Rigorous mean-field limit and cross-diffusion

The mean-field limit in a weakly interacting stochastic many-particle system for multiple population species in the whole space is proved. The limiting system consists of cross-diffusion equations, modeling the segregation of populations. The mean-field limit is performed in two steps: First, the many-particle system leads in the large population limit to an intermediate nonlocal diffusion system. The local cross-diffusion system is then obtained from the nonlocal system when the interaction potentials approach the Dirac delta distribution.The global existence of the limiting and the intermediate diffusion systems is shown for small initial data, and an error estimate is given.


Introduction
Cross-diffusion models are systems of quasilinear parabolic equations with a nondiagonal diffusion matrix. They arise in many applications in cell biology, multicomponent gas dynamics, population dynamics, etc. [21]. To understand the range of validity of these diffusion systems, it is important to derive them from first principles or from more general models. In the literature, cross-diffusion systems were derived from random walks on lattice models [40], the kinetic Boltzmann equation [5], reaction-diffusion systems [9,19], or from stochastic many-particle systems [36]. We derive in this paper rigorously the n-species cross-diffusion system where σ i > 0 and a ij are real numbers, starting from a stochastic many-particle system for multiple species. System (1) describes the diffusive dynamics of populations subject to segregation effects modeled by the term on the right-hand side [14].
X k,N η,i (0) = ξ k i , i = 1, . . . , n, k = 1, . . . , N, (2) where (W k i (t)) t≥0 are d-dimensional Brownian motions, σ i are the constant diffusion coefficients, the initial data ξ 1 i , . . . , ξ N i are independent and identically distributed random variables with the common probability density function u 0 i , and the interaction potential V η ij is given by Here, V ij is a given smooth function and η > 0 the scaling parameter. The scaling is chosen in such a way that the L 1 norm of V η ij stays invariant and V η ij → a ij δ in the sense of distributions as η → 0, where δ denotes the Dirac delta distribution.
The mean-field limit N → ∞, η → 0 has to be understood in the following sense. For fixed η > 0, the many-particle model (2) is approximated for N → ∞ by the intermediate stochastic system where u η,j is the probability density function ofX k η,j , satisfying the nonlocal diffusion system Observe that we consider N independent copiesX k η,i (t), k = 1, . . . , N, of the solution to (3). In particular, the intermediate system depends on N only through the initial datum. Then, passing to the limit η → 0 in the intermediate system, the limit ∇V η ij * u η,j → a ij ∇u j in L 2 leads to the limiting stochastic system X k i (0) = ξ k i , i = 1, . . . , n, k = 1, . . . , N, (5) and the law of X k i is a solution to the limiting cross-diffusion system (1). We remark that also here, we consider N independent copies X k i (t), k = 1, . . . , N, of the solution to (5). The main result of this paper is the proof of the error estimate sup k=1,...,N if we choose η and N such that η −(2d+4) ≤ ε log N holds, where ε > 0.

State of the art
Particle limits from systems of stochastic differential equations were investigated intensively in the last decades to derive, for instance, reaction-diffusion equations [12] or McKean-Vlasov equations [10,15]. For details, we refer to the reviews [16,20] and the classical works of Sznitman [37,38]. Oelschläger [27] considered in the 1980s a weakly interacting particle system of N particles and proved that in the limit N → ∞, the stochastic system converges to a deterministic nonlinear process. Later, he generalized his approach to systems of reaction-diffusion equations [28]. The analysis of quasilinear diffusion systems started more recently. The chemotaxis system was derived by Stevens [36] from a stochastic many-particle system with a limiting procedure that is based on Oelschläger's work. Reaction-diffusion systems with nonlocal terms were derived in [22] as the mean-field limit of a master equation for a vanishing reaction radius; also see [18]. The two-species Maxwell-Stefan equations were found to be the hydrodynamic limit system of two-component Brownian motions with singular interactions [34]. Nonlocal Lotka-Volterra systems with cross-diffusion were obtained in the large population limit of point measure-valued Markov processes by Fontbona and Méléard [13]. Moussa [26] then proved the limit from the nonlocal to the local diffusion system but only for triangular diffusion matrices. A derivation of a space discretized version of this system from a Markov chain model was presented in [11]. Another nonlocal mean-field model was analyzed in [7].
Related cross-diffusion systems have been derived also by using different techniques, e.g., the relative entropy method. It was employed in hydrodynamic limits [39] to derive the Stokes and incompressible Euler equations [33]. This technique was recently applied to multicomponent Euler flows to derive crossdiffusion systems of Maxwell-Stefan type [17].
Our system (1) is different from the aforementioned Shigesada-Kawasaki-Teramoto system (neglecting for a moment the simple diffusion terms) [13,26]. Our derivation produces the first term on the right-hand side. The reason for the difference is that in [13], the diffusion coefficient σ i in (2) is assumed to depend on the convolutions W ij * u j for some functions W ij -yielding the last term in the previous equation-while we have assumed a constant diffusion coefficient. It is still an open problem to derive the general Shigesada-Kawasaki-Teramoto system; the approach of Moussa [26] requires that a ij = 0 for j < i.
System (1) was also investigated in the literature. A formal derivation from the intermediate diffusion system (4) was performed by Galiano and Selgas [14], while probabilistic representations of (1) were presented in [2]. A rigorous derivation from the stochastic many-particle system (2) is still missing in the literature. In this paper, we fill this gap by extending the technique of [8] to diffusion systems. Compared to [8], the argument to derive the uniform estimates is more involved and involves a nonlinear Gronwall argument (see Lemma 17 in Appendix A).
It is well known that stochastic many-particle systems may lead in the large population limit to the porous-medium equation with quadratic diffusion [29]. Therefore, the quadratic structure with the porousmedium term a ii u i ∇u i in system (1) is not surprising. In [24], the quadratic porous-medium equation was approximated by the nonlocal equation ∂ t u = div(uW η ), where W η = ∇u * V η and V η approximates the Dirac delta distribution. In this context, the nonlocal system (4) is just a system version of this approximation.
The global existence of solutions to (1) for general initial data and coefficients a ij ≥ 0 is an open problem. The reason is that we do not know any entropy structure of (1). For the two-species system, Galiano and Selgas [14] proved the global existence of weak solutions in a bounded domain with noflux boundary conditions under the condition 4a 11 a 22 > (a 12 + a 21 ) 2 . The idea of the proof is to show that H(u) = u i (log u i − 1)dx is a Lyapunov functional (entropy). The condition can be weakened to a 11 a 22 > a 12 a 21 using the modified entropy H 1 (u) = (a 21 u 1 (log u 1 − 1) + a 12 u 2 (log u 2 − 1))dx. If a 11 a 22 > a 12 a 21 and u 1 u 2 > 0, system (1) is parabolic in the sense of Petrovskii which gives the localin-time existence of classical solutions [1]. The existence of the so-called segregated solutions to (1) with a 11 = a 12 = 1 and a 21 = a 22 = k > 0 was shown in [3,4], writing the equations as a parabolic-hyperbolic system. We use the following notation throughout the paper. We write · L p and · H s for the norms of L p = L p (R d ) and H s = H s (R d ), respectively. Furthermore, |u| 2 = n i=1 u 2 i for u = (u 1 , . . . , u n ) ∈ R n and u 2 L p = n i=1 u i 2 L p for functions u = (u 1 , . . . , u n ). We use the notation u(t) = u(·, t) for functions depending on x and t, and C > 0 is a generic constant whose value may change from line to line.

Main results
The first two results are concerned with the solvability of the nonlocal diffusion system (4) and the limiting cross-diffusion system (1). The existence results are needed for our main result, Theorem 3 below.
We impose the following assumptions on the interaction potential.
As the potential may be negative (and a ij may be negative too), we introduce where σ = min i=1,...,n σ i > 0 and C * > 1 is a constant only depending on s and d. Then there exists a unique global solution u η = (u η,1 , . . . , u η,n ) to problem (4) We may interpret (6) also as a smallness condition for the nonlinear diffusion: For the diffusion system at hand, cross-diffusion should help to create segregation effects. Therefore, the smallness condition may prevent nonconstant steady states. Since we do not use the structure of the equations, we can only expect the global existence of solutions for sufficiently small initial data. The proof of this result is based on the Banach fixed-point theorem and a priori estimates and is rather standard. We present it for completeness.

Proposition 2. (Existence for the limiting cross-diffusion system)
Moreover, let u η be the solution to problem (4). Then the following error estimate holds for any T > 0: for some constant C(T ) > 0. The proposition is proved by performing the limit η → 0 in (4) which is possible in view of the uniform estimate (7). The error estimate (9) follows from the uniform bounds and the smallness condition (6).
For our main result, we need to make precise the stochastic setting. Let (Ω, F, (F t ) t≥0 , P) be a filtered probability space and let (W k i (t)) t≥0 for i = 1, . . . , n, k = 1, . . . , N be d-dimensional F t -Brownian motions that are independent of the random variables ξ k i . We assume that the Brownian motions are independent and that the initial data ξ 1 i , . . . , ξ N i are independent and identically distributed random variables with the common probability density function u 0 i . We prove in Sect. 4 that if s > d/2 + 2 and the initial density u 0 satisfies the smallness condition (6), the stochastic differential systems (2), (3), and (5) have pathwise unique strong solutions; also see Remark 11. (6) hold. Let X k,N η,i and X k i be solutions to problems (2) and (5), respectively. Furthermore, let 0 < ε < 1 be sufficiently small and choose

Theorem 3. (Error estimate for the stochastic system)
where the constant C(t) depends on t, n, D 2 V ij L ∞ , and the initial datum u 0 .
The idea of the proof is to derive error estimates for the differences The expectations on the right-hand side are estimated by taking the difference of the solutions to the corresponding stochastic differential equations, exploiting the Lipschitz continuity of ∇V η ij , and observing that V η ij * ∇u j − a ij ∇u j L 2 (0,T ;L 2 ) ≤ Cη. Usually, population models are accompanied by birth-death terms, e.g., Lotka-Volterra terms. In the context of the Keller-Segel model, source terms have been derived in [36] via suitable Poisson processes. We expect that a similar approach can be used to derive Lotka-Volterra terms also in (1).
The paper is organized as follows. Sections 2 and 3 are concerned with the proof of Propositions 1 and 2, respectively. The existence of solutions to the stochastic systems is shown in Sect. 4. The main result (Theorem 3) is then proved in Sect. 5. Finally, the appendix recalls some auxiliary results needed in our analysis.

Existence for the nonlocal diffusion system (4)
We show Proposition 1 whose proof is split into several lemmas.
Proof. The idea is to apply the Banach fixed-point theorem. For this, we introduce where T * > 0 will be determined later. It is quite standard to check that the space Y is complete with this metric. The fixed-point operator where u is the unique solution to the Cauchy problem By standard parabolic theory, this linear advection-diffusion problem has a weak solution We prove that sup 0<t<T * u(·, t) 2 H s ≤ M for sufficiently small values of T * > 0. Then u ∈ Y and S : Y → Y is well defined. We apply the differential operator D α for an arbitrary multi-index α ∈ N d of order |α| ≤ s to (10), multiply the resulting equation by D α u i , and integrate over R d : We sum these equations from i = 1, . . . , n, apply the Cauchy-Schwarz inequality to the integral on the right-hand side, and the Moser-type calculus inequality (Lemma 15): where we recall that σ = min i=1,...,n σ i > 0 and ε is any positive number. The last term on the right-hand side can be absorbed by the second term on the left-hand side if ε ≤ σ/(2n). Hence, summing over all multi-indices α of order |α| ≤ s, using Young's convolution inequality (Lemma 16), and the inequality Gronwall's inequality then yields Note that the time T * depends on M and hence on u 0 in such a way that T * becomes smaller if u 0 H s is large but T * > 0 is bounded from below if u 0 H s is small. It remains to show that the map S : Y → Y is a contraction, possibly for a smaller value of T * > 0.
Let v, w ∈ Y and take the difference of the equations satisfied by Sv and Sw, respectively: Multiplying these equations by (Sv) i − (Sw) i , summing from i = 1, . . . , n, integrating over R d , and using the Cauchy-Schwarz inequality leads to 1 2 We deduce from Young's convolution inequality that 1 2 By definition of the metric on Y , we have shown that The constants C 1 and C 2 depend on M (and hence on u 0 ) in such a way that they become larger if u 0 H s is large but they stay bounded for small values of u 0 H s . Thus, because of v(0) = w(0), Gronwall's inequality gives and the definition of the metric leads to Then, choosing T * > 0 such that C 2 (σ, η, M )T * e C1(σ,η,M )T * ≤ 1/2 shows that S : Y → Y is a contraction. Again, T * depends on u 0 but it is bounded from below for small values of u 0 H s . Thus, we can apply the Banach fixed-point theorem, finishing the proof.

Lemma 5. (A priori estimates)
Let assumption (6) hold. For the local solution u η to problem (4), the uniform estimate (7) holds. In particular, the solution u η can be extended to a global one.
Proof. We proceed similarly as in the proof of Lemma 4. We choose α of order |α| ≤ s, apply the operator D α on both sides of (4), multiply the resulting equation by D α u η,i , and integrate over R d . By the Cauchy-Schwarz inequality, the Moser-type calculus inequality, and Young's convolution inequality and writing u i instead of u η,i , we obtain where C M is the constant from Lemma 15, C * > 1 depends on C M and the constant of the embedding H s (R d ) → L ∞ (R d ), and we have used V η ij L 1 = A ij . Summation of all |α| ≤ s leads to 1 2 which can be written as This inequality holds for all t ∈ [0, T ], where T < T * . By Lemma 17, applied to f (t) = u(t) 2 H s , g(t) = ∇u(t) H s , a = σ, and b = C * n i,j=1 A ij , we find that u(t) 2 H s ≤ (a/b) 2 for t ∈ [0, T ]. Here, we use Assumption (6). We deduce that (d/dt) u 2 H s ≤ 0 and consequently u(t) H s ≤ u 0 H s for t ∈ [0, T ]. Now, we take u(T ) as the initial datum for problem (4). We deduce from Lemma 4 the existence of a solution u to (4) defined on [T, T + T * ). Here, T * > 0 can be chosen as the same end time as before since the norm of the initial datum u(T ) H s is not larger as u 0 H s . Note that T * becomes smaller only when the initial datum is larger in the H s norm. Hence, u(t) exists for t ∈ [T, 2T ] and inequality (11) holds. As before, we conclude from Lemma 17 that u(t) H s ≤ u 0 H s for t ∈ [T, 2T ]. This argument can be continued, obtaining a global solution satisfying u(t) H s ≤ u 0 H s for all t > 0. Then, under the stronger assumption (6), which leads to (7), finishing the proof.
Proof. Let u and v be two solutions to (4) with the same initial data. We multiply the difference of the equations satisfied by u i and v i by u i − v i , sum from i = 1, . . . , n, and integrate over R d . Then, for all 0 < t < T and some T > 0, By assumption, Thus, using the Cauchy-Schwarz inequality, it follows that 1 2 if we choose ε ≤ γ/n 2 . Observe that the norm u L 2 (0,∞;H s+1 ) is bounded. This allows us to apply the Gronwall inequality, and together with the fact that (u − v)(0) L 2 = 0, we infer that (u − v)(t) L 2 = 0, concluding the proof.

Existence for the cross-diffusion system (1)
We prove Proposition 2 whose proof is split into two lemmas.

Lemma 7. (Existence and uniqueness of solutions) Let the assumptions of Proposition 2 hold. Then there exists a unique solution to (1) satisfying (8).
Proof. Let u η be the solution to (4). We prove that a subsequence of (u η ) converges to a solution to problem (1). In view of the uniform estimate (7), there exists a subsequence of (u η ), which is not relabeled, such that, as η → 0, u η u weakly in L 2 (0, T ; H s+1 (R d )).
We show that u is a weak solution to problem (1). First, we claim that To prove this statement, we observe that V η ij * ψ → a ij ψ strongly in L 2 (0, T ; L 2 (R d )) for any φ ∈ L 2 (0, T ; L 2 (R d )) [32, Theorem 9.10] and ∇u η,j ∇u j weakly in L 2 (0, T ; L 2 (R d )). Therefore, for all which proves the claim. Estimate (7) and the embedding and consequently, for any T > 0, The weak formulation of (4) reads as where . Since the ball B R (0) is bounded and the embedding H 1 (B R (0)) → L 2 (B R (0)) is compact, the Aubin-Lions lemma [35] gives the existence of a subsequence of (u η ), which is not relabeled, such that u η → u strongly in L 2 (0, T ; L 2 (B R (0))) as η → 0, and the limit coincides with the weak limit in (12). We deduce that Estimate (13) shows that this convergence even holds in L 2 (0, T ; L 2 (B R (0))). We can perform the limit in (14), which shows that the limit u is a solution to the cross-diffusion problem (1). The uniform estimates (8) follow from (7) using the lower semicontinuity of the norm. Next, we show the uniqueness of solutions. Let u and v be two solutions to (1) with the same initial data. Taking the difference of the equations satisfied by u i and v i , multiplying the resulting equation by ZAMP Rigorous mean-field limit and cross-diffusion Page 11 of 21 122 u i − v i , summing from i = 1, . . . , n, integrating over R d , and using the Cauchy-Schwarz inequality leads to In view of estimate (8), this becomes and the constant C(t) > 0 is integrable (as it depends on u(t) H s+1 ). Gronwall's inequality then implies that (u − v)(t) = 0 for t > 0.

Lemma 8. (Error estimate)
Let the assumptions of Proposition 2 hold. Let u be the solution to (1) and u η be the solution to (4). Then the error estimate (9) holds.
Proof. We take the difference of Eqs. (4) and (1), Multiplying this equation by u η,i − u i , summing from i = 1, . . . , n, integrating over R d , using the Cauchy-Schwarz inequality, and the estimate |a ij | ≤ A ij , we find that We estimate the right-hand side term by term. First, by the continuous embedding H s (R d ) → L ∞ (R d ), To estimate I 2 , let g ∈ L 2 (R d ; R n ). Since supp V η ij ⊂ B η (0), the fundamental theorem of calculus shows that and consequently, Finally, by Assumption (6), Therefore, , and Gronwall's lemma gives the conclusion.

Existence of solutions to the stochastic systems
Let the initial data ξ 1 , . . . , ξ n be square integrable. We prove the solvability of the stochastic ordinary differential systems (2), (3), and (5). ZAMP Rigorous mean-field limit and cross-diffusion Page 13 of 21 122 Lemma 9. (Solvability of the stochastic many-particle system) For any fixed η > 0, problem (2) has a unique strong solution X k,N η,i that is F t -adapted.  (2).

Remark 11.
We have shown in Proposition 2 that if u 0 ∈ H s (R d ; R n ) and the smallness condition (6) holds, there exists a unique solution u η ∈ L ∞ (0, ∞; H s (R d ; R n )). The regularity for u η , required in Lemma 10, is fulfilled if s > d/2 + 2, thanks to the embedding Proof of Lemma 10. We proceed as in the proof of Lemma 3.2 of [8].
By assumption, ∇V η ij * u η,j = V η ij * ∇u η,j is bounded and Lipschitz continuous. Therefore, has a unique strong solutionX k η,i . By [6], the stochastic processX k η,i is absolutely continuous with respect to the Lebesgue measure, thus there exists the probability density functionū η,i ofX k η,i , which is Lipschitz continuous. Let φ i be a smooth test function. Then Itô's lemma implies that Applying the expectation to the previous expression yields This is the weak formulation of the linear problem The solutionū η,i of this linear problem is Lipschitz continuous. Thus, by regularity theory [41,Theorem 1.3.11], it holds thatū η,i ∈ L ∞ (0, T ; H s (R d )) ∩ L 2 (0, T ; H s+1 (R d )). But since u η,i is the unique solution to problem (4) in that solution class, this implies thatū η,i = u η,i . This finishes the proof.
By the same technique, the solvability of the limiting stochastic system can be proved.

Lemma 12.
(Solvability of the limiting stochastic system) Let u be the unique solution to problem (1) satisfying |∇u| ∈ L ∞ (0, ∞; W 1,∞ (R d ; R n )). Then there exists a unique strong solution X k i with probability density function u i .

Proof of Theorem 3
First, we show an estimate for the difference X k,N η,i −X k η,i . Lemma 13. Let the assumptions of Theorem 3 hold. Then, for any t > 0, where the constant C(t) depends on t, n, D 2 V ij L ∞ , and the initial datum u 0 .
Proof. We set for k = 1, . . . , N, The difference of Eqs. (2) and (3), satisfied by X k,N η,i andX k η,i , respectively, equals Taking the supremum in (0, t) and the expectation and using the Cauchy-Schwarz inequality with respect to t yields sup k=1,...,N Rigorous mean-field limit and cross-diffusion Page 15 of 21 122 We estimate the terms J 1 , J 2 , and J 3 separately.
Furthermore, by similar arguments, For the third term, we set and write the square as a product of two sums Z k,1 i,j , . . . , Z k,N i,j : We show that the last term vanishes. Indeed, sinceX k η,i ,X η,j , andX m η,j are independent by construction (since we consider N independent copies) with distribution functions u η,i , u η,j , and u η,j , respectively, the joint distribution equals u η,i ⊗ u η,j ⊗ u η,j . This gives for = m and i = j or, when i = j, k = , k = m, If = m, i = j, and k = or k = m, it holds by a similar calculation that E(Z k, i,j (s)Z k,m i,j (s)) = 0, and thus, E(Z k, i,j (s)Z k,m i,j (s)) = 0 for all = m. Also the expectation of Z k, i,j vanishes since We have shown that for = 1, . . . , N, Z k, i,j are uncorrelated. Therefore, using the estimates |∇V η ij | ≤ we deduce that For fixed ε ∈ (0, 1) and η ∈ (0, 1), we choose N ∈ N such that ε log N ≥ η −2d−4 . Using η −2d−2 ≤ η −2d−4 ≤ exp(Cη −2d−4 ) for C ≥ 1, we obtain sup k=1,...,N This proves the result.
Next, we prove an estimate for the differenceX k η,i − X k i .

Lemma 14.
Let the assumptions of Theorem 3 hold. Then for t > 0, Proof. SinceX k η,i and X k η,i are N independent copies, we can omit the index k in the proof. We use similar arguments as in the proof of Lemma 13. Taking the difference of Eqs. (3) and (5), satisfied bȳ X η,i and X i , respectively, and setting (V η ij * ∇u η,j )( X i (s), s) − (V η ij * ∇u η,j )(X η,i (s), s) ds =: K 1 + K 2 + K 3 .
Using (15), the first two terms on the right-hand side are estimated according to a ij ∇u j − V η ij * ∇u j L 2 (0,t;L 2 ) u i L 2 (0,t;L 2 ) ≤ C(n)η D 2 u L 2 (0,t;L 2 ) ≤ Cη, where we have used Lemma 16 (ii) and the error estimate from Lemma 13. Finally, the term K 3 can be controlled by We need the assumption s > d/2 + 2 for the continuous embedding H s (R d ) → W 2,∞ (R d ), which allows us to estimate D 2 u η in L ∞ (R d ). This shows that E(U t ) ≤ Cη + C