξ/ − ( x − q ( t )) µ ǫ ( x )1 x ∈ [ ξ/ , − ξ/ ≥ ξ µ ǫ ( x ) sign ( x − / + x/ ∈ [ ξ/ , − ξ/ − C = ξ | µ ǫ ( x ) sign ( x − / | x/ ∈ [ ξ/ , − ξ/ − ξ µ ǫ ( x ) sign ( x − / − x/ ∈ [ ξ/ , − ξ/ − C ≥ ξ | µ ǫ ( x ) | x/ ∈ [ ξ/ , − ξ/ − C ≥ ξ | µ ǫ ( x ) | − C. From the reflection condition on k we also get ( X it − q ( t ))d k it ≥ ξ d | k i | . Proof.
By Itˆo formula, we haved | X i − q ( t ) | (23)= 2( X i − q ( t ))( − µ ǫ ( X i ) + 1 N N X j =1 µ ǫ ( X j ))d t + 2 σ ( X i − q ( t ))(d W i − N N X j =1 d W j )+ σ (1 − N )d t + 2( X i − q ( t ))( − d k i + 1 N N X j =1 d k j ) . We average over i . For the interaction term with N P Nj =1 µ ǫ ( X j ), the condition N P Ni =1 X it = q ( t )implies 1 N N X i =1 ( X i − q ( t )) 1 N N X j =1 µ ǫ ( X j )d t = 0and similarly for the other interaction terms (with N P Nj =1 d W j and with N P Nj =1 d k j ). Hence we21et d 1 N N X i =1 | X i − q ( t ) | = − N N X i =1 ( X i − q ( t )) µ ǫ ( X i )d t + 2 1 N N X i =1 σ ( X i − q ( t ))d W i + σ (1 − N )d t − N N X i =1 ( X i − q ( t ))d k i . Now we apply Remark 19 and obtain1 N N X i =1 | X iT − q ( T ) | + ξ N N X i =1 Z T | µ ǫ ( X i ) | d r + 2 ξ N N X i =1 Z T d | k i | r ≤ N N X i =1 | X iT − q ( T ) | + 2 1 N N X i =1 Z T ( X i − q ( t )) µ ǫ ( X i )d t + CT + 2 1 N N X i =1 Z T ( X i − q ( t ))d k i ≤ N N X i =1 | X i − q (0) | + CT + 2 σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 Z T ( X ir − q ( r ))d W ir (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + σ T ≤ C + 2 σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 Z T ( X ir − q ( r ))d W ir (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) By Burkholder-Davis-Gundi inequality (and boundedness of X and q ), we arrive at C ′ E N N X i =1 Z T | µ ǫ ( X i ) | d r ! p + C ′ E N N X i =1 Z T d | k i | r ! p ≤ C + C E N N X i =1 Z T | X ir − q ( r ) | d r ! p/ ≤ C, which is the desired bound.Thanks to the previous Lemma, we can conclude a uniform BV estimate on the drift. Lemma 20.
For every ≤ p < ∞ , it holds sup N,ǫ,i =1 ...N E Z T | µ ǫ ( X ǫ,N,i ) | d r ! p + sup N,ǫ,i =1 ...N E Z T | d k N,ǫ,ir | ! p < + ∞ . roof. We start as before from formula (23), for fixed i , and use Remark 19, getting | X iT − q ( T ) | + ξ Z T | µ ǫ ( X i ) | d r + 2 ξ Z T d | k i | r ≤ | X i − q (0) | + CT + 2 Z T | X ir − q ( r ) | N N X j =1 | µ ǫ ( X jr ) | d r + 2 σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ( X ir − q ( r ))(d W i − N N X j =1 d W j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + σ T + 2 Z T | X ir − q ( r ) | N N X j =1 d k jr ≤ C + 2 Z T N N X j =1 | µ ǫ ( X jr ) | d r + 2 σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ( X ir − q ( r ))(d W i − N N X j =1 d W j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 Z T N N X j =1 d k jr . By Burkholder-Davis-Gundi inequality, we get C ′ E Z T | µ ǫ ( X i ) | d r ! p + C ′ E Z T d | k i | r ! p ≤ C + C E Z T N N X j =1 | µ ǫ ( X jr ) | d r p + C + C E Z T N N X j =1 d k jr p . Here we use Lemma 18 and conclude C ′ E Z T | µ ǫ ( X i ) | d r ! p + C ′ E Z T d | k i | r ! p ≤ C. The proof is complete.
In this paragraph we use a similar strategy to estimate the H¨older norm of X i , first taking theaverage over i to remove the interaction term, then using the estimate on the average to control theinteraction term. In order to bound the H¨older norm, we take the Itˆo differential of | X it − q ( t ) − X is + q ( s ) | (instead of just | X it − q ( t ) | ).Throughout this subsection, we will assume Conditions 1-(i,ii) on µ , 2 on q and 3 on X .We start with a preliminary result which will be used in the next estimates: Lemma 21.
For every ≤ p < ∞ , it holds for some C p ≥ independent of s, t , sup N,ǫ,i E (cid:18)Z ts [( X ir − q ( r ) − X is + q ( s )) µ ǫ ( X ir )] − d r (cid:19) p ≤ C p | t − s | p , sup N,ǫ,i E (cid:18)Z ts [( X ir − q ( r ) − X is + q ( s )) n ( X ir )] − | d k ir | (cid:19) p ≤ C p | t − s | p . roof. We start with the first inequality and we fix δ > ǫ and N . Usingthe elementary inequality [ a + b ] − ≤ | a | + [ b ] − for a, b ∈ R , we have[( X ir − q ( r ) − X is + q ( s )) µ ǫ ( X ir )] − ≤ | q ( r ) − q ( s ) || µ ǫ ( X ir ) | + [( X ir − X is ) µ ǫ ( X ir )] − ≤ | q ( r ) − q ( s ) || µ ǫ ( X ir ) | + | X ir − X is | max [ δ, − δ ] | µ ǫ | + (1 X ir <δ,X ir ≤ X is + 1 − δ 0. Therefore[( X ir − q ( r ) − X is + q ( s )) n ( X ir )] − | d k ir | = [( X ir − q ( r ) − X is + q ( s )) n ( X ir )] − X ir ∈ ∂ ]0 , | d k ir |≤ | q ( r ) − q ( s ) || d k ir | . The Lipschitz property of q and Lemma 20 give E (cid:18)Z ts | q ( r ) − q ( s ) || d k ir | (cid:19) p ≤ C | t − s | p E Z T | d k ir | ! p ≤ C p | t − s | p and we arrive at the second estimate. 24 emark 22. Only in the above proof we use Condition 1-(ii). If µ diverged at the boundaries like x − α for some α > , then a similar result to Lemma 21 should hold, but with ( X ir − q ( r ) − X is + q ( s )) α in place of ( X ir − q ( r ) − X is + q ( s )) . However, such result would not be enough, since in the nextLemma 23 the power- factor ( X ir − q ( r ) − X is + q ( s )) appears and is needed to cancel the interactionterm when taking the average. We also expect, for µ diverging like x − α with α > , that the particlesshould not even touch the boundaries (as it is without interaction), but we do not focus on this point. We estimate the H¨older norm of the average of the drift over i : Lemma 23. There exists < α ≤ / such that, for every ≤ p < ∞ , it holds, for some C p ≥ independent of s, t , sup N,ǫ E N N X i =1 Z ts | µ ǫ ( X ǫ,N,i ) | d r ! p + sup N,ǫ E N N X i =1 Z ts | d k ǫ,N,ir | ! p ≤ C p | t − s | αp . Proof. We start estimating the H¨older norm of N P Ni =1 | X i − q | . For this we fix s and we have,for t > s , d | X it − q ( t ) − X is + q ( s ) | (24)= 2( X it − q ( t ) − X is + q ( s ))( − µ ǫ ( X it ) + 1 N N X i =1 µ ǫ ( X jt ))d t + 2 σ ( X it − q ( t ) − X is + q ( s ))(d W it − N N X j =1 d W jt ) + σ (1 − N )d t + 2( X it − q ( t ) − X is + q ( s ))( − d k it + 1 N N X j =1 d k jt ) . Similarly to the argument in Lemma 18, averaging over i we get rid of the interaction terms N P Ni =1 µ ǫ ( X jt ), N P Nj =1 d W jt and N P Nj =1 d k jt :d 1 N N X i =1 | X it − q ( t ) − X is + q ( s ) | = − N N X i =1 ( X it − q ( t ) − X is + q ( s )) µ ǫ ( X it )d t + 2 σ N N X i =1 ( X it − q ( t ) − X is + q ( s ))d W it + σ (1 − N )d t − X it − q ( t ) − X is + q ( s ))d k it . 25e take the p -power and obtain N N X i =1 | X it − q ( t ) − X is + q ( s ) | ! p ≤ C p N N X i =1 Z ts [( X ir − q ( r ) − X is + q ( s )) µ ǫ ( X ir )] − d r ! p + C p σ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 Z ts ( X ir − q ( r ) − X is + q ( s ))d W ir (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + C p σ p | t − s | p + C p N N X i =1 Z ts [( X ir − q ( r ) − X is + q ( s )) n ( X ir )] − | d k ir | ! p . The first addend of the RHS is controlled via Lemma 21 and Jensen inequality (applied to theaverage over i ): E N N X i =1 Z ts [( X ir − q ( r ) − X is + q ( s )) µ ǫ ( X ir )] − d r ! p ≤ sup N,ǫ,i E (cid:18)Z ts [( X ir − q ( r ) − X is + q ( s )) µ ǫ ( X ir )] − d r (cid:19) p ≤ C p | t − s | p . Similarly for the last addend. The second addend is controlled via Burkholder-Davis-Gundi in-equality and Jensen inequality: E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 Z ts ( X ir − q ( r ) − X is + q ( s ))d W ir (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ sup N,ǫ,i E (cid:12)(cid:12)(cid:12)(cid:12)Z ts ( X ir − q ( r ) − X is + q ( s ))d W ir (cid:12)(cid:12)(cid:12)(cid:12) p ≤ C p sup N,ǫ,i E (cid:18)Z ts | X ir − q ( r ) − X is + q ( s ) | d r (cid:19) p/ ≤ C p sup N,ǫ,i E sup t | X it − q ( t ) | p | t − s | p/ ≤ C p | t − s | p/ Therefore we have E N N X i =1 | X it − q ( t ) − X is + q ( s ) | ! p ≤ C p | t − s | p/ . Now we recall the following elementary inequality (consequence of Cauchy-Schwarz inequality), forevery two sequences of real numbers a i , b i : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 ( a i − b i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N N X i =1 | a i − b i | ! / N N X i =1 | a i + b i | ! / . a i = X it − q ( t ), b i = X is − q ( s ) and using Jensen inequality, we get theH¨older bound on N P Ni =1 | X i − q | : E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 | X it − q ( t ) | − | X is − q ( s ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ E N N X i =1 | X it − q ( t ) − X is + q ( s ) | ! p/ N N X i =1 | X it − q ( t ) + X is − q ( s ) | ! p/ ≤ E N N X i =1 | X it − q ( t ) − X is + q ( s ) | ! p ! / E N N X i =1 | X it − q ( t ) + X is − q ( s ) | p ! / ≤ C p | t − s | p/ sup N,ǫ,i ( E sup t | X it − q ( t ) | p ) / ≤ C p | t − s | p/ . (25)On the other hand, averaging (23) and using again the cancellation of the interaction terms, we getthe equation for N P Ni =1 | X i − q | :1 N N X i =1 | X it − q ( t ) | − | X is + q ( s ) | = − N N X i =1 Z ts ( X ir − q ( r )) µ ǫ ( X ir )d r + 2 1 N N X i =1 Z ts σ ( X ir − q ( r ))d W ir + σ (1 − N )( t − s ) − N N X i =1 Z ts ( X ir − q ( r ))d k ir , and so, by Remark 19, we obtain E N N X i =1 Z ts | µ ǫ ( X ir ) | d r ! p + E N N X i =1 Z ts | d k ir | ! p ≤ C p E N N X i =1 Z ts ( X ir − q ( r )) µ ǫ ( X ir )d r ! p + E N N X i =1 Z ts ( X ir − q ( r ))d k ir ! p + C p ( t − s ) p ≤ C p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 | X it − q ( t ) | − | X is + q ( s ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + C p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 Z ts σ ( X ir − q ( r ))d W ir (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + C p ( t − s ) p . We control the first addend in the RHS by (25) and the second addend by Burkholder-Davis-Gundiinequality (and Jensen inequality on the average over i ): E N N X i =1 Z ts | µ ǫ ( X ir ) | d r ! p + E N N X i =1 Z ts | d k ir | ! p ≤ C p | t − s | p/ + C p sup N,ǫ,i ( E sup t | X ir − q ( t ) | p ) | t − s | p/ + C p ( t − s ) p ≤ C p | t − s | p/ , which is the desired estimate with α = 1 / 4. 27ow we can prove the uniform H¨older bound: Lemma 24. With the notation of the previous Lemma, for every ≤ p < ∞ , it holds, for some C p ≥ independent of s, t , sup N,ǫ,i E | X it − X is | p ≤ C p | t − s | αp/ . Proof. By Jensen inequality, it is enough to prove the estimate for p ≥ 2. We start again with theequation (24) for a fixed i . Taking the p/ | X it − q ( t ) − X is + q ( s ) | p ≤ C p (cid:18)Z ts [( X ir − q ( r ) − X is + q ( s )) µ ǫ ( X ir )] − d r (cid:19) p/ + C p Z ts | X ir − q ( r ) − X is + q ( s ) | N N X j =1 | µ ǫ ( X jt ) | d r p/ + C p σ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ts ( X ir − q ( r ) − X is + q ( s ))d( W ir − N N X j =1 W jr ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p/ + C p σ p | t − s | p/ + C p (cid:18)Z ts [( X ir − q ( r ) − X is + q ( s )) n ( X ir )] − | d k ir | (cid:19) p/ + C p Z ts | X ir − q ( r ) − X is + q ( s ) | N N X j =1 | d k jr | p/ The first addend of the RHS is controlled again via Lemma 21: E (cid:18)Z ts [( X ir − q ( r ) − X is + q ( s )) µ ǫ ( X ir )] − d r (cid:19) p/ ≤ C p | t − s | p/ . Similarly for the fourth addend. The previous Lemma 23 allows to control the second addend: E Z ts | X ir − q ( r ) − X is + q ( s ) | N N X j =1 | µ ǫ ( X jt ) | d r p/ ≤ C p E Z ts N N X j =1 | µ ǫ ( X jt ) | d r p/ ≤ C p | t − s | αp/ Similarly for the fifth addend. The third addend is controlled via Burkholder-Davis-Gundi inequal-28ty: E Z ts ( X ir − q ( r ) − X is + q ( s ))d( W ir − N N X j =1 W jr ) p/ ≤ C p sup N,ǫ,i E (cid:18)Z ts | X jr − q ( r ) − X is + q ( s ) | d r (cid:19) p/ ≤ C p | t − s | p/ Putting all together we get E | X it − q ( t ) − X is + q ( s ) | p ≤ C p | t − s | αp/ . Using the Lipschitz continuity of q , we obtain the desired bound. Remark 25. We have shown that the H¨older exponent is α/ / (since we can take α = 1 / ).This is a consequence of our argument, but we expect that the optimal H¨older exponent is still / . We conclude with a H¨older estimate on the total variation of the drift: Lemma 26. For every ≤ p < ∞ , it holds for some C p ≥ independent of s, t sup N,ǫ,i E (cid:18)Z ts | µ ǫ ( X ǫ,N,ir ) | d r (cid:19) p + sup N,ǫ,i E (cid:18)Z ts | d k ǫ,N,ir | (cid:19) p ≤ C | t − s | αp/ . Proof. The equation (23) for | X t − q ( t ) | implies Z ts X i − q ( r )) µ ǫ ( X i )d r + Z ts X i − q ( r ))d k ir ≤ | X is − q ( s ) | − | X it − q ( t ) | + C Z ts N N X j =1 | µ ǫ ( X j ) | d r + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ts σ ( X i − q ( t ))(d W i − N N X j =1 d W j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + σ ( t − s ) + C Z ts N N X i =1 | d k jr | . and so, by Remark 19, Z ts | µ ǫ ( X i ) | d r + Z ts | d k ǫ,N,ir |≤ C | X is − X it − q ( s ) + q ( t ) || X is + X it − q ( s ) − q ( t ) | + C Z ts N N X j =1 | µ ǫ ( X j ) | d r + C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ts σ ( X i − q ( t ))(d W i − N N X j =1 d W j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Cσ ( t − s ) + C Z ts N N X j =1 | d k jr | . 29y Burkholder-Davis-Gundi inequality, we get E (cid:18)Z ts | µ ǫ ( X i ) | d r (cid:19) p + E (cid:18)Z ts | d k ǫ,N,ir | (cid:19) p ≤ C E | X is − X it − q ( s ) + q ( t ) | p + C E Z ts N N X j =1 | µ ǫ ( X j ) | d r p + C ( t − s ) p + C E Z ts N N X j =1 | d k jr | p . Lemma 23 and Lemma 24 allow to conclude the desired bound. In this Subsection we show the convergence of the regularized particle system (20) to the McKean-Vlasov SDE (21), by a compactness argument; as a consequence, we get the existence of a solutionto (21). We are given a probability space (Ω , A , P ) and independent Brownian motions W i , i ≥ 1, on a (right-continuous complete) filtration ( F t ) t . For each N , we are given ( X ,N , . . . X N,N ) F -measurable random variable and we let ( X i,N,ǫ , k i,N,ǫ ) be the corresponding solution to theregularized N -particle system (20). Through all the section, we assume Conditions 1, 2 and 3.In the following we use the notation C t = C ([0 , T ]; R d ), C t, [0 , = C ([0 , T ]; [0 , W β,pt = W β,p ([0 , T ]) for the fractional Sobolev space of order 0 < β < ≤ p < ∞ ,with norm k f k pW β,pt = Z T | f ( t ) | p d t + Z T Z T | f ( t ) − f ( s ) | p | t − s | βp d s d t. We consider the Polish space E = C t × C t, [0 , × C t , endowed with its Borel σ -algebra. We denote ageneric element of E as γ = ( γ , γ , γ ) or (for reasons that will be clear later) ( W, X, Z ); with a littleabuse of notation, we use W, X, Z also to denote the canonical projections on E . We consider alsothe space P ( E ) of probability measures on E with the topology of weak convergence of probabilitymeasure (also endowed with its Borel σ -algebra). The space P ( E ) is a Polish space as well [AGS08,Remark 7.1.7]. For a measure ν and a function g on E , we use the notation ν ( g ) = R E g d µ (whenthe integral makes sense). We take the random empirical measures on E given by L N,ǫ = 1 N N X i =1 δ ( W i ( ω ) ,X i,N,ǫ ( ω ) , − R · µ ǫ ( X i,N,ǫr )d r − k i,N,ǫ ) , which are random variables on P ( E ). Proposition 27. Assume Conditions 1, 2 and 3 (actually, Condition 1-(iii) is not needed). Thenthe family ( Law ( L N,ǫ )) N,ǫ (probability measures on P ( E ) ) is tight. Remark 28. In view of the proof, we recall the following standard/known facts: • To prove that a family of probability measures ( P n ) n on a metric space χ is tight, it is enoughto find a nonnegative function F on χ , such that F is coercive (that is, with compact sublevelsets) and R χ F d P n is bounded uniformly in n . When χ = P ( E ) for E as above (and more generally for every Polish space E ), endowedwith the topology of weak convergence, we can take F ( ν ) = R E g d ν as nonnegative coercivefunction on P ( E ) , provided that g : E → R is a nonnegative coercive function on E . Indeed,every sublevel set { F ≤ C } is compact: for any sequence ( ν n ) n of measures on E , if all ν n belong to { F ≤ C } , then, by the previous point applied to g , ( ν n ) n is tight, hence there existsa subsequence which is weakly convergent to some measure ν on E , and µ also belongs to { F ≤ C } by Fatou lemma. • By Sobolev embedding, there exists C > such that, for every γ in E = C t × C t, [0 , × C t , k γ k C αt ≤ C k γ k W β,pt for α = β − /p , provided that β − /p > . By Ascoli-Arzel`a theorem, the norm k · k C αt iscoercive on E for α > . Therefore, to show that a certain family of probability measures P n on P ( E ) is tight, it is enough to show that sup n E P n Z E k γ k W β,pt µ (d γ ) < ∞ for some β > , p ≥ with β − /p > (here E P n denotes the expectation under P n ).Proof. By the previous Remark 28, it is enough to verify that, for some β > p ≥ β > /p ,for h = 1 , , 3, sup n E Z E k γ h k W N,ǫt L ǫ,N (d γ ) < ∞ . For h = 1, that is the Brownian motion component, we have E Z E k γ k W β,pt L N,ǫ (d γ ) = E N N X i =1 k W i k W β,pt = E k W k W β,pt < ∞ for any β < / p ≥ h = 2, that is the solution component, by Lemma 24 we get, for some β > 0, for every p ≥ < δ < β , for every i = 1 , . . . N , E [ k X i,N,ǫ k pW β − δ,pt ]= Z T E | X i,N,ǫt | p d t + Z T Z T E | X i,N,ǫt − X i,N,ǫs | p | t − s | β − δ ) p d s d t ≤ T + Z T Z T | t − s | − (1 − δp ) d s d t sup s,t E | X i,N,ǫt − X i,N,ǫs | p | t − s | βp ≤ C for some C > N and ǫ . It follows that E k γ k W β − δδ,pt = E N N X i =1 k X ǫ,N,i k W β − δ,pt ≤ C. A similar argument, using Lemma 26 in place of Lemma 24, works for h = 3. The proof iscomplete. 31rom now on, we assume that N P Ni =1 δ X i,N converges in law to some probability measureLaw( X ).In the following, we fix a limit point Q of Law( L N,ǫ ) and a P ( E )-valued random variable L withlaw Q . With a little abuse of notation, we do not re-label the subsequence of Law( L N,ǫ ) convergingto Q , and we assume that L is also defined on the same probability space (Ω , A , P ) (this is only fornotational simplicity). We call Q N,ǫ,e , Q e the (deterministic) probability measures on E obtainedaveraging resp. L N,ǫ , L , namely, for every Borel bounded or nonnegative function g on E , E Q ǫ,N,e [ g ] = E [ L ǫ,N ( g )] , E Q e [ g ] = E [ L ( g )] , where E Q denotes the expectation with respect to Q . Remark 29. We recall some useful facts of convergence in law of random probability measures. • Let H : E → ˜ E a continuous map with values in some Polish space ˜ E , then the P ( ˜ E ) -valuedrandom variables H L N,ǫ converge in law to H L . This follows from the continuity of themap ν H ν , which in turn follows from the continuity of H . • Let g be in C b ( E ) , then the real-valued random variables L N,ǫ ( g ) converge in law to L ( g ) .Similarly to the previous point, this follows from the continuity of the map ν ν ( g ) . • The probability measures Q N,ǫ,e converge weakly to Q e : indeed, for every g in C b ( E ) , E [ L N,ǫ ( g )] converge to E [ L ( g )] . • For any Borel set B of E , it holds L N,ǫ ( B ) = 1 P -a.s. if and only if Q N,ǫ,e ( B ) = 1 ; similarlyfor L and Q ǫ . Now we show that, for a.e. L , W (the first component in E ) is a Brownian motion under L and X is a generalized solution to the McKean-Vlasov equation, with the right initial condition, under L . The proof is in two parts. In the first part, we prove that the expectation of X t under L is q ( t )and that the law of the initial condition X and W is Law( X ) ⊗ Wiener measure. In the secondpart we idenfity the reflection term and prove its properties. Lemma 30. It holds P -a.s.: • For every t , E L [ X t ] = q ( t ) . • Under the measure L , the random path t X t − X − σW t − Z t is actually L -a.s. deterministic; that is, the law of this path under L is a Dirac delta.Proof. For the first point, we start fixing t . The function on E defined by ( W, X, Z ) X t iscontinuous and bounded (as X takes values in [0 , E L N,ǫ [ X t ] converge in law to E L [ X t ]. On the other hand, equation (20) gives, for every( N, ǫ ), P -a.s., E L N,ǫ [ X t ] = 1 N N X i =1 X i,N,ǫ = q ( t ) . E L [ X t ] is δ q ( t ) , that is E L [ X t ] = q ( t ) on a P -full measure set Ω t , which maydepend on t . To make the exceptional set independent on t , we note that, by dominated convergencetheorem, t E L [ X t ] is continuous for every ω and that q is also continuous by assumption, hencewe have the equality for every t in the full-measure set Ω ′ = ∩ s ∈ Q ∩ [0 ,T ] Ω s . The proof of the firstpoint is complete.For the second point, we have to prove that ( X − X − σW − Z ) L is a Dirac delta P -a.s..By Remark 29, the P ( C t )-valued random variables ( X − X − σW − Z ) L N,ǫ converge in law to( X − X − σW − Z ) L . On the other hand, equation (20) gives, for every ( N, ǫ ), P -a.s.: for every i = 1 , . . . N , ( X − X − σW − Z ) δ ( W i ,X i,N,ǫ , − R · µ ǫ ( X i,N,ǫr )d r − k i,N,ǫ ) = δ X i,N,ǫ − X i,N,ǫ − σW it + R · µ ǫ ( X i,N,ǫr )d r + k i,N,ǫ = δ q ( t ) − q (0)+ N P Nj =1 [ R · µ ǫ ( X j,N,ǫr )d r − σW jt + k j,N,ǫ ] =: δ γ N,ǫ , note that γ N,ǫ is independent of i . Averaging over i , we get( X − X − σW − Z ) L N,ǫ = δ γ N,ǫ , in particular ( X − X − σW − Z ) L N,ǫ is concentrated on the subset { δ γ | γ ∈ C t } of P ( C t ). Weclaim that { δ γ | γ ∈ C t } is a closed set in P ( C t ). Hence, since ( X − X − σW − Z ) L N,ǫ convergesin law to ( X − X − W − Z ) L , also ( X − X − σW − Z ) L is concentrated on { δ γ | γ ∈ C t } , thatis the law of X − X − σW − Z under L is a Dirac delta.It remains to prove the above claim. If δ γ n converge to a measure ν , then, by tightness of δ γ n ,there exists a compact set K in C t such that δ γ n ( K ) > / γ n belong to K , for every n .Therefore there exists a subsequence γ n k converging to some element γ in K , hence δ γ nk convergeto δ γ and so ν = δ γ belongs to { δ γ | γ ∈ C t } , which is then closed. The proof of the second pointis complete. Lemma 31. It holds P -a.s.: under L , the C t × R -valued random variable ( W, X ) has law P W ⊗ Law ( X ) , where P W is the Wiener measure.Proof. The map from E to C t × R defined by ( W, X, Z ) ( W, X ) is continuous. Therefore, byRemark 29, the random empirical measures( W, X ) L N,ǫ = 1 N N X i =1 δ ( W i ,X i,N ) converge in law to ( W, X ) L . On the other hand, the above random measures converge in law to P W ⊗ Law( X ) (see e.g. [CDFM20, Lemma 29]). Hence the law of Law L ( W, X ) is δ P W ⊗ Law( X ) ,that is Law L ( W, X ) = P W ⊗ Law( X ) P -a.s.. The proof is complete.Next we define the process k by k t = k ( X, Z ) t = − Z t X r / ∈ ]0 , d Z r , (26)if Z is a BV path on [0 , T ], k t = 0 otherwise. We call | k | the total variation process associated with k . 33 emma 32. It holds P -a.s.: • The processes Z , R · µ ( X r )d r , k have BV trajectories L -a.e. and their BV norms are integrablewith respect to L . • It holds L -a.e.: for every t ≥ , Z t + k t = Z t X r ∈ ]0 , d Z r = − Z t µ ( X r )1 X r ∈ ]0 , d r. • It holds L -a.e.: the process k satisfies the condition d | k | t = 1 X t ∈{ , } d | k | t , d k t = n ( X t )d | k | t . (27) Proof. For all statements but the integrability of the BV norms, by Remark 29, it is enough toprove these statements Q e -a.e. (recall Q e is the average of L ) instead of L -a.e. (provided we workwith Borel sets/properties, as the proof will do); it is also enough to prove integrability of the BV norms of Z , R · µ ( X r )d r , k with respect to Q e . Again by Remark 29, the measures Q N,ǫ,e convergein law to Q e , hence we can work with Q N,ǫ,e and Q e only. BV property of Z and k . By Lemma 20, we have E Q N,ǫ,e k Z k BV ≤ E N N X i =1 Z T | µ ǫ ( X i,N,ǫr ) | d r + | k i,N,ǫ | T ≤ C (28)for some constant C independent of ǫ and N . Now the BV norm is lower semi-continuous in C t ,since it can be written as k γ k BV = sup π X [ t i ,t i +1 [ ∈ π | γ ( t i +1 ) − γ ( t i ) | , the sup being over all partitions π of [0 , T ]. Therefore it holds E Q e k Z k BV ≤ C, in particular Z and so k have BV paths Q e -a.s.. Support property of | k | . By definition, k is concentrated on { t ∈ [0 , T ] | X t ∈ { , }} , whichis a closed set, hence also its total variation process | k | is concentrated on this set and we concludethat d | k | t = 1 X t ∈{ , } d | k | t .BV property of R · µ ( X r )d r . Since µ ǫ ≥ µ δ for ǫ < δ , by monotone convergence theorem we34ave E Q e Z T | µ ( X r ) | d r = sup δ E Q e Z T | µ δ ( X r ) | d r = sup δ lim N,ǫ E Q N,ǫ,e Z T | µ δ ( X r ) | d r = sup δ lim N,ǫ E N N X i =1 Z T | µ δ ( X N,ǫ,ir ) | d r ≤ lim inf N,ǫ E N N X i =1 Z T | µ ǫ ( X N,ǫ,ir ) | d r < ∞ , in particular also R · µ ( X r )d r has BV trajectories, with Q e -integrable BV norm. Representation formulae for Z + k and k . We take a > δ > ϕ : [0 , → R C functionwith support in [ δ, − δ ], ˜ n ; [0 , → R a continuous extension of the outer normal n with supporton ] δ, − δ [ c and with ˜ n ≥ − δ, 1] and ˜ n ≤ , δ ], g, h : [0 , T ] → R continuous with g non-negative. We consider the set A = A a,ϕ, ˜ n,h,g = { ( W, X, Z ) ∈ E | k Z k BV ≤ a, Z T h ( r ) ϕ ( X r )d Z r = − Z T h ( r ) ϕ ( X r ) µ ( X r )d r, Z T g ( r )˜ n ( X r )d Z r ≤ } . Lemma 33. The set A is closed in E .Proof. Let ( W n , X n , Z n ) be a sequence in A converging to ( W, X, Z ) uniformly. Since the BV normof Z n is bounded by a for every n , up to taking a subsequence we can assume that dZ n convergesweakly-* to a measure ν with total variation k ν k T V ≤ a . Passing to the limit in the chain rule for Z , we find that, for every ψ in C ∞ ([0 , T ]), Z T ψ d ν = ψ ( T ) Z T − ψ (0) Z − Z T ψ ′ Z d r. Hence ν is the distributional derivative of Z , which therefore satisfies k Z k BV ≤ a .Concerning the stability of the conditions involving µ and ˜ n , note that h ( r ) ϕ ( X nr ) → h ( r ) ϕ ( X r )uniformly and also h ( r ) ϕ ( X nr ) µ ( X nr ) → h ( r ) ϕ ( X r ) µ ( X r ) uniformly (since µ is C on [ δ, − δ ]).This fact and the weak-* convergence of Z n implies that Z [0 ,T ] h ( r ) ϕ ( X r )d Z r = lim n Z T h ( r ) ϕ ( X nr )d Z nr = − lim n Z T h ( r ) ϕ ( X nr ) µ ( X nr )d r = − Z T h ( r ) ϕ ( X r ) µ ( X r )d r. Reasoning similarly for ˜ n , we find Z T g ( r )˜ n ( X r )d Z r = lim n Z T g ( r )˜ n ( X nr )d Z nr ≤ . This proves that ( W, X, Z ) is in A . Hence A is closed.35ow the equation for X ǫ,N and Condition 1-(iii) imply, for δ < ρ : for ǫ < δ , under Q ǫ,N,e itholds a.s. Z T h ( r ) ϕ ( X r )d Z r = − Z T h ( r ) ϕ ( X r ) µ ( X r )d r, Z T g ( r )˜ n ( X r )d Z r = − Z T g ( r )˜ n ( X r )( µ ǫ ( X r )d r + d k r ) ≤ . Moreover the uniform bound (28) implies Q ǫ,N,e {k Z k BV > a } ≤ a E Q ǫ,N,e k Z k BV ≤ C/a. Therefore, for δ < ρ , for any a , Q ǫ,N,e ( A ) ≥ − C/a . Since A is closed, we conclude that Q e ( A ) ≥ − C/a . Hence Q e is concentrated on the set B ϕ, ˜ n,h,g = { ( W, X, Z ) ∈ Ω | k Z k BV < ∞ , Z T h ( r ) ϕ ( X r )d Z r = − Z T h ( r ) ϕ ( X r ) µ ( X r )d r, Z T g ( r )˜ n ( X r )d Z r ≤ } , for every ϕ , ˜ n , h , g as above. Now we take: ϕ = ϕ m tending pointwise to 1 ]0 , and uniformlybounded in m ; ˜ n = ˜ n m tending pointwise to n ( x )1 { , } and uniformly bounded in m ; h in acountabledense set D in C t , g in D + countable dense set D + in { g ∈ C t | g ≥ } . Then Q e is concentratedon ˜ B = ∩ m,h ∈ D,g ∈ D + B ϕ m , ˜ n m ,h,g ∩ { ( W, X, Z ) | Z · µ ( X )d r ∈ BV } . Lemma 34. For every ( W, X, Z ) in ˜ B , it holds: Z t X r ∈ ]0 , d Z r = − Z t µ ( X r )1 X r ∈ ]0 , d r ∀ t, (29)d k t = n ( X t )d | k | t . (30) Proof. For every ( W, X, Z ), for every fixed h in D and g in D + , the BV property of Z (and so of k ) and of R · µ ( X r )d r implies, via dominated convergence theorem, Z T h ( r ) ϕ m ( X r )d Z r → Z T h ( r )1 X ∈ ]0 , d Z r , Z T h ( r ) ϕ m ( X r ) µ ( X r )d r → Z T h ( r )1 X ∈ ]0 , µ ( X r )d r, Z T g ( r )˜ n m ( X r )d Z r → − Z T g ( r ) n ( x )d k r . Therefore, if ( W, X, Z ) is in ˜ B , passing to the limit in m in the definition of B ϕ, ˜ n,h,g we get Z T h ( r )1 X ∈ ]0 , d Z r = − Z T h ( r ) µ ( X r )1 X ∈ ]0 , d r, − Z T g ( r ) n ( x )d k r ≤ . for all h in D , g in D + . By the density of D and D + we obtain (29) and (30).36e conclude that, for Q e -a.e. ( W, X, Z ), the representation formulae (29) and (30) hold. Theproof of Lemma 32 is complete. Remark 35. Only in the above proof we use Condition 1-(iii). If σ = 0 , we expect that, by asuitable version of Girsanov theorem on domains, the time spent by X on the boundary has zeroLebesgue measure, Q -a.s.. Morally this should allow to remove or relax the condition 1-(iii). We are ready to prove: Proposition 36. It holds P -a.e.: under L , ( X, k ) is a generalized solution to the McKean-Vlasovproblem (21) starting from X , with initial distribution Law ( X ) (more precisely, ( E, B ( E ) , W, X, k, L ) is a weak generalized solution with initial distribution Law ( X ) ).Proof. By Lemma 31, P -a.s., W is a Brownian motion under L and X is independent of W . As aconsequence of Lemma 30 it holds, P -a.s., under L : for every t , X t = X − Z t + σW t + q ( t ) − q (0) − E L Z t , (31)where we have used E L W t = 0. By Lemma 32, it holds, P -a.s., under L : R · µ ( X r )1 X r ∈ ]0 , d r and k are in BV with integrable BV norms and, for every t , Z t = Z t µ ( X r )1 X r ∈ ]0 , d r + k t = Z t µ ( X r )d r + k t , (32)where k t satisfies (27) and where we have used that µ (0) = µ (1) = 0. Therefore ( X, k ) is ageneralized solution.We deduce, via Yamada-Watanabe, the existence of a strong solution to (21), that is the exis-tence part of Theorem 9, as well as uniqueness in law: Corollary 37. It holds P -a.s.: under L , ( X, k ) is a strong solution to the SDE (21) and thelaw of X under L coincide with the unique law Law ( ¯ X ) of any solution to (21) starting fromLaw ( ¯ X ) = Law ( X ) .Proof. We have P -a.s.: the couple ( X, k ) is a weak generalized solution under L , hence it is a strongsolution, via Proposition 17. Proposition 16 gives uniqueness in law for the X component.Finally we arrive at the convergence result, that is Theorem 10: Corollary 38. The family ( N P Ni =1 δ X ǫ,N,i = X L ǫ,N ) ǫ,N of random probability measures on C ([0 , T ]; [0 , converges in probability, as ǫ → and N → ∞ , to the law of the McKean-Vlasovsolution ¯ X (starting from Law ( X ) ).Proof. Since the limit Law( ¯ X ) is deterministic (and P ( E ) is a metric space), it is enough to proveconvergence in law. Since P ( P ( E )) is a metric space and the family (Law( X L ǫ,N ) ǫ,N ) ǫ,N is rela-tively compact (that is tight), it is enough to prove that every limit point of (Law( X L ǫ,N ) ǫ,N ) ǫ,N is actually δ Law( ¯ X ) . This is an immediate consequence of Corollary 37. The proof is complete.37 .5 Pathwise analysis This subsection is dedicated to the proof of Proposition 12; we assume in this Subsection theconditions of Proposition 12. We use a pathwise approach developed e.g. in [CDFM20], we explainfirst briefly the core idea behind it. Let (Ω , A , P ) be a probablity space and W : Ω → C ([0 , T ] , R )a random variable on this space. Note that at this point we do not impose that W is a BrownianMotion. Consider the SDEd X t = [ − µ ( X t ) + E [ µ ( X t )]]d t + ˙ q t d t + σ d W t − d k t − σ E [d W t ] + E [d k t ] X ∈ C ([0 , T ]; [0 , , k ∈ C ([0 , T ]; R ) , d | k | = 1 X t ∈{ , } d | k | , d k = n ( X t )d | k | . (33)If we endow the probability space with a (right-continuous complete) filtration ( F t ) t ≥ and assumethat W is a Brownian Motion with respect to this filtration, clearly equation (33) is exactly theMcKean-Vlasov equation (21).On the other hand, let ( X ( N ) , k ( N ) ) be the solution of the interacting particle system (20). Let ω ∈ Ω be fixed. On a suitable discrete prabability space endowed with the point counting measurethe process ( X ( N ) , k ( N ) )( ω ) = ( X i,N ( ω ) , k i,N ( ω )) i =1 ,...,N is a random variable in the variable i andas such a solution to equation (33). The mean with respect to the point counting measure is exactlythe empirical average.This is the main idea behind the proof of the Lemma 40. First we recall the definition ofWasserstein distance. Definition 39. Let ( E, d ) be a polish space. Let P ( E ) be the space of probability measures on E with finite second moment. The -Wasserstein distance on P ( E ) is defined as W ,E ( µ, ν ) := inf ((cid:18)Z E × E d ( x, y ) m ( dx, dy ) (cid:19) | m coupling of µ, ν ) µ, ν ∈ P ( E ) . From now on, we work under the assumptions of Proposition 12. Lemma 40. Let ( ¯ X, ¯ k ) be the solution to equation (21) with initial condition ¯ X with law ν . Let ( X ( N ) , k ( N ) ) be a solution to the interacting particle system (20) with initial condition X ( N ) =( X ,N , . . . , X N,N ) . Assume that ( ¯ X , X ( N )0 ) is independent on the noise W ( N ) = ( W , . . . , W N ) .For every t ∈ [0 , T ] , we have P − a.s. , W , R ( Law ( ¯ X t ) , N N X i =1 δ X i,Nt ) ≤ C E Z T d | ¯ k s | ! + 1 N N X i =1 Z T d | k i,Ns | ! · W ,C ([0 ,T ] , R ) ( Law ( W ) , N N X i =1 δ W i ) + W , [0 , ( ν , N N X i =1 δ X i,N ) . Proof. For simplicity of notation, we take σ = 1 (the argument is the same for general σ ∈ R ).Call ν := Law( W ) the Wiener measure on C t = C ([0 , T ] , [0 , ω ∈ Ω, we consider theempirical measure L N,ω := N P Ni =1 δ ( W i ( ω ) ,X i,N ( ω )) as a law on E = ( C t × [0 , , B ( C t ) × B ([0 , P ω ∈ P ( E × E ) be any coupling of ν ⊗ ν and L N,ω . It is easy to verify that P ω can be seen asa measure on Ω ω := ( E × { ( W ( ω ) , X ,N ( ω )) , . . . , ( W N ( ω ) , X N,N ( ω )) } ), endowed with the product38 -algebra A ω := B ( C t ) × { ( W ( ω ) ,X ,N ( ω )) ,..., ( W N ( ω ) ,X N,N ( ω )) } . Indeed, for every Borel bounded testfunction ϕ : E × E → R , P ω ( ϕ ) = Z E × E ϕ ( x, y ) P ω (d x, y ) L N,ω (d y )= 1 N N X i =1 Z E ϕ ( x, ( W i ( ω ) , X i,N ( ω )) P ω (d x, d( W i ( ω ) , X i,N ( ω )))= Z Ω ω ϕ ( x, y ) P ω (d x, y ) L N,ω (d y ) . On the space (Ω ω , A ω , P ω ) we define the projections (Π , Π ) ∼ ν ⊗ ν and (Π , Π ) ∼ L N,ω onthe first and second marginal space, respectively (in particular, Π ∼ ν and Π ∼ N P Ni =1 δ W i ( ω ) ).Since the law of Π is the Wiener measure ν , we have that Π is a Brownian motion, and if we plugit as the driver of equation (33) we obtain a strong unique solution ( ¯ X, ¯ k ) thanks to Theorem 9.Let ( X ( N ) , k ( N ) ) be the solution of equation (20) given by Proposition 5. There exists a setof full measure Ω ⊂ Ω such that for every ω ∈ Ω and every 1 ≤ i ≤ N , ( X i,N ( ω ) , k i,N ( ω ))satisfies equation (20). Defining ( ˜ X, ˜ k )( W i ( ω ) , X i,N ( ω )) := ( X i,Nt , k i,Nt )( ω ), we have that, forevery t ∈ [0 , T ], E P ω [˜ k t ] = N P Nj =1 k j,Nt ( ω ) andd ˜ X t = ( µ ( ˜ X t ) − E P ω [ µ ( ˜ X t )])d t + d q t + dΠ t − d˜ k t − d E P ω [Π t ] + d E P ω [˜ k t ] , on Ω . We define b ( ¯ X t ) := µ ( ¯ X t ) − E P ω [ µ ( ¯ X t )] and we estimate the following12 d( ¯ X t − ˜ X t − (Π t − Π t ) + E P ω [Π t − Π t ] + Z t [ b ( ¯ X s ) − b ( ˜ X s )]d s ) = − ( ¯ X t − ˜ X t )d¯ k t + ( ¯ X t − ˜ X t )d˜ k t + ( ¯ X t − ˜ X t )d E P ω [¯ k t ] − ( ¯ X t − ˜ X t )d E P ω [¯ k t ] + (Π t − Π t )d¯ k t − (Π t − Π t )d˜ k t − (Π t − Π t )d E P ω [¯ k t ] + (Π t − Π t )d E P ω [˜ k t ] − E P ω [Π t − Π t ]d¯ k t + E P ω [Π t − Π t ]d˜ k t + E P ω [Π t − Π t ]d E P ω [¯ k t ] − E P ω [Π t − Π t ]d E P ω [˜ k t ] + (cid:20)Z t [ b ( ¯ X s ) − b ( ˜ X s )]d s (cid:21) d( E P ω [¯ k t ] − E P ω [˜ k t ]) . The first and second term on the right-hand side are always negative by the conditions on theboundaries. If we take expectation under P ω on both sides, we have that the third and fourth termon the right-hand side vanish, because E P ω [ ¯ X t ] = E P ω [ ˜ X t ] = q t . Similarly, the expectation of thelast term vanishes. Hence, we have that E P ω [ | ¯ X t − ˜ X t − (Π − Π ) − (Π t − Π t ) + E P ω [Π t − Π t ] + Z t [ b ( ¯ X t ) − b ( ˜ X s )]d s | ] ≤ E P ω [ sup t ∈ [0 ,T ] | Π t − Π t | ] E P ω Z T d | ¯ k s | ! + E P ω Z T d | ˜ k s | ! + E P ω [ | Π − Π | ]The proof is concluded by first using Gronwall’s lemma and then choosing P ω = P ωW ⊗ P ω , where P ωW (resp. P ω ) is the optimal coupling in W ,C t ( ν, N P Ni =1 δ W i ) (resp. W , [0 , ( ν , N P Ni =1 δ X i,Nt ))39hanks to the previous proposition, it is immediate to derive the convergence of the particlesystem to the McKean-Vlasov equation, provided that we have convergence at time 0 and a boundon the second moment of k N . Proof of Proposition 12. By Lemma 40 and using H¨older inquality, we have E " sup t ∈ [0 ,T ] W , [0 , (Law( ¯ X t ) , N N X i =1 δ X i,Nt ) ≤ C E Z T d | ¯ k s | ! + 1 N N X i =1 E Z T d | k i,Ns | ! · E " W ,C ([0 ,T ] , R ) (Law( W ) , N N X i =1 δ W i ) + E " W , [0 , (Law( ¯ X ) , N N X i =1 δ X i,N ) . The first term on the right-hand side is uniformly bounded in N thanks to Lemma 18. Theempirical measure of independent random variables distributed as the the Wiener measure convegresin Wasserstein metric to the Wiener measure as O (1 / p log( N )), see [BLG14]. The Wassersteindistance of the intial conditions converges faster. Remember X i,N = Y i + P Nj =1 δ Y j + q (0), where( Y i ) i ∈ N is a family of independent and identically distributed random variables. We see that thespeed of convergnce of N P Ni =1 δ X i,N is the same as the the speed of convergence of P Ni =1 δ Y j , whichis 1 / √ N , see [FG15]. For a fixed ω ∈ Ω, take an optimal coupling m = m ( ω ) between Law( ¯ X )and N P Ni =1 δ Y i , we have that ( x, y − E m [ y ] − E m [ x ]) m ( dx, dy ) is a coupling between Law( ¯ X )and N P Ni =1 δ X i . We can compute W , [0 , (Law( ¯ X ) , N N X i =1 δ X i,N ) ≤ E m (cid:2) | X − Y + E m Y − E m X | (cid:3) = Var m ( X − Y ) ≤ E m | X − Y | = W , [0 , (Law( ¯ X ) , N N X i =1 δ Y i, ) . Taking the square roots and the expectation under P concludes the proof. The system (20) can be seen as an SDE on the moving domain H t ∩ [0 , N , where H t = { x ∈ R N | N N X i =1 x i = q ( t ) } , with normal boundary conditions. Indeed, formally, for each i = 1 , . . . N and m = 0 , 1, on theboundary x i = m , the direction of reflection ( − m ( e i − N − (1 , . . . e i being the i -th vector of40he canonical basis) is orthogonal to the face H t ∩ { x | x i = m } . Here we use this fact to showwell-posedness of the system (20).We introduce some notation. In the following, we fix N and omit the superscripts N and ǫ inthe notation. We call H = { x ∈ R N | N P Ni =1 x i = 0 } , 1 = (1 , , . . . ∈ R N , Π : R N → R N theprojector on H , that is Π x = x − N − ( x · A : H → R N − a linear isometry and wecall D t = A Π( H t ∩ [0 , N ). For i = 1 , . . . N , m = 0 , 1, we call ∂ i,m [0 , N = { x ∈ [0 , N | x i = m } , ∂ i,m D t = A Π( H t ∩ ∂ i,m [0 , N ), γ i,m = ( − m ( e i − N − 1) the direction of reflection of (20) on theface ∂ i,m [0 , N and ν i,m = Aγ i,m . For x in ∂ [0 , N = ∪ i,m ∂ i,m [0 , N , we callΓ( x ) = { X i,m c i,m γ i,k x ∈ ∂ i,k [0 , N | c i,m ≥ ∀ i = 1 , . . . N, m = 0 , } . Similarly, for y in ∂D t = ∪ i,m ∂ i,m D t , we call N t ( y ) = { X i,m c i,m ν i,k y ∈ ∂ i,k D t | c i,m ≥ ∀ i = 1 , . . . N, m = 0 , } , note that N t ( y ) = A Π( x ) if x is in ∂ i,m [0 , N .We consider the following SDE on D t :d Y t = A Π b ( t, A − Y t + q ( t )1)d t + Πd W t + d h t ,Y t ∈ D t ∀ t, P -a.s. , d | h | t = 1 Y t ∈ ∂D t d | h | t , d h t = ν t d | h | t , ν t ∈ N t ( Y t ) , (34)where ( Y, h ) is the solution, W is an N -dimensional Brownian motion with respect to a (complete,right-continuous) filtration ( F t ) t and b is the drift of the system (20). This is an SDE on a movingdomain D t with reflection at the boundary. As we will see, the SDE (34) is, up to the isometry A ,the system (20). Lemma 41. Under Condition 2 on q and the Lipschitz continuity of µ ǫ , given a probability space (Ω , A , P ) and a Brownian motion W on a (complete right-continuous) filtration ( F t ) t , there existsa unique (strong) solution to the SDE (34) .Proof. The existence and uniqueness result is a consequence of [NO15, Theorem 1.7] for SDEson moving domains with reflecting boundaries, provided that the assumptions of that theoremhold. We focus on two key assumptions, namely: a) the fact that N t ( y ) is the cone of inwardnormal vectors of D t at y , for every t and every y ∈ ∂D t ; b) relation (1.16) in [NO15]. The otherassumptions of [NO15, Theorem 1.7] are easy to verify.Concerning assumption a), we observe that, for each i = 1 , . . . N , m = 0 , 1, the vector γ i,m is theinward normal, in the N − H t ∩ [0 , N , of the corresponding face H t ∩ ∂ i,m [0 , N : indeed γ i,m belongs to H and, for every v in H ∩ ∂ i,m [0 , N , we have γ i,m · v = 0.Since A is an isometry, the vector ν i,m = Aγ i,m is the inward normal, in the convex polyhedron D t ,of the corresponding face ∂ i,m D t = A Π( H t ∩ ∂ i,m [0 , N ). Now N t ( y ) is the convex cone generatedby ν i,m , with i, m such that y ∈ ∂ i,m D t . Hence N t ( y ) is the convex cone of inward normal vectors(in the sense of [NO15, Definition 2.2]), see e.g. formula (4.23) in [Cos92].Assumption b) reads as follows. Define a s,z ( ρ, η ) = max u ∈ R N − , | u | =1 min s ≤ t ≤ t + η min y ∈ ∂D t , | y − z |≤ ρ min ν ∈ N t ( y ) , | ν | =1 ( ν · u ) . (35)41ondition (1.16) in [NO15] readslim η → lim ρ → inf s ∈ [0 ,T ] inf z ∈ ∂D s a s,z ( ρ, η ) = a > . (36)In order to show this condition, we take I = I s,z ( ρ, η ) = ∪ s ≤ t ≤ t + η ∪ y ∈ ∂D t , | y − z |≤ ρ { ( i, m ) | y ∈ ∂ i,m D t } ,u s,z ( ρ, η ) = c I X ( i,m ) ∈ I s,z ( ρ,η ) ν i,m , where c I > | u s,z ( ρ, η ) | = 1; note that min I c I = c N > 0. Wealso note that, for suitable ρ > η > s and z ), for every ρ < ρ and η < η ,for every s and z , for every j = 1 , . . . N , at most one element between ( j, 0) and ( j, 1) belongs to I s,z ( ρ, η ). Moreover, since the average of y is in [ ξ, − ξ ] (for all y ∈ ∂D t for all t ), y i cannot be all0, nor they can be all 1, hence I s,z cannot be { (1 , , (2 , , . . . ( N, } or { (1 , , (2 , , . . . ( N, } .As a consequence,if ( i, m ) ∈ I, then there exist at most N − j = i with ( j, m ) ∈ I. (37)We compute the scalar products among ν i,m , using the isometry property of A : | ν i,m | = | γ i,m | = 1 − N − ,ν i,m · ν j,m = γ i,m · γ j,m = − N − for i = j,ν i,m · ν j,n = γ i,m · γ j,n = N − for i = j, m = n. We call ˆ ν i,m = (1 − N − ) − / ν i,m . For η < η and ρ < ρ , we get by (37), for every ( i, m ) in I = I s,z ( ρ, η ),ˆ ν i,m · u I = c I (1 − N − ) − / − N − − X ( j,m ) ∈ I,j = i N − + X ( j,n ) ∈ I,j = i,m = n N − = c I (1 − N − ) − / (1 − N − − ( N − N − ) ≥ c N N − (1 − N − ) − / . Now, for every s, z , for every t ∈ [ s, s + η ] and y ∈ ∂D t with | y − z | ≤ ρ , N t ( y ) is contained in theconvex cone generated by ˆ ν i,m , ( i, m ) ∈ I s,z ( η, ρ ). Therefore, for η < η and ρ < ρ , for every s and z , we have ν · u I ≥ c N N − (1 − N − ) − / , for every ν as in (35) . and so a s,z ( ρ, η ) ≥ c N N − (1 − N − ) − / > 0, in particular (36) holds. The proof is complete.Now we show that the SDE (34) is equivalent to the system (20). We introduce some notation.We take a Borel map G : { ( t, y, v ) | t ∈ [0 , T ] , y ∈ ∂D t , v ∈ N t ( y ) } → [0 , + ∞ ) N × , ( t, x, v ) ( c i,m ) i =1 ,...N,m =0 , , c i,m = 0 if y does not belong to ∂ i,m D t , and v = X ( i,m ) ,y ∈ ∂ i,m D t c i,m ν i,m . [Note that this map G exists but is not uniquely determined: indeed, if y belongs to ∂ i, D t ∪ ∂ i, D t for each i (that is, y = A Π x for some x with x i ∈ { , } for each i ), then ν i,m are not linearlyindependent.] For a solution ( Y, h ) to (34), with d h = ν d | h | , we call X Y,ht = A − Y t + q ( t )1 ,k Y,ht = Z t X ( i,m ) G i,m ( r, Y r , ν r ) n ( X ir ) e i X ir = m d | h | r , recall that n ( m ) = − ( − m is the outward normal of [0 , 1] in m = 0 , Lemma 42. Assume that ( Y, h ) is a solution to (34) . Then ( X Y,h , k Y,h ) is a solution to the system (20) .Proof. Let ( Y, h ) be a solution to (34), take ( X, k ) = ( X Y,h , k Y,h ). By the definition of X Y,h and k Y,h , P -a.s. X is has continuous paths with values in [0 , 1] and k has continuous paths, and, foreach i , | k i | is concentrated on { t | X i ∈ { , }} and has direction n ( X i ). Hence the second andthird lines of (20) are satisfied. We haved h t = X ( i,m ) G i,m ( t, Y t , ν t )1 Y t ∈ ∂ i,m D t ν i,m d | h | t = X ( i,m ) G i,m ( t, Y t , ν t )1 X it = m A ( − m ( e i − N − | h | t = − X ( i,m ) G i,m ( t, Y t , ν t )1 X it = m n ( X it ) Ae i d | h | t + N − X ( i,m ) G i,m ( t, Y t , ν t )1 X it = m n ( X it ) A | h | t = − A d k t + N − A (d k · A ( − d k t + N − X i d k it . Hence, applying the transformation X t = A − Y t + q ( t )1 to the first line of (34), we obtain the firstline of (20). Therefore ( X Y,h , k Y,h ) satisfies (20). The proof is complete.By Lemmas 41 and 42, we get existence of a solution to (20). Remark 43. We expect also the converse of Lemma 42 to hold, namely, if ( X, k ) solves (20) , then ( Y = A Π X, h = − A Π k ) solves (34) . In particular, from this converse we would get uniqueness for (20) . However showing the third line of (34) is not immediate, hence we do not follow this strategy. We conclude the proof of Proposition 5 by showing uniqueness for (20): Lemma 44. Strong uniqueness (in X ) holds for the SDE (20) .Proof. The proof follows the line of Proposition 13, replacing the expectation with the empiricalaverage. Let ( X, k X ), ( Y, k Y ) two solutions to (20) with the same initial condition X = Y . In43his proof we call d K X = N P Nj =1 [ µ ( X j )d t − d W j + d k X,j ] and similarly for K Y . By Itˆo formulafor continuous semimartingales, we have, for every i = 1 , . . . N ,d | X i − Y i | = 2( X i − Y i )( − µ ( X i ) + µ ( Y i ))d t + 2( X i − Y i )d K X − X i − Y i )d K Y − X i − Y i )d k X,i + 2( X i − Y i )d k Y,i . The one-side Lipschitz condition of µ implies( X i − Y i )( − µ ( X i ) + µ ( Y i )) ≤ c | X i − Y i | . and the orientation of k (as the outward normal) implies − Z t ( X i − Y i )d k X,i ≤ X i − Y i )d k Y,i . For the addends with K , we average over i and use that K doesnot depend on i and that N P i X it = N P i Y it = q ( t ): we obtain1 N X i ( X i − Y i )d K X = 0and similarly for ( X i − Y i )d K Y . 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