Quantitative estimates of propagation of chaos for stochastic systems with W 1, kernels

Transcription

1 oname manuscrit o. will be inserted by the editor) Quantitative estimates of roagation of chaos for stochastic systems with W, kernels Pierre-Emmanuel Jabin Zhenfu Wang Received: date / Acceted: date Abstract We derive quantitative estimates roving the roagation of chaos for large stochastic systems of interacting articles. We obtain exlicit bounds on the relative entroy between the joint law of the articles and the tensorized law at the limit. We have to develo for this new laws of large numbers at the exonential scale. But our result only requires very weak regularity on the interaction kernel in the negative Sobolev sace Ẇ,, thus including the Biot-Savart law and the oint vortices dynamics for the 2d incomressible avier-stokes. Keywords Proagation of chaos Relative entroy Law of large numbers 2d incomressible avier-stokes Mathematics Subject Classification 2) 35Q3 6F7 6H 76R99 P.E. Jabin is artially suorted by SF Grant 3242 and by SF Grant RMS Ki- et) Z. Wang is artially suorted by SF Grant 3242 and Ann G. Wylie Dissertation Fellowshi P.E. Jabin CSCAMM and Det. of Mathematics, University of Maryland, College Park, MD 2742, USA. jabin@cscamm.umd.edu Z. Wang CSCAMM and Det. of Mathematics, University of Maryland, College Park, MD 2742, USA. zwang423@math.umd.edu

2 2 Pierre-Emmanuel Jabin, Zhenfu Wang Introduction. Motivation We consider large systems of indistinguishable oint-articles given by the couled stochastic differential equations SDEs) dx i = F X i ) dt+ KX i X j ) dt+ 2σ dwt i, i =,, ) j i where for simlicity X i Π d, the d-dimensional torus, the W i are indeendent standard Wiener Processes Brownian motions) in R d and the stochastic term in ) should be understood in the Itô sense. The interaction term is normalized by the factor /, corresonding to the mean field scaling. For a fixed our goal is hence to derive exlicit, quantitative estimates comaring System ) to the mean field limit ρ solving t ρ + div x ρ [F + K x ρ]) = σ ρ. 2) Such estimates in articular imly the roagation of chaos in the limit. But recisely because they are quantitative, they also characterize the reduction of comlexity of System ) for large and finite. A guiding motivation of interaction kernel K in our work is given by the Biot-Savart law in dimension 2, namely Kx) = α x x 2 + K x), 3) where x denotes the rotation of vector x by π/2 and where K is a smooth correction to eriodize K on the torus reresented by [ /2, /2] d. If ωx) L Π d ) with, then u = K x ω solves ) curl u = curl K x ω = α ω, div u = div K ω =. Π d ω If F =, the limiting equation 2) becomes t ω + K x ω x ω = σ ω, 4) where we now write on ωt, x), using the classical notation for the vorticity of a fluid. Eq. 4) is invariant by the addition of a constant ω ω + C. We may hence assume that Π d ω = and Eq. 4) is then equivalent to the 2d incomressible avier-stokes system on ut, x) s.t. ω = curl u, t u + u x u = x + σ u, div u =. 5) The system of articles ) now corresonds to a system of interacting oint vortices with additive noise. Because we resent our method in the simlest

3 Quantitative estimates of roagation of chaos 3 framework where articles are indistinguishable, all oint vortices necessarily have the same vorticity in this setting. Our main result rovides an exlicit estimate quantifying that the system ) is within O /2 ) from the limit 2) in an aroriate statistical sense. However the method that we develo is able to handle general kernels K which are not necessarily singular only at the origin or may not even be functions. The rest of the introduction is organized as follows: We state recisely our main result in the next subsection. We devote subsection.3 to a longer discussion of various examles of kernels K that are covered by our main theorem. While our main focus concerns systems with non-vanishing diffusion, the tools that are develoed in this article can also be alied to settings with vanishing diffusion. We give an examle of such a result in subsection.4. The last subsection in the introduction sketches the roof of our basic a riori estimates. Section 2 resents the roof of our main results, assuming that one has two critical estimates, Theorems 3 and 4, which are sort of modified laws of large numbers. We establish some reliminary combinatorics notations in section 3. This enables us to easily rove Theorem 3 in section 4. The roof of Theorem 4 is considerably more difficult; it is erformed in section 5 which is the main technical contribution of this article..2 Main results We start by recalling the recise definition of the sace Ẇ, Π d ) which is used both in Pro. and in Theorem and which is critical to our alications. Definition A function f with f = belongs to Π Ẇ, Π d ) iff there d exists a vector field g in L Π d ) s.t. f = div g. Similarly a vector field K with K = belongs to Π Ẇ, Π d ) iff there exists a matrix field V in L Π d ) d s.t. K = div V or K α = β βv αβ. We then denote and similarly f Ẇ, K Ẇ, = inf g L, with f = div g, g = inf V L, with K = div V. V Following the basic aroach introduced in [29], our main idea is to use relative entroy methods to comare the couled law ρ t, x,..., x ) of the whole system ) to the tensorized law ρ t, x,..., x ) = ρ = Π i= ρt, x i ), consisting of indeendent coies of a rocess following the law ρ, solution to the limiting equation 2).

4 4 Pierre-Emmanuel Jabin, Zhenfu Wang As our estimates carry over ρ, we do not consider directly the system of SDEs ) but instead work at the level of the Liouville equation t ρ + div xi ρ F x i ) + i= Kx i x j ) = σ xi ρ, 6) where and hereafter we use the convention that K) =. The law ρ encomasses all the statistical information about the system. Given that it is set in with >>, the observable statistical information is tyically contained in the marginals j= ρ,k t, x,..., x k ) = ρ t, x,..., x ) dx k+... dx. Π d k) 7) Our final goal is to obtain exlicit bounds on ρ,k ρ k, where ρ k = Π k i= ρt, x i). Those bounds will follow from a relative entroy estimate between ρ and a solution ρ to 6). But for this, we cannot use any weak solution to the Liouville 6) and instead require Definition 2 Entroy solution) A density ρ L ), with ρ and ρ dx =, is an entroy solution to Eq. 6) on the time interval [, T ], iff ρ solves 6) in the sense of distributions, and for a.e. t T t Πd ρ t, X ) log ρ t, X ) dx xi ρ 2 + σ dx ds i= ρ ρ log ρ dx t div F x i ) + div Kx i x j )) ρ dx ds, i, j= i= 8) where for convenience we use in the article the notation X = x,, x ). In general it can be difficult to obtain the well osedness of an advectiondiffusion equation such as 6) under very weak regularity of the advection field K, such as is our case here. We refer to [3] for an examle of such study. In our case though, we do not need the well osedness and it is in fact straightforward to check that there exists at least one entroy solution to 6). Proosition Assume that ρ log ρ <, σ σ >, and that F, div F L. Assume finally that K Ẇ, with as well div K

5 Quantitative estimates of roagation of chaos 5 Ẇ,. Then there exists an entroy solution ρ satisfying ρ t, X ) log ρ t, X ) dx + σ Π 2 d Πd ρ log ρ dx + i= t Πd xi ρ 2 ρ t div K 2 Ẇ, + t div F L. 2 σ Moreover for any φ L 2 [, T ], W, Π 2d )) with φ L 2 t Wx, t φt, x, x 2 ) Kx x 2 ) ρ,2 t, x, x 2 ) dx dx 2 dt K Ẇ, Π 2d + t + 2 Πd ρ log ρ dx + σ so that the roduct K ρ is well defined. t div K 2 Ẇ, σ 2 + t dx ds 9) 2 div F L, σ ) Our method revolves around the control of the rescaled relative entroy H ρ ρ )t) = ρ t, X ) log ρ t, X ) Π ρ d t, X ) dx, ) while our main result is the exlicit estimate Theorem Assume that div F L Π d ), that K Ẇ, Π d ) with div K Ẇ,. Assume that σ σ >. Assume moreover that ρ is an entroy solution to Eq. 6) as er Def. 2. Assume finally that ρ L [, T ], W 2, Π d )) for any < solves Eq. 2) with inf ρ > and ρ =. Then Π d H ρ ρ )t) e M K + K 2 ) t H ρ ρ ) + + M + t + K 2 )) σ σ ), where we denote K = K Ẇ, + div K Ẇ, and M is a constant which only deends on M 2 ) ρ L d, σ, inf ρ, ρ W,, su, ρ log ρ, div F L. Remark There is no exlicit regularity assumtion on F in the revious theorem. evertheless some regularity on F is imlicitly required, in articular to obtain W 2, solution ρ to 2).

6 6 Pierre-Emmanuel Jabin, Zhenfu Wang Remark 2 While our results are resented for simlicity in the torus Π d, they could be extended to any bounded domain Ω with aroriate boundary conditions. The ossible extension to unbounded domains however aears highly non-trivial, in articular in view of the assumtion inf ρ > which could not hold anymore. The roof of Theorem strongly relies on the roerties of the relative entroy over tensorized saces such as. Those roerties are also critical to derive aroriate control on the observables or marginals ρ,k. In articular the sub-additivity imlies that the relative entroy of the marginals is bounded by the total relative entroy or H k ρ,k ρ k ) = k Π d k ρ,k log ρ,k ρ k dx... dx k H ρ ρ ), 2) for which we refer to [24, 38, 39] where estimates quantifying the classical notion of roagation of chaos are thoroughly investigated. It is then ossible to derive from Theorem the strong roagation of chaos as er Corollary Under the assumtions of Theorem, if H ρ ρ ) as, then over any fixed time interval [, T ] H ρ ρ ), as. As a consequence considering any finite marginal at order k, one has the strong roagation of chaos ρ,k ρ k L [, T ], L Π d k )). Finally in the articular case where su H ρ ρ ) = H <, and where su σ σ = S <, then one has that, for some constant C deending only on k, H, S, T and K and M defined in Theorem, ρ,k ρ k L [, T ], L Π d k )) C. 3) Remark 3 The rate of convergence in / in 3) is widely considered to be otimal as it corresonds to the size of stochastic fluctuations. We refer for examle to [36] where entroy methods are used in this context for smooth interaction kernels; see also the rior [] and [3, 9]. Proof Corollary follows directly from Theorem by using inequality 2) and the Csiszár-Kullback-Pinsker inequality see for instance [47]) for any f and g functions on Π d k f g L Π d k ) 2k H k f g).

7 Quantitative estimates of roagation of chaos 7 The starting stes in the roof of Theorem, such as the relative entroy and the reduction to a modified law of large numbers, had already been exosed in [29]. However the resent contribution exands much on the basic ideas and techniques introduced in [29]: First we make better use of the diffusion, which was instead mostly considered as a erturbation in [29]. This is the main reason why we are essentially able to gain one full derivative in our assumtion on K with resect to the K L in [29]. The main technical contribution in the resent article, namely the modified law of large numbers stated in Theorem 4, is considerably more difficult to rove than any equivalent in [29]. This has lead to several new ideas in the combinatorics aroach, detailed in the roof of Theorem 4 in section 5. Theorem 4 is also much more general and we believe that it can be of further and wider use. The imortance of law of large numbers for the roagation of chaos or the mean field limit has of course long been recognized, at least since Kac, see [3] or [45]. We also refer to [2] for an examle where the classical law of large numbers is used but which is limited to Lischitz kernels K. The relative entroy at the level of the Liouville equation does not seem to have been widely used yet with [48], in the context of hydrodynamics of Ginzburg-Landau, being maybe the closest to the aroach develoed here. We also refer to [5] for a different, trajectorial, view on the role of the entroy in SDEs..3 Alications We delve in this section into some examles of kernels K that our method can handle and discuss at the same time where our result stands in comarison to the existing literature. In general quantitative estimates of roagation of chaos were reviously only available for smooth, Lischitz, kernels K such as in the classical result [37]; see also [,3,9,36] for more on the classical Lischitz case. Gronwall-like estimates with Lischitz force fields, but a fixed number of SDEs, were also at the basis of [27]. System ) retains simle additive interactions, contrary to the more comlex structure found for examle in [4,4]; but it still includes a large range of first order models, such as swarming, oinion dynamics or aggregation equations, see for instance [2,5,,2] or [32]. The list of examles given below is hence by no means exhaustive and we refer to our recent survey [3] for a more thorough discussion of current imortant questions. The 2d viscous vortex model where K satisfies 3). As mentioned in the introduction, the mean field limit 2) is then the 2d incomressible avier- Stokes equation written in vorticity form, Eq. 4). We can write K = div V, V = [ φ arctan x ] x 2 + ψ, φ arctan x2 x + ψ 2

8 8 Pierre-Emmanuel Jabin, Zhenfu Wang where one can choose φ smooth with comact suort in the reresentative /2, /2) 2 of Π 2 and ψ, ψ 2 ) a corresonding smooth correction to eriodize V. Therefore K satisfies the assumtions of Theorem. The convergence of the systems of oint vortices ) to the limit 4) had first been established in [43] for a large enough viscosity σ. The well osedness of the oint vortices dynamics has been roved globally in [42]; see also [4]. Finally the convergence to the mean field limit has been obtained with any ositive viscosity σ in the recent [6]. However those results rely on a comactness argument based on a control of the singular interaction rovided by the dissiation of entroy in the system. As far as we know, this article is the first to rovide a quantitative rate of roagation of chaos for the 2d viscous vortex model. Hamiltonian structure. If the dimension d is even then the revious examle can be generalized to include any Hamiltonian structure. In that case one has d = 2n, x = q, ) with q, Π n and for some Hamiltonian H : Π 2n R, K = H, q H). i,j V q i q j ). This is formally easy by Theorem now alies if H L Π 2n ), though this may not be the otimal condition see the discussion below). The theorem rovides roagation of chaos for such systems with diffusion with much weaker assumtions than any comarable result in the literature. We are nevertheless somewhat limited by our framework here. One would for examle tyically want to aly this to the classical ewtonian dynamics where H = i 2 i /2 + choosing the aroriate function F in the system of articles ). The first issue is that the momentum should be unbounded instead of having Π n ; as we mentioned in one of the remarks after Theorem, such an extension of our result to R n for examle would be non-trivial... The second issue concerns the diffusion which for such models usually alies only to the momentum. This leads to a degenerate diffusion whereas we absolutely require it in every variable. Collision-like interactions. We can even handle extremely singular interactions where some sort of collision event occurs at some fixed horizon. Consider for examle any function φ L Π d ), any smooth field Mx) of matrices and define K = div M I φ ), or K α x) = β β M αβ x) I φx) ). It is straightforward to choose M s.t. div K Ẇ, or even div K = : A simle examle is simly to take M anti-symmetric. As M I φ L, Theorem alies. This articular choice of K means that two articles i and j will interact exactly when φx i X j ) =. An obvious examle

9 Quantitative estimates of roagation of chaos 9 is φx) = x 2 2R) 2 in which case the articles can be seen as balls of radius R which interact when touching. But in the context of swarming, one could have birds, or other animals, which interact as soon as they can see each other; this is different from the cone of vision tye of interaction found for examle in [6] where the interaction is much less singular bounded). Micro-organisms such as bacteria may also have comlicated, non-smooth shaes. In all those cases {φ } is not a ball in general and may even be a singular set. Since Mx) is smooth, one could interret K as being suorted on the measure δ φ=. But in fact we do not need any regularity on φ, not even φ BV and here K may not even be a measure... Gradient flow structure. The dual to the Hamiltonian case is to take K = ψ for some otential ψ. This lets us see the system of articles ) as a gradient flow with diffusion and it endows the mean field limit 2) with the derived and nonlinear gradient flow structure. When ψ is convex, but not necessarily smooth, it is ossible to strongly use this gradient flow structure. This is in articular the key to obtain the well osedness of Eq. 2), even without diffusion, as in [7,8] and in [2] for the mean field limit. However it does not seem easy for our aroach to fully make use of such gradient flows. This is seen on the assumtions of Theorem where having K Ẇ, is not very demanding, ψ L would be enough, while the condition div K Ẇ, actually forces us to consider Lischitz otentials ψ. Of course any ψ convex is Lischitz so that Theorem still extends the known theory for general ψ. But it is clearly not erforming as well as in the Hamiltonian case. A very good examle of this is the 2d Patlak-Keller-Segel model of chemotaxis where one would like to have K = α x/ x 2 + K x). This choice of K is just a rotation of π/2 from the 2d avier-stokes kernel given by 3). Therefore we still have that K Ẇ, by using a rotation of the matrix V that we wrote in the avier-stokes setting. But unfortunately div K is now one full derivative away from Ẇ, and Theorem cannot be alied. By studying the secific roerties of the system though, a convergence result to measure-valued solutions was obtained in [26] while the convergence to weak solutions was achieved in [7] see also [8] for the sub-critical case). We also refer to [35] for general Coulomb interactions. Those results are not quantitative though and a major oen roblem remains to find an equivalent of Theorem in this case. We wish to conclude this subsection about kernels K to which Theorem alies, by discussing more in details the assumtion K Ẇ,. We first come back to the vortex dynamics for 2d avier-stokes and the kernel K given by the Biot-Savart law 3). Since div K =, the classical way

10 Pierre-Emmanuel Jabin, Zhenfu Wang to reresent K is by K = curl ψ with ψx) = α log x + ψ x), with again ψ a smooth correction to eriodize ψ. Obviously ψ is not bounded which at first glance suggests that K does not belong to Ẇ,. This is incorrect as the right choice of V above demonstrates but it means that knowing whether K Ẇ, is not as simle as it may seem. The distinction is rather technical but it is critical for us as it allows us to handle the crucial examle of the vortex model. It also turns out to be connected with a fundamental difficulty in our roof. Our estimates directly use a reresentation K = div V and the most difficult term would vanish if V were anti-symmetric, which is the case if we take K = curl ψ. The fact that we cannot take K = curl ψ with ψ L is resonsible for the main technical difficulty in this article and in articular this is what requires Theorem 4 whose roof takes all of section 5. We refer to the more secific comments that we make in subsection 2.. In general the study of the K for which there exists a matrix field V L s.t. div V = K turns out to be a very comlex mathematical question. This can be done coordinate by coordinate obviously so the question is equivalent to finding the scalar field φ for which there exists a vector field u L s.t. div u = φ. The difficulty is that for a given K, there does not exist a unique matrix field V s.t. div V = K. Of course in dimension d = 2 if div K =, then there exists a unique ψ u to a constant, s.t. K = curl ψ. In dimension d > 2, if div K =, there exists an anti-symmetric matrix V s.t. K = div V. The anti-symmetric matrix V is not unique in general though with the well known issue of the gauge choice for vector otential if d = 3. But even in dimension 2, there is no reason why ψ L if K Ẇ,. This is indeed connected to the fact that the Riesz transforms are unbounded on L and the kernel K of 3) is the classical examle of this. Instead one only has in general that ψ BMO. However even in this simle case, it is not known if ψ BMO is equivalent to K Ẇ,. This question is connected to the classical reresentation of BMO functions in []. For any ψ BMO, [] showed that there exists ψ, ψ, ψ 2 L s.t. ψ = ψ + R ψ + R 2 ψ 2 with R i, i =, 2, the Riesz transforms. If it were always ossible to take ψ = then we would have the equivalence but that seems at best) highly non-trivial. Instead the ositive results that we have are much more recent and limited. This line of investigation was started in the seminal [4] which roved that if K L d Π d ) then K Ẇ, Π d ). If K is known to be a signed measure then this was extended in [44] to find that K = div V with V L iff there exists C s.t. for any Borel set U Kdx) C U. 4) This result in [44] hence has the direct consequence U

11 Quantitative estimates of roagation of chaos Proosition 2 If d > and K belongs to the Lorentz sace L d, Π d ) then K Ẇ,. Proof Assuming K L d, then for a constant C, we have that Decomose now dyadically U {x Π d, Kx) M} C M d. Kx) dx U + k 2 k+ {x U, Kx) 2 k }. Define k s.t. 2 d k+) d k U 2 and bound {x U, Kx) 2 k } U for k k, {x U, Kx) 2 k } {x Π d, Kx) 2 k } This leads to Kx) dx U + U k k 2 k+ U + C k>k 2 d) k+ C 2 d k for k > k. U + 2 k+2 U + C 2 d) k+ C U d d, by using the definition of k. By the isoerimetric inequality, there exists a constant C d s.t. U d d C d U so that we verify the condition 4) which concludes the roof. Pro. 2 not only alies to K given by 3) but roves in general that any K with Kx) C/ x belongs to Ẇ,. The original result in [4] is not constructive, and it is even roved that the V L s.t. K = div V cannot be obtained linearly from K. The develoment of constructive algorithms to obtain V is a current imortant field of research, see [46]..4 The case with vanishing diffusion While we are mostly interested in Eq. 6) when the viscosity does not asymtotically vanishes, a nice and essentially free) consequence of the method develoed here is to also rovide a result with vanishing viscosity. The result is of course weaker and requires K L with div K L. Obtaining an entroy solution to 6) in the sense of 2) is even more straightforward in this case as there is no need for integration by arts. Moreover we

12 2 Pierre-Emmanuel Jabin, Zhenfu Wang also directly obtain the following bound, which relaces in that case the one rovided by Pro.. ρ t, X ) log ρ t, X ) dx + σ i= ρ log ρ dx + t div K L + div F L ). t Πd xi ρ 2 Under those stronger assumtions on K, we have the following result ρ dx ds 5) Theorem 2 Assume that div F L Π d ), that K L Π d ) with also div K L Π d ). Assume moreover that ρ is an entroy solution to Eq. 6) as er Def. 2. Assume finally that ρ L [, T ], W, Π d )) solves Eq. 2) with Π d ρ =. Then H ρ ρ )t) e M 2 K t H ρ ρ ) + + M 2 + K t) σ σ ), where we now denote K = K L + div K L and M 2 is a constant which only deends on ) log ρ L M 2 σ, log ρ W,, su ρ dx), ρ log ρ, div F L. Remark 4 The constant M 2 is in the above comlex form simly because we include all cases σ σ. For instance if σ σ, then M 2 only exlicitly deend on M 2 su M 2 su ) log ρ L ρ dx). For the vanishing viscosity case σ σ =, M2 only exlicitly deends on ). log ρ L ρ dx), log ρ W, See the roof of Theorem 2 in subsection 2.6 for more details. Remark 5 To control the error caused by the difference σ σ, we need log ρ L Π d ). This can be relaced to moment assumtions like log ρx) C x k so that the result can easily be extended to the whole sace R d.

13 Quantitative estimates of roagation of chaos 3 Theorem 2 is obviously mostly only useful in comarison to our main result if σ σ =, including otentially the urely deterministic setting where σ = or cases where the viscosity is degenerate in some directions. But it also requires less regularity on the limit ρ and could also be of use in such a situation. In articular it does not require that inf ρ > and is hence easy to extend to unbounded domains contrary to Theorem. Because of its usefulness for degenerate viscosities, it is rather natural to comare Theorem 2 to results for kinetic mean field limits based on the 2nd order dynamics dq i = P i dt, dp i = KQ i Q j ) dt + 2σ dwt i. 6) j= We refer to [9,28] for an introduction to the mean field question in this kinetic setting. The best results so far have been obtained in [23] for a singular kernel K with Kx) C x α, Kx) C x α with α < ; in [22] for Hölder continuous K. The most classical case is again the Poisson kernel Kx) = γ d x/ x d which is unfortunately out of reach so far excet in dimension as in [25]). It is ossible to treat truncated kernels such as Kx) = γ d x/ x +ε ) d with the most realistic ε obtained in [33,34]. However none of the techniques in those articles seems, so far, to be able to handle any diffusion and esecially vanishing or degenerate diffusion as in 6). In the case of 6) where the limiting equation is often called Vlasov-Fokker-Planck, we refer for examle to [3] which requires more regularity on K. We remark that in comarison, the theory of mean field limits for urely st order systems without viscosity is much more advanced, with the limit of oint vortices already obtained in [2]. However as we noticed before, the techniques develoed there do not seem to be comatible with any vanishing) diffusion. An obvious oint of comarison for Theorem 2 is our revious result in [29]. This revious result covered the case of 6) with the same assumtion K L ; it also introduced the basic ideas for the method used here, based on the relative entroy and combinatorics estimates. However [29] was relying strongly on the simlectic structure of the dynamics in 6). Extending the method to general kernels K which may not even be Hamiltonian, as is done by Theorem 2, changes the scoe of the result. It has also been roved to be quite comlex: From a technical oint of view, the whole combinatorics estimates of [29] can be summarized in section 3 of the resent article while the new estimates are considerably longer, see section 5..5 Sketch of the roof of Proosition The roof follows very classical ideas: Consider a regularized interaction kernel K ε. Eq. 6) with K ε now has a unique solution ρ,ε for any initial measure

14 4 Pierre-Emmanuel Jabin, Zhenfu Wang ρ. The goal is to take the limit ε, by extracting weak-* converging subsequences of ρ,ε, and to derive 6) for the limiting kernel K and the various estimates such as 8) and 9). The only small) difficulty in this rocedure is to obtain adequate uniform bounds. For this reason we only exlain here how to derive those bounds for any weak solution ρ to 6) which also satisfies 8). The first ste is to rove from 8) that t Πd xi ρ 2 dx ds C ρ log ρ dx. ρ i= Observe that if div K Ẇ,, that is div K = div ψ with ψ L = div K Ẇ,, then t div Kx i x j ) ρ dx ds On the other hand t i= i, j= div K Ẇ, xi ρ dx ds σ 2 div K Ẇ, + div K Ẇ, 2 σ i= i= t div K Ẇ, 2 σ + This imlies that i, j= t t t i= t Πd xi ρ 2 ρ ρ dx ds σ 2 div K Ẇ, t div K 2 Ẇ, + σ 2 σ 2 xi ρ dx ds dx ds i= t div Kx i x j ) ρ dx ds i= Introducing this bound in 8) shows that ρ t, X ) log ρ t, X ) dx + σ Π 2 d Πd ρ log ρ dx + t Πd xi ρ 2 Πd xi ρ 2 i= t ρ ρ dx ds. Πd xi ρ 2 ρ t div K 2 Ẇ, 2 σ + t div F L, dx ds. dx ds

15 Quantitative estimates of roagation of chaos 5 which since σ σ exactly roves 9). From [38] by convexity, we know that t Π2 x ρ,2 2 dx dx 2 d ρ,2 t Πd xi ρ 2 i= ρ dx ds. If K Ẇ,, i.e. if Kx) = div V x) or using coordinates K α x) = d β= βv αβ x) with V a matrix-valued field, then for any φ W, Kx x 2 ) φx, x 2 ) ρ,2 dx dx 2 Π 2 d = V x x 2 ) φ x ρ,2 + x φ ρ,2 ) dx dx 2 Π 2 d V L φ L + V L φ L Π2 d x ρ,2 2 which leads to ) using that inf V V L = K Ẇ,. Finally, we note that div x x γ = d x γ γ α x α x α x γ+2 = d γ x γ, ρ,2 dx dx 2 ) /2, so that with the same aroach it would be ossible to derive the bound ρ,2 Π x 2 d x 2 γ dx Π2 x ρ,2 2 dx 2 d γ) 2 dx dx 2, ρ d,2 for any γ < 2 if d = 2 and for γ = 2 if d > 2, which has roved critical in the revious derivation and studies of the 2d incomressible avier-stokes for instance see [4, 6] and [43]. 2 Proofs of Theorems and 2 2. Sketch of the roof of Theorem Our goal in this subsection is to resent the main stes of the roof. For this reason, we make several simlifying assumtions that allow us to focus on the main ideas. First of all, we assume that F =, div K =, K α = β β V αβ with V L Π d ) δ, for δ small in terms of some norms of ρ. We also assume that ρ C with inf ρ > and that ρ solution to 6) so that we may easily maniulate this equation. Finally we assume that σ = σ =. is a classical

16 6 Pierre-Emmanuel Jabin, Zhenfu Wang Following our revious discussion about the criticality of the assumtion K = div V with V L, we refer the readers in articular to the end of ste 2 after formula 9) and to ste 5 in the following roof. That ste requires the use of Theorem 4 whose roof contains the main technical difficulties of the article. If instead one would assume that V is anti-symmetric then the term B in ste 5 vanishes and as we mentioned above, we would have a much simler roof. Unfortunately this would not let us handle our most imortant kernel K = x / x 2 corresonding to the 2d incomressible avier-stokes system. Ste : Time evolution of the relative entroy. First of all it is straightforward to derive an equation on ρ from the limiting equation 2) t ρ + Kx i x j ) xi ρ = σ xi ρ i= j= i= + Kx i x j ) K x ρx i ) xi ρ. i= j= Combining this with the Liouville equation 6), one obtains that d dt H ρ ρ )t) 2 ρ Kx i x j ) K x ρx i )) xi log ρ dx i, j= i= ρ x i log ρ 2 ρ. 7) A full justification of this calculation is given later in the main roof in Lemma 2. Ste 2: Using K = div V. As the kernel K is not bounded but we only have that K = div V with V L, the next ste is to integrate by arts to make V exlicit in our estimates 2 = i, j= i, j= ρ Kx i x j ) K x ρx i )) xi log ρ dx 2 x ρ V x i x j ) V x ρx i )) : i ρ dx Π ρ d ρ V x i x j ) V x ρx i )) : xi ρ xi dx. Π ρ d i, j= Observe that we were careful in writing that ρ xi log ρ = ρ ρ xi ρ,

17 Quantitative estimates of roagation of chaos 7 so that after integration by arts, the second term involves a derivative of ρ / ρ which can be controlled thanks to the dissiation term in 7). More recisely by Cauchy-Schwartz ρ 2 V x i x j ) V x ρx i )) : xi ρ xi dx i, j= Π ρ d ρ x i 2 ρ 2 i= Π ρ d dx ρ 2 + Πd xi ρ 2 ρ ρ 2 V x i x j ) V x ρx i )) dx. i= j= Of course ρ x i 2 ρ 2 ρ = ρ x i log ρ 2 ρ ρ so that the first term is actually bounded by the dissiation of entroy. On the other hand xi ρ 2 ρ 2 = x i ρx i ) 2 ρx i ) 2. Hence we obtain that d dt H ρ ρ )t) A + B, 8) where 2 A = C ρ ρ V x i x j ) V x ρx i )) i= dx, j 9) B = 2 i, j= ρ V x i x j ) V x ρx i )) : 2 x i ρx i ) ρx i ) dx, and C ρ is a constant deending only on the smoothness of ρ. We oint out here that 2 x i ρ is a symmetric matrix. Hence, if V is antisymmetric, then the term B comletely vanishes: B =. Ste 3: Change of law from ρ to ρ. The two revious terms A and B can be seen as the exectations of the corresonding random variables with resect to the law ρ. Obviously we do not know the roerties of ρ and would much refer having exectations with resect to the tensorized law ρ. We hence use the following Lemma For any two robability densities ρ and ρ on, and any Φ L ), one has that η > Φ ρ dx H ρ ρ ) + Π log ρ e Φ dx. d

18 8 Pierre-Emmanuel Jabin, Zhenfu Wang Proof We give the short) roof for the sake of comleteness. Define f = λ e Φ ρ, λ = ρ e Φ dx. otice that f is a robability density as f and f =. Hence by the convexity of the entroy ρ log f dx ρ log ρ dx. Π d On the other hand, one can easily check that ρ log f dx = ρ Φ dx + ρ log ρ dx log λ Π d Π, d which concludes the roof of the lemma. To aly Lemma to A, we first exand A coordinate by coordinate as A C ρ d i= α,β= ρ V α,β x i x j ) V α,β x ρx i )) j= ow alying Lemma first to each in A and then to Φ α,β = 2 V α,β x i x j ) V α,β x ρx i )), j= 2 dx. Φ = 2 V x i x j ) V x ρx i )) : i,j= 2 x i ρx i ), ρx i ) in B, we obtain that with à = C ρ 2 B = log d i= α,β= A + B 2 H ρ ρ )t) + à + B, log ρ e ex j= V α,β x i x j ) V α,β x ρx i )) i, j= V xi xj) V x ρxi)) : 2 x i ρx i ) ρx i ) dx. 2 ρ dx, 2)

19 Quantitative estimates of roagation of chaos 9 Observe that the cost to erform this change of law is, unfortunately, severe as we now have exonential factors in à and B. That is the reason why we need L or almost L ) bounds on V. Ste 4: Bounding à through a law of large number at the exonential scale. By symmetry of ermutation, we may take i = in Ã. Define ψ α,β z, x) = V α,β z x) V α,β x ρz), so that j= = V α,β x x j ) V α,β x ρx )) j,j 2= ψ αβ x, x j )ψ αβ x, x j2 ). 2 We remark that each ψ has vanishing exectation with resect to ρ Π d ψ αβ z, x) ρx) dx =. Theorem 3 Consider any ρ L Π d ) with ρ and ρx) dx =. Π d Assume that a scalar function ψ L with ψ L < 2e, and that for any fixed z, ψz, x) ρx) dx = then Π d ρ ex C = 3 + where ρ t, X ) = Π i= ρt, x i) j,j 2= ψx, x j )ψx, x j2 ) dx 5α α) 3 + β β ), 2) α = e ψ L ) 4 <, β = 2e ψ L ) 4 <. 22) We give a straightforward roof of Theorem 3 in section 4, using the combinatorics techniques develoed in the article. But note that this theorem is essentially a variant of the well known law of large numbers at exonential scales; the main difference being that ψx, x j )ψx, x j2 ) does not have vanishing exectation if j = j 2, j = or j 2 =. Technically Theorem 3 is hence rather simle, contrary to Theorem 4 below. Using Theorem 3 and by taking V L small enough, we deduce that à C ρ. 23)

20 2 Pierre-Emmanuel Jabin, Zhenfu Wang Ste 5: Bound on B through a new modified law of large numbers. We now define 2 x ρx) φx, z) = V x z) V ρx)) :, ρx) and we aly to B the following result Theorem 4 Consider ρ L Π d ) with ρ and Π d ρ dx =. Consider further any φx, z) L with γ := C su ) 2 su z φ., z) L ρ dx) <, where C is a universal constant. Assume that φ satisfies the following cancellations φx, z) ρx) dx = Π d z, φx, z) ρz) dz = Π d x. Then ρ ex φx i, x j ) dx 3 <, 24) γ i,j= where we recall that ρ t, X ) = Π i= ρt, x i). Theorem 4 is by far the main technical difficulty in this article. Observe that contrary to classical laws of large numbers, it requires two recise cancellations on φ, searately in x where φx, z) ρx) dx = div Kx z) div K x ρx)) ρx) dx =, Π d Π d as div K = and in z where we use the classical cancellation Π d V x z) V x ρx)) ρz) dz =. Choosing δ so that V L is small enough, Theorem 4 again imlies that B C ρ. 25) While Theorem 4 looks similar to the modified law of large numbers that was at the heart of our revious result [29], it is considerably more difficult to rove. In [29], we relied a lot on the natural simlectic structure of the roblem, which is comletely absent here. The roof Theorem 4 is therefore the main technical difficulty and contribution of the article, erformed in Section 5. As we noticed earlier, if V were anti-symmetric, then one would have φ = and in turn B =. The main technical difficulty here is due to the need for a V without symmetries, which is required to handle 2d incomressible avier- Stokes.

21 Quantitative estimates of roagation of chaos 2 Final ste: Conclusion of the roof. Inserting 23) and 25) in 8), we deduce that d dt H ρ ρ ) 2 H ρ ρ ) + C ρ, allowing to conclude through Gronwall s lemma. There are several additional difficulties in the general roof. The fact that V L is not small forces us to carefully rescale all our estimates. Similarly since ρ is only an entroy solution to the Liouville Eq. 6), we have to roceed more carefully in estimating the relative entroy. 2.2 Time evolution of the relative entroy The first ste in the roof is to estimate the time evolution of the relative entroy, Lemma 2 Assume that ρ is an entroy solution to Eq. 6) as er Def. 2. Assume that ρ W, [, T ] Π d ) solves Eq. 2) with inf ρ > and ρ = Π d. Then H ρ ρ )t) = ρ t, X ) log ρ t, X ) Π ρ d t, X ) dx H ρ ρ ) t 2 ρ Kx i x j ) K x ρx i )) xi log ρ dx ds i, j= t 2 ρ div Kx i x j ) div K x ρx i )) dx ds i, j= σ t x i log ρ 2 ρ + C t σ σ, i= ρ where we recall that ρ t, X ) = Π i= ρt, x i) and with C = t 2 Πd ρ log ρ +2 log ρ σ div K 2 2W Ẇ, +, σ div F L. σ Proof From the limiting equation 2), one can readily check that log ρ solves t log ρ + + i= i= F x i ) + Kx i x j )) xi log ρ = σ x i ρ ρ j= i= Kx i x j ) K x ρx i ) xi log ρ j= div F x i ) + div K x ρx i )). i= 26)

22 22 Pierre-Emmanuel Jabin, Zhenfu Wang Remark that log ρ W, [, T ] ) since ρ W, [, T ] Π d ) and ρ is bounded from below. Therefore log ρ can be used as a test function against ρ in Eq. 6). This imlies that ρ log ρ dx = ρ log ρ dx Π d t + t log ρ + F x i )+Kx i x j )) xi log ρ dx ds σ ρ t i= i,j= xi log ρ xi ρ dx ds. Using the equation 26) on log ρ, we obtain ρ log ρ dx = ρ log ρ dx Π d t + Kx i x j ) K x ρx i ) xi log ρ dx ds + i= t i= ρ t ρ j= div F x i ) + div K x ρx i )) dx ds i= σ ρ xi ρ ρ ) σ xi ρ x i ρ ρ dx ds. Using the entroy dissiation for ρ given by 8), we have that H ρ ρ )t) H ρ ρ )) + D 2 2 t i, j= t i, j= ρ Kx i x j ) K x ρx i )) xi log ρ dx ds ρ div Kx i x j ) div K x ρx i )) dx ds, 27) with D = i= t σ ρ xi ρ ρ + σ xi ρ x i ρ xi ρ 2 ) σ. ρ ρ

23 Quantitative estimates of roagation of chaos 23 By integration by arts xi ρ ρ + xi ρ x i ρ x i ρ 2 ) Π ρ d ρ ρ xi ρ = ρ 2 Π ρ 2 2 xi ρ x i ρ + x i ρ 2 ) d ρ ρ = x i log ρ 2 ρ. ρ 28) On the other hand, Of course xi ρ σ σ ) ρ i= Π ρ d = σ σ ) xi ρ x i ρ xi ρ 2 ) + ρ Π ρ d ρ 2. i= Πd xi ρ 2 ρ ρ 2 i= while by Cauchy-Schwartz = Πd xi ρx i ) 2 ρ i= ρx i ) 2 log ρ 2 W,, i= t xi ρ x i ρ Π ρ d t log ρ 2 W, + 2 σ + t 2 div F L, σ t log ρ 2 W, + Πd ρ log ρ + by Pro. based on the entroy dissiation. This leads to i= t t div K 2 Ẇ, σ 2 Πd xi ρ 2 ρ σ σ ) i= t xi ρ ρ Π ρ d [ σ σ t 2 log ρ 2 W + div K 2 Ẇ,, σ ) ρ log ρ. σ + 2 div F L σ ] 29)

24 24 Pierre-Emmanuel Jabin, Zhenfu Wang Finally combining 29) with 28) D σ + t [ i= t ρ x i log ρ ρ 2 log ρ 2 W, + div K 2 Ẇ, σ 2 which inserted in 27) concludes the roof. 2 + σ σ + 2 ρ log ρ σ ] ) 2 div F L, σ 2.3 Bounding the interaction terms: The bounded divergence term We now have to obtain the main estimates, starting with the case where the kernel belongs to Ẇ, Π d ) and has bounded divergence. Lemma 3 Assume that ρ W 2, Π d ) for any <, then for any kernel L Ẇ, Π d ) with div L L, one has that 2 2 i, j= ρ Lx i x j ) L x ρx i )) xi log ρ dx ρ div Lx i x j ) div L x ρx i )) dx i, j= σ 4 i= where C is a universal constant and ρ xi log ρ ρ 2 dx + C M L H ρ ρ ) + ), M L = d 3 ρ 2 W, L 2 Ẇ, σ inf ρ) 2 + L Ẇ, inf ρ 2 ρ L su + div L L. Proof Remark that in this estimate, time is now only a fixed arameter and will hence not be secified in this roof. Denote V L Π d ) s.t. L = div V. By the definition of Ẇ, we assume that V L 2 L Ẇ,. By integration by arts 2 2 i, j= ρ Lx i x j ) L x ρx i )) xi log ρ dx ρ div Lx i x j ) div L x ρx i )) dx = A + B, i, j=

25 Quantitative estimates of roagation of chaos 25 with A = 2 B = 2 i, j= ρ V x i x j ) V x ρx i )) : xi ρ xi dx, Π ρ d [ 2 x ρ V x i x j ) V x ρx i )) : i ρ Π ρ d ] i, j= div Lx i x j ) + div L x ρx i ) dx. We treat indeendently A and B. The bound on A. First by Cauchy-Schwartz and by using a b a 2 /4 + b 2 A σ 4 + d σ i= i= ρ 2 ρ xi 2 dx ρ ρ 2 xi ρ V x i x j ) V x ρx i )) ρ j= 2 ρ dx. Remark that xi ρ ρ 2 = xi log ρx i ) 2 ρ 2 W, inf ρ) 2. Hence one has that A σ 4 i= + d ρ 2 W, σ inf ρ) 2 ρ xi log ρ ρ 2 dx d i= α,β= V α,β x i x j ) V α,β x ρx i )) j= 2 ρ dx, 3) where V α,β is the corresonding coordinate of the matrix field V. For some η > to be chosen later, we aly Lemma with Φ = 2 η V α,β x i x j ) V α,β x ρx i )), j=

26 26 Pierre-Emmanuel Jabin, Zhenfu Wang to find d i= α,β= V α,β x i x j ) V α,β x ρx i )) j= 2 ρ dx d2 η 2 H ρ ρ ) + 2 η 2 By symmetry i= α,β= i= log = log log ρ e j η V α,βx i x j) V α,β x ρx i))) 2 dx. ρ e ρ e j η V α,βx i x j) V α,β x ρx i))) 2 dx j ηv α,βx x j) V α,β x ρx ))) 2 dx. 3) Define ψz, x) = η V α,β z x) η V α,β ρz). Choose η = /4 e V L ) and note that ψ L 4 e and that for a fixed z, ρx) ψz, x) dx =. Since 2 η V α,β x x j ) V α,β x ρx )) = j= j,j 2= ψx, x j ) ψx, x j2 ), we may aly Theorem 3 to obtain that ρ e j ηv α,βx x j) V α,β x ρx ))) 2 dx C, for some exlicit universal constant C. Combining 3)-3) with this bound yields the final estimate on A A σ 4 i= + C d 3 ρ 2 W, V 2 L σ inf ρ) 2 again for some universal constant C. ρ xi log ρ ρ 2 dx H ρ ρ ) + ), 32) The bound on B. Define φx, z) = V x z) V x ρx)) : 2 x ρx) div Lx z)+ div L x ρx), 33) ρx)

27 Quantitative estimates of roagation of chaos 27 so that B = 2 i,j= ρ [ V x i x j ) V x ρx i )) : div Lx i x j ) + div L x ρx i ) ] dx 2 x i ρ ρ = 2 i,j= Aly Lemma with so that ρ φx i, x j ) dx. Φ = 2 η φx i, x j ), i,j= B η H ρ ρ ) + ρ e i,j η φxi,xj) dx. 34) η Observe that φx, z) ρz) dz =. While by integration by arts Π d 2 x ρx) V x z) V x ρx)) : ρx) dx Π ρx) d = div Lx z) div L x ρx)) ρx) dx, Π d imlying that Π d φx, z) ρx) dx =. ote as well from 33) that Hence choosing su φ., z) L ρ dx) 2 V L z inf ρ 2 ρ L + 2 div L L. η = C V L inf ρ su 2 ρ L + div L L we may aly Theorem 4 to bound ρ e i,j η φxi,xj) dx C, for some universal constant C. Hence from 34), we conclude that V L 2 ) ρ L B C su + div L L H ρ ρ ) + ). 35) inf ρ ), To finish the roof of the lemma, we simly have to add 32) and 35), recalling that V L 2 L Ẇ,.

28 28 Pierre-Emmanuel Jabin, Zhenfu Wang 2.4 Bounding the interaction terms: The divergence term only in Ẇ, Lemma 4 Assume that ρ W, Π d ) for any <, then for any kernel L L Π d ) with div L Ẇ,, one has that 2 2 ρ Lx i x j ) L x ρx i )) xi log ρ dx i, j= ρ div Lx i x j ) div L x ρx i )) dx i, j= σ 4 i ρ xi log ρ ρ 2 dx + C M 2 L where C is a universal constant and H ρ ρ ) + ), M 2 L = L L + div L Ẇ, ) ρ L inf ρ + d σ div L 2 Ẇ,. Proof The roof follows similar ideas to the roof of Lemma 3 but now we have to integrate by arts the term with div L instead of the term with L. Denote L L s.t. div L = div L and div L Ẇ, = L L. Write 2 Hence 2 2 i, j= = i, j= ρ div Lx i x j ) div L x ρx i )) dx i, j= i, j= xi ρ ρ ρ Lxi x j ) L x ρx i )) ρ dx Lxi x j ) L x ρx i )) xi log ρ dx. ρ Lx i x j ) L x ρx i )) xi log ρ dx ρ div Lx i x j ) div L x ρx i )) dx = A + B, i, j= 36) with A = 2 i, j= xi ρ ρ Lxi x j ) L x ρx i )) ρ dx,

29 Quantitative estimates of roagation of chaos 29 and B = 2 i, j= ρ Lxi x j ) L x ρx i ) ) xi log ρ dx, for L = L L. Bound for A. We start with Cauchy-Schwartz to bound A σ 4 i= + σ ρ xi 2 ρ2 Π ρ d ρ d ρ i= α= L α x i x j ) L α x ρx i ) j= 2 dx, where L α is the α coordinate of L. Denote ψz, x) = η L α z x) L α ρz)), and use Lemma for Φ = j= ψx i, x j ) 2 to obtain i= ρ L α x i x j ) L α x ρx i ) j= η 2 H ρ ρ ) + 2 η 2 i= Of course Π d ψz, x) ρx) dx = so that taking 2 dx log ρ e j ψxi,xj) 2 dx. and alying Theorem 3, we find A σ 4 η = =, 4 e L L 4 e div L Ẇ, i= ρ xi log ρ ρ 2 dx + C d div L 2 Ẇ, σ H ρ ρ ) + ). 37) Bound for B. We follow the same stes as before, define φx, z) = Lx z) L x ρx) ) x log ρx), and first aly Lemma with Φ = η 2 i,j= φx i, x j ) to find B η H ρ ρ ) + η log ρ e i,j φxi,xj) dx.

30 3 Pierre-Emmanuel Jabin, Zhenfu Wang Since div L = div L div L =, we have that φx, z) ρz) dz = φx, z) ρx) dx =. Π d Π d Choose η = C L L su log ρ L ρ dx) and aly now Theorem 4 to conclude that B C L L + div L Ẇ, ) ρ L inf ρ Combining 37) and 38) concludes the roof. inf ρ =, C L L + div L Ẇ, ) ρ L H ρ ρ ) + ). 38) 2.5 Conclusion of the roof of Theorem The roof of Theorem follows from the revious estimates through a careful decomosition of the kernel K. By the assumtion of Theorem, we have that K α = β V αβ where V L Π d ) is a matrix field, and that there exists K L s.t. div K = div K and K L Π d ) 2 div K Ẇ,. For convenience, we use the notation K = K L Π d ) + V L Π d ) 2 div K Ẇ, + 2 K Ẇ,. Define K = div V K. ote that div K = and obviously since K L and we can choose K s.t. K =, then K Ẇ, with K Ẇ, C d K. We combine Lemma 2 with Lemma 3 for L = K, and finally with Lemma 4 for L = K. We obtain H ρ ρ )t) H ρ ρ ) + C t σ σ t + C M K + M 2 K ) H ρ ρ )s) + ) ds. Because of our secific bounds M K d 3 K 2 ρ 2 W, Ẇ, σ inf ρ) 2 + K Ẇ, 2 ρ L su, inf ρ M 2 K K ) ρ L L + div K Ẇ, + d inf ρ σ div K 2. Ẇ, 39) To kee calculations simle, we do not try here to obtain fully exlicit bounds which would still be ossible) and simlify 39) in H ρ ρ )t) H ρ ρ ) + M + t + K 2 )) σ σ t + M K + K 2 ) H ρ ρ )s) + ), 4)

31 Quantitative estimates of roagation of chaos 3 where we only ket exlicit a simlified deendence on K and where the constant M deends only on M d, σ, inf ρ, ρ W,, su 2 ρ L, By Gronwall lemma, 4) imlies that H ρ ρ )t) e M K + K 2 ) t ) ρ log ρ, div F L. H ρ ρ ) + + M + t + K 2 )) σ σ ), which concludes the roof of Theorem. 2.6 Proof of Theorem 2 The roof of our result for vanishing viscosity is in fact now straightforward as it uses our revious analysis. First of all, we have an direct equivalent of Lemma 2 H ρ ρ )t) = ρ t, X ) log ρ t, X ) Π ρ d t, X ) dx H ρ ρ ) t 2 ρ Kx i x j ) K x ρx i )) xi log ρ dx ds i, j= t 2 ρ div Kx i x j ) div K x ρx i )) dx ds + α i, j= where when σ σ =, 4) α = σ 4 i= t ρ xi log ρx i ) 2 dx ds t σ σ log ρ 2 W,, while when σ σ > as, we can take σ = σ/2 as in Lemma 2 which gives α = C 2 t σ σ with C 2 given by C 2 = 2 ρ log ρ + 2 σ t Π σ div K L + 2 σ div F L + 2 log ρ 2 W,. d

32 32 Pierre-Emmanuel Jabin, Zhenfu Wang There is no need for any integration by art on the other terms in 4). Denote for some η > η φx, z) = Kx z) K x ρx)) x log ρx)+ div Kx z) div K x ρx). We directly aly Lemma to and find 2 2 t i, j= t i, j= Φ = 2 φx i, x j ), i,j= ρ Kx i x j ) K x ρx i )) xi log ρ dx ds ρ div Kx i x j ) div K x ρx i )) dx ds ρ ex φx i, x j ) dx. η H ρ ρ ) + η log i,j= We use Theorem 4 and observe for this φ satisfies the required assumtions and in articular ) 2 su z φ., z) L γ = C su ρ dx) ) 2 <, rovided that C η 2 K 2 η < and where we recall that su log ρ L ρ dx) C K su log ρ L ρ dx) K = K L + div K L. By Theorem 4, we hence have for some universal constant C > 2 2 t i, j= t i, j= M 2 K t ρ Kx i x j ) K x ρx i )) xi log ρ dx ds ρ div Kx i x j ) div K x ρx i )) dx ds H ρ ρ )s) + ) ds.,

33 Quantitative estimates of roagation of chaos 33 Inserting this in 4), we find that H ρ ρ ) H ρ ρ ) + M 2 K t + M 2 + t K ) σ σ for some constant M 2 deending only on M 2 σ, log ρ W,, su H ρ ρ )s) + ) ds log ρ L ρ dx), ρ log ρ, div F L This concludes the roof by Gronwall lemma. ). 3 Preliminary of combinatorics Before the roof of the main estimates Theorem 3 and Theorem 4, we list some useful combinatorics results used throughout this article. We first recall Stirling s formula n! = λ n 2πn n e ) n, 42) where < λ n < and λ n as n. We have the elementary bound following from 42) Lemma 5 For any q, one has ) q e q. One also has the basic combinatorics on -tules Lemma 6 For any q, one has {b,..., b ) l b l q and b + b b = q} = ) q. Proof Proof of Lemma 6) When =, the lemma trivially holds true with the convention ) = if = q =. We thus assume 2 in the following. Since each tule b, b 2,, b ) uniquely determines a ) tule c, c 2,, c ) and recirocally via it suffices to verify that c = b, c 2 = b + b 2,, c = b + b b, {c, c 2,, c ) c < c 2 < < c q } = ) q. This is simly obtained by choosing distinct integers from the set {, 2,, q } and assigning the smallest one to c, the second smallest to c 2, and so on.

34 34 Pierre-Emmanuel Jabin, Zhenfu Wang Much of the combinatorics that we handle is based only on the multilicity in the multi-indices. It is therefore convenient to know how many multi-indices can have the same multilicity signature Lemma 7 For any a,..., a q s.t. a + + a q =, then the set of multiindices I = i,..., i ) with i k q and corresonding multilicities has cardinal { i,..., i ) {,..., q} l a l = {k, i k = l} }! = a! a q!. Proof This is the basic multinomial relation: We have to choose a times among ositions, 2 a 2 times among the remaining ositions and so on...! a! a q! is the coefficient of xa... x aq q Similarly as for the binomial coefficients, in the exansion of x + + x q ) leading to the obvious estimate a,...,a q, a + +a q=! a! a q! = q. 43) Let us fix some notations here. We write the integer valued tule as I = i,, i ). The overall set T q, of those indices is defined as T q, = {I = i,, i ) i ν q, for all ν }. 44) We thus define the multilicity function Φ q, : T q, {,,, } q with Φ q, I ) = A q = a, a 2,, a q ), where a l = { ν i ν = l}. In many of our roofs, we use cancellations so that any I which has an index of multilicity exactly leads to a vanishing term. This leads to the definition of the effective set E q, by E q, = {I T q, Φ q, I ) = A q = a,, a q ) One has the following combinatorics result Lemma 8 Assume that q. Then l= with a ν for any ν q.} 2 ) q E q, l ) q l 2 2 ) 2 2 e 2 2 q 2. 45) 2)

35 Quantitative estimates of roagation of chaos 35 Proof Proof of Lemma 8) Pick any multi-index I = i,, i ) E q, and recall that I = {i,, i }. The fact that I E q, imlies that the multilicity of each integer cannot be one. Hence I 2. Indeed, if I 2 +, then 2 ) ) 2 + > =, which is imossible. If =, then E q, =. The estimate 45) holds trivially. In the following we assume that 2. Denote l = I which can be, 2,, 2. Consequently, one has by summing all ossible choices for I 2 E q, = {I E q, I = l}. l= For a fixed I = l, there are q l) many choices of numbers l from S = {, 2,, q} to comose I. Having already chosen those l numbers from S, without loss of generality we may assume that I as a set coincides with {, 2,, l}. The total choices of tule I can be bounded by l trivially since each i ν has at most l choices. Remark that l 2 q 2, so that ) ) q q l 2. Hence one has 2 ) q E q, l ) q l 2 2 ) 2. l= The last inequality in 45) is now ensured by Lemma 5, in articular the following inequality ) q 2 e 2 q 2 2 ) 2. This finishes the roof of Lemma 8. 4 Proof of Theorem 3 The goal here is to bound ρ ex ψx, x j ) ψx, x j2 ) dx, j,j 2=

36 36 Pierre-Emmanuel Jabin, Zhenfu Wang for any bounded ψ with vanishing average against ρ. Since exa) = k! Ak 3 2k)! A2k, k= k= it suffices only to bound the series with even terms ρ ex ψx, x j ) ψx, x j2 ) dx 3 k= j,j 2= ρ 2k)! j,j 2= ψx, x j ) ψx, x j2 ) where in general the k th even term can be exanded as 2k ρ ψx, x j ) ψx, x j2 ) dx 2k)! Π d = 2k)! j,j 2= 2k j,,j 4k = 2k dx, ρ ψx, x j ) ψx, x j4k ) dx. 46) 47) We divide the roof in two different cases: Where k is small comared to and in the simler case where k is comarable to or larger than. Case: 4 4k First observe that for any articular choice of indices j,..., j 4K, one has ρ ψx, x j ) ψx, x j4k ) dx ψ 4k L. 48) The whole estimate hence relies on counting how many choices of multi-indices j,..., j 4k ) lead to a non-vanishing term. Denote hence,4k the set of multiindices j,, j 4k ) s.t. ρ ψx, x j ) ψx, x j4k ) dx. Denote by a,..., a ) the multilicity for j,..., j 4k ), a l = {ν {,..., 4k}, j ν = l}. If there exists l s.t. a l =, then the variable x l enters exactly once in the integration. Assume for simlicity that j = l then ρ ψx, x j ) ψx, x j4k ) dx = ψx, x j2 ) ψx, x j4k ) Π i l ρx i ) dx i Π d ) ρx j ) ψx, x j ) dx j =, Π d