Rigorous mean-field limit and cross-diffusion

Abstract

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.

1 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

$$\begin{aligned} \begin{aligned} \partial _t u_i - \sigma _i\Delta u_i&= {\text {div}}\bigg (\sum _{j=1}^n a_{ij}u_i\nabla u_j\bigg )\quad \text{ in } \mathbb {R}^d,\quad \ t>0, \\ u_i(0)&= u_i^0, \quad i=1,\ldots ,n, \end{aligned} \end{aligned}$$

(1)

where $\sigma _i>0$ 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].

1.1 Setting of the problem

We consider n subpopulations of interacting individuals moving in the whole space $\mathbb {R}^d$ with the particle numbers $N_i\in \mathbb {N}$, $i=1,\ldots ,n$. We take $N_i=N$ to simplify the notation. The individuals are represented by the stochastic processes $X_{\eta ,i}^{k,N}(t)$ evolving according to

$$\begin{aligned} \begin{aligned} \mathrm{d}X_{\eta ,i}^{k,N}(t)&= -\sum _{j=1}^n\frac{1}{N}\sum _{\ell =1}^N \nabla V_{ij}^\eta \big (X_{\eta ,i}^{k,N}(t)-X_{\eta ,j}^{\ell ,N}(t)\big )\mathrm{d}t + \sqrt{2\sigma _i}\mathrm{d}W_i^k(t), \\ X_{\eta ,i}^{k,N}(0)&= \xi _i^k, \quad i=1,\ldots ,n,\quad \ k=1,\ldots ,N, \end{aligned} \end{aligned}$$

(2)

where $(W_i^k(t))_{t\ge 0}$ are d-dimensional Brownian motions, $\sigma _i$ are the constant diffusion coefficients, the initial data $\xi _i^1,\ldots ,\xi _i^N$ are independent and identically distributed random variables with the common probability density function $u_i^0$, and the interaction potential $V_{ij}^\eta$ is given by

$$\begin{aligned} V_{ij}^{\eta }(x) = \frac{1}{\eta ^d}V_{ij}\bigg (\frac{|x|}{\eta }\bigg ), \quad x\in \mathbb {R}^d,\quad \ i,j=1,\ldots ,n. \end{aligned}$$

Here, $V_{ij}$ is a given smooth function and $\eta >0$ the scaling parameter. The scaling is chosen in such a way that the $L^1$ norm of $V_{ij}^\eta$ stays invariant and $V_{ij}^\eta \rightarrow a_{ij}\delta$ in the sense of distributions as $\eta \rightarrow 0$, where $\delta$ denotes the Dirac delta distribution.

The mean-field limit $N\rightarrow \infty$, $\eta \rightarrow 0$ has to be understood in the following sense. For fixed $\eta >0$, the many-particle model (2) is approximated for $N\rightarrow \infty$ by the intermediate stochastic system

$$\begin{aligned} \begin{aligned} d\bar{X}_{\eta ,i}^k(t)&= -\sum _{j=1}^n(\nabla V_{ij}^\eta * u_{\eta ,j}) (\bar{X}_{\eta ,i}^k(t),t)\mathrm{d}t + \sqrt{2\sigma _i} dW_i^k(t), \\ \bar{X}_{\eta ,i}^k(0)&= \xi _i^k, \quad i=1,\ldots ,n,\quad \ k=1,\ldots ,N, \end{aligned} \end{aligned}$$

(3)

where $u_{\eta ,j}$ is the probability density function of $\bar{X}_{\eta ,j}^k$, satisfying the nonlocal diffusion system

$$\begin{aligned} \begin{aligned} \partial _t u_{\eta ,i}&= \sigma _i\Delta u_{\eta ,i} + {\text {div}}\bigg (\sum _{j=1}^n u_{\eta ,i} \nabla V_{ij}^\eta * u_{\eta ,j}\bigg )\quad \text{ in } \mathbb {R}^d,\quad \ t>0, \\ u_{\eta ,i}(0)&= u_{i}^0, \quad i=1,\ldots ,n, \end{aligned} \end{aligned}$$

(4)

Observe that we consider N independent copies $\bar{X}_{\eta ,i}^k(t)$, $k=1,\ldots ,N$, of the solution to (3). In particular, the intermediate system depends on N only through the initial datum. Then, passing to the limit $\eta \rightarrow 0$ in the intermediate system, the limit $\nabla V_{ij}^\eta * u_{\eta ,j} \rightarrow a_{ij}\nabla u_j$ in $L^2$ leads to the limiting stochastic system

$$\begin{aligned} \begin{aligned} d\widehat{X}_i^k(t)&= -\sum _{j=1}^n a_{ij}\nabla u_j(\widehat{X}_i^k(t),t)\mathrm{d}t + \sqrt{2\sigma _i}dW_i^k(t), \\ \widehat{X}_i^k(0)&= \xi _i^k, \quad i=1,\ldots ,n,\quad \ k=1,\ldots ,N, \end{aligned} \end{aligned}$$

(5)

and the law of $\widehat{X}_i^k$ is a solution to the limiting cross-diffusion system (1). We remark that also here, we consider N independent copies $\widehat{X}_{i}^k(t)$, $k=1,\ldots ,N$, of the solution to (5). The main result of this paper is the proof of the error estimate

$$\begin{aligned} {\sup _{k=1,\ldots ,N}}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t}\big |X_{\eta ,i}^{k,N}(s) - \widehat{X}_i^k(s)\big |\bigg ) \le C(t)\eta , \end{aligned}$$

if we choose $\eta$ and N such that $\eta ^{-(2d+4)}\le \varepsilon \log N$ holds, where $\varepsilon >0$.

1.2 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 $N\rightarrow \infty$, 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 cross-diffusion 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)

$$\begin{aligned} \partial _t u_i = \Delta \bigg (u_i\sum _{j=1}^n a_{ij}u_j\bigg ) = {\text {div}}\bigg (\sum _{j=1}^n a_{ij}u_i\nabla u_j\bigg ) + {\text {div}}\bigg (\sum _{j=1}^n a_{ij}u_j\nabla u_i\bigg ) \end{aligned}$$

derived in [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 $\sigma _i$ 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 porous-medium term $a_{ii}u_i\nabla u_i$ in system (1) is not surprising. In [24], the quadratic porous-medium equation was approximated by the nonlocal equation $\partial _t u={\text {div}}(uW_\eta )$, where $W_\eta =\nabla u*V_\eta$ and $V_\eta$ 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 $a_{ij}\ge 0$ 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 no-flux boundary conditions under the condition $4a_{11}a_{22}>(a_{12}+a_{21})^2$. The idea of the proof is to show that $H(u)=\int u_i(\log u_i-1) \mathrm{d}x$ is a Lyapunov functional (entropy). The condition can be weakened to $a_{11}a_{22}>a_{12}a_{21}$ using the modified entropy $H_1(u)=\int (a_{21}u_1(\log u_1-1)+a_{12}u_2(\log u_2-1))\mathrm{d}x$. If $a_{11}a_{22}>a_{12}a_{21}$ and $u_1u_2>0$, system (1) is parabolic in the sense of Petrovskii which gives the local-in-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 $\Vert \cdot \Vert _{L^p}$ and $\Vert \cdot \Vert _{H^s}$ for the norms of $L^p=L^p(\mathbb {R}^d)$ and $H^s=H^s(\mathbb {R}^d)$, respectively. Furthermore, $|u|^2 = \sum _{i=1}^n u_i^2$ for $u=(u_1,\ldots ,u_n)\in \mathbb {R}^n$ and $\Vert u\Vert _{L^p}^2=\sum _{i=1}^n\Vert u_i\Vert _{L^p}^2$ for functions $u=(u_1,\ldots ,u_n)$. We use the notation $u(t)=u(\cdot ,t)$ for functions depending on x and t, and $C>0$ is a generic constant whose value may change from line to line.

1.3 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. Let $V_{ij}\in C_0^2(\mathbb {R}^d)$ be such that ${\text {supp}}(V_{ij}) \subset B_1(0)$ for $i,j=1,\ldots ,n$. Then $V_{ij}^\eta (x)=\eta ^{-d}V_{ij}(|x|/\eta )$ for $\eta >0$ satisfies ${\text {supp}}(V_{ij}^\eta )\subset B_\eta (0)$ and

$$\begin{aligned} a_{ij} := \int \limits _{\mathbb {R}^d} V_{ij}(|x|)\mathrm{d}x = \int \limits _{\mathbb {R}^d}V_{ij}^\eta (x)\mathrm{d}x, \quad i,j=1,\ldots ,n. \end{aligned}$$

As the potential may be negative (and $a_{ij}$ may be negative too), we introduce

$$\begin{aligned} A_{ij} := \Vert V_{ij}\Vert _{L^1} = \Vert V_{ij}^\eta \Vert _{L^1}, \quad i,j=1,\ldots ,n. \end{aligned}$$

Proposition 1

(Existence for the nonlocal diffusion system) Let $u^0=(u_1^0,\ldots ,u_n^0)\in$ $H^s(\mathbb {R}^d;\mathbb {R}^n)$ with $s>d/2+1$ and $u_i^0\ge 0$ in $\mathbb {R}^d$ and assume that for some $0<\gamma <\sigma$, it holds that

$$\begin{aligned} \Vert u^0\Vert _{H^s} \le \frac{\sigma -\gamma }{C^*\sum _{i,j=1}^n A_{ij}}, \end{aligned}$$

(6)

where $\sigma =\min _{i=1,\ldots ,n}\sigma _i>0$ and $C^*>1$ is a constant only depending on s and d. Then there exists a unique global solution $u_\eta =(u_{\eta ,1},\ldots ,u_{\eta ,n})$ to problem (4) such that $u_{\eta ,i}(t)\ge 0$ in $\mathbb {R}^d$, $t>0$, $u_\eta \in L^\infty (0,\infty ;H^s(\mathbb {R}^d;\mathbb {R}^n)\cap L^2(0,\infty ;H^{s+1}(\mathbb {R}^d;\mathbb {R}^n))$, and

$$\begin{aligned} \sup _{t>0}\Vert u_\eta (t)\Vert _{H^s}^2 + \gamma \Vert \nabla u_\eta \Vert _{L^2(0,\infty ;{H^{s}})}^2 \le \Vert u^0\Vert _{H^s}^2. \end{aligned}$$

(7)

We may interpret (6) also as a smallness condition for the nonlinear diffusion:

$$\begin{aligned} \sum _{i,j=1}^n A_{ij} \le \frac{\sigma }{C^*\Vert u^0\Vert _{H^s}}. \end{aligned}$$

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) Let $u^0\in H^s(\mathbb {R}^d;\mathbb {R}^n)$ with $s>d/2+1$ be such that $u_i^0\ge 0$ in $\mathbb {R}^d$ and let (6) hold. Then there exists a unique global solution $u=(u_1,\ldots ,u_n)$ to problem (1) satisfying $u_i(t)\ge 0$ in $\mathbb {R}^d$, $t>0$, $u\in L^\infty (0,\infty ;H^s(\mathbb {R}^d;\mathbb {R}^n))\cap L^2(0,\infty ;H^{s+1}(\mathbb {R}^d;\mathbb {R}^n))$, and

$$\begin{aligned} \sup _{t>0}\Vert u(t)\Vert _{H^s}^2 + \gamma \Vert u\Vert _{L^2(0,\infty ;H^{s+1})}^2 \le \Vert u^0\Vert _{H^s}^2. \end{aligned}$$

(8)

Moreover, let $u_\eta$ be the solution to problem (4). Then the following error estimate holds for any $T>0$:

$$\begin{aligned} \Vert u_\eta -u\Vert _{L^\infty (0,T;L^2)} + \Vert \nabla (u_\eta -u)\Vert _{L^2(0,T;L^2)} \le C(T)\eta \end{aligned}$$

(9)

for some constant $C(T)>0$.

The proposition is proved by performing the limit $\eta \rightarrow 0$ 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 $(\Omega ,{\mathcal {F}},({\mathcal {F}}_t)_{t\ge 0},{\mathcal P})$ be a filtered probability space and let $(W_i^k(t))_{t\ge 0}$ for $i=1,\ldots ,n$, $k=1,\ldots ,N$ be d-dimensional ${\mathcal {F}}_t$-Brownian motions that are independent of the random variables $\xi _i^k$. We assume that the Brownian motions are independent and that the initial data $\xi _i^1,\ldots ,\xi _i^N$ are independent and identically distributed random variables with the common probability density function $u_i^0$.

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.

Theorem 3

(Error estimate for the stochastic system) Let $u^0\in H^s(\mathbb {R}^d;\mathbb {R}^n)$ with $s>d/2+2$ be such that $u_i^0\ge 0$ in $\mathbb {R}^d$ and let (6) hold. Let $X_{\eta ,i}^{k,N}$ and $\widehat{X}_{i}^k$ be solutions to problems (2) and (5), respectively. Furthermore, let $0<\varepsilon <1$ be sufficiently small and choose $N\in \mathbb {N}$ such that $\varepsilon \log N\ge \eta ^{-2d-4}$. Then, for any $t>0$,

$$\begin{aligned} {\sup _{k=1,\ldots ,N}}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |X_{\eta ,i}^{k,N}(s) - \widehat{X}_{i}^k(s)\big |\bigg ) \le C(t)\eta , \end{aligned}$$

where the constant C(t) depends on t, n, $\Vert D^2 V_{ij}\Vert _{L^\infty }$, and the initial datum $u^0$.

The idea of the proof is to derive error estimates for the differences $X_{\eta ,i}^{k,N}-\bar{X}_{\eta ,i}^k$ and $\bar{X}_{\eta ,i}^k-\widehat{X}_i^k$ (where $\bar{X}_{\eta ,i}^{k,N}$ solves (3)) and to use

$$\begin{aligned} \sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |X_{\eta ,i}^{k,N} - \widehat{X}_i^k\big |\bigg )&\le \sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |X_{\eta ,i}^{k,N}-\bar{X}_{\eta ,i}^k\big |\bigg ) \\&\quad + \sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |\bar{X}_{\eta ,i}^k-\widehat{X}_i^k\big |\bigg ). \end{aligned}$$

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 $\nabla V_{ij}^\eta$, and observing that ${\Vert V_{ij}^\eta * \nabla u_j - a_{ij}\nabla u_j\Vert _{L^2(0,T;L^2)}\le C\eta }$.

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.

2 Existence for the nonlocal diffusion system (4)

We show Proposition 1 whose proof is split into several lemmas.

Lemma 4

(Local existence of solutions) Let $u^0=(u_1^0,\ldots ,u_n^0)\in H^s(\mathbb {R}^d;\mathbb {R}^n)$ with $s>d/2+1$ and $u_i^0\ge 0$ in $\mathbb {R}^d$. Then there exists $T^*>0$ such that (4) possesses the unique solution $u_\eta =(u_{\eta ,1},\ldots ,u_{\eta ,n})\in L^\infty (0,T^*;H^s(\mathbb {R}^d;\mathbb {R}^n))\cap L^2(0,T^*;H^{s+1}(\mathbb {R}^d;\mathbb {R}^n))$ satisfying $u_{\eta ,i}(t)\ge 0$ in $\mathbb {R}^d$ for $t>0$. The time $T^*>0$ depends on $\Vert u^0\Vert _{H^s}$ such that $T^*\rightarrow 0$ if $\Vert u^0\Vert _{H^s}\rightarrow \infty$.

Proof

The idea is to apply the Banach fixed-point theorem. For this, we introduce

$$\begin{aligned} Y = \Big \{v\in {L^\infty (0,T^*;H^s(\mathbb {R}^d;\mathbb {R}^n))}: \sup _{0<t<T^*}\Vert v(\cdot ,t)\Vert _{H^s}^2 \le M:= 1 + \Vert u^0\Vert _{H^s}^2\Big \}, \end{aligned}$$

endowed with the metric ${\text {dist}}(u,w)=\sup _{0<t<T^*}\Vert (u-w)(t)\Vert _{L^2}$, 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 $S:Y\rightarrow Y$ is defined by $Sv=u$, where u is the unique solution to the Cauchy problem

$$\begin{aligned} \partial _t u_i = \sigma _i\Delta u_i + {\text {div}}\bigg (\sum _{j=1}^n u_i\nabla V_{ij}^\eta * v_j\bigg ), \quad u_i(0)=u_i^0 \text{ in } \mathbb {R}^d. \end{aligned}$$

(10)

By standard parabolic theory, this linear advection–diffusion problem has a weak solution $u_i\in C^0([0,T];L^2(\mathbb {R}^d))\cap L^2(0,T;H^1(\mathbb {R}^d))$. Bootstrapping the regularity result of [41, Theorem 1.3.11], we arrive at $u_i\in C^0([0,T];H^s(\mathbb {R}^d))\cap L^2(0,T;H^{s+1}(\mathbb {R}^d))$.

We prove that $\sup _{0<t<T^*}\Vert u(\cdot ,t)\Vert _{H^s}^2 \le M$ for sufficiently small values of $T^*>0$. Then $u\in Y$ and $S:Y\rightarrow Y$ is well defined. We apply the differential operator $D^\alpha$ for an arbitrary multi-index $\alpha \in \mathbb {N}^d$ of order $|\alpha |\le s$ to (10), multiply the resulting equation by $D^\alpha u_{i}$, and integrate over $\mathbb {R}^d$:

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|D^\alpha u_i|^2 \mathrm{d}x + \sigma _i\int \limits _{\mathbb {R}^d}|\nabla D^\alpha u_i|^2 \mathrm{d}x = -\int \limits _{\mathbb {R}^d}\sum _{j=1}^n D^\alpha (u_i\nabla V_{ij}^\eta * v_j)\cdot \nabla D^\alpha u_i \mathrm{d}x. \end{aligned}$$

We sum these equations from $i=1,\ldots ,n$, apply the Cauchy–Schwarz inequality to the integral on the right-hand side, and the Moser-type calculus inequality (Lemma 15):

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|D^\alpha u|^2 \mathrm{d}x + \sigma \int \limits _{\mathbb {R}^d}|\nabla D^\alpha u|^2 \mathrm{d}x \le \sum _{i,j=1}^n \big \Vert D^\alpha (u_i\nabla V_{ij}^\eta * v_j)\big \Vert _{L^2} \Vert \nabla D^\alpha u_i\Vert _{L^2} \\&\quad \le C(\varepsilon )\sum _{i,j=1}^n \big \Vert D^\alpha (u_i\nabla V_{ij}^\eta * v_j)\big \Vert _{L^2}^2 + \varepsilon n\Vert \nabla D^\alpha u\Vert _{L^2}^2 \\&\quad \le C(\varepsilon )\sum _{i,j=1}^n\Big (\Vert u_i\Vert _{L^\infty } \big \Vert D^s(\nabla V_{ij}^\eta * v_j)\big \Vert _{L^2} + \Vert D^s u_i\Vert _{L^2} \Vert \nabla V_{ij}^\eta * v_j\Vert _{L^\infty }\Big )^2 \\&\qquad + \varepsilon n\Vert \nabla D^\alpha u\Vert _{L^2}^2, \end{aligned}$$

where we recall that $\sigma =\min _{i=1,\ldots ,n}\sigma _i>0$ and $\varepsilon$ 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 $\varepsilon \le \sigma /(2n)$. Hence, summing over all multi-indices $\alpha$ of order $|\alpha |\le s$, using Young’s convolution inequality (Lemma 16), and the inequality $\Vert \nabla V_{ij}^\eta \Vert _{L^1}\le C(\eta )$, we find that

$$\begin{aligned} {\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{d}t}\Vert u\Vert _{H^s}^2 + \frac{\sigma }{2}\Vert \nabla u\Vert _{H^s}^2&\le C\sum _{i,j=1}^n\Big (\Vert u\Vert _{L^\infty }\Vert \nabla V_{ij}^\eta \Vert _{L^1}\Vert D^s v_j\Vert _{L^2} + \Vert D^s u_i\Vert _{L^2}\Vert \nabla V_{ij}^\eta \Vert _{L^1}\Vert v_j\Vert _{L^\infty }\Big )^2 \\&\le C(\eta )\Vert u\Vert _{H^s}^2\Vert v\Vert _{H^s}^2, \end{aligned}$$

where in the last step we have taken into account the continuous embedding $H^s(\mathbb {R}^d)\hookrightarrow L^\infty (\mathbb {R}^d)$. As $v\in Y$ and consequently $\Vert v(t)\Vert _{H^s}^2\le M$, we infer that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert u\Vert _{H^s}^2 \le C(\eta ) M\Vert u\Vert _{H^s}^2. \end{aligned}$$

Gronwall’s inequality then yields

$$\begin{aligned} \Vert u(t)\Vert _{H^s}^2 \le \Vert u^0\Vert _{H^s}^2 e^{C(\eta )Mt} = (M-1)e^{C(\eta )MT^*} \le M, \quad 0<t<T^*, \end{aligned}$$

if we choose $T^*>0$ so small that $e^{C(\eta )MT^*}\le M/(M-1)$. We conclude that $u\in Y$.

Note that the time $T^*$ depends on M and hence on $u^0$ in such a way that $T^*$ becomes smaller if $\Vert u^0\Vert _{H^s}$ is large but $T^*>0$ is bounded from below if $\Vert u^0\Vert _{H^s}$ is small.

It remains to show that the map $S:Y\rightarrow Y$ is a contraction, possibly for a smaller value of $T^*>0$. Let $v,w\in Y$ and take the difference of the equations satisfied by Sv and Sw, respectively:

$$\begin{aligned}&\partial _t\big ((Sv)_i-(Sw)_i\big ) - \sigma _i\Delta \big ((Sv)_i-(Sw)_i\big ) \\&\quad = {\text {div}}\bigg (\sum _{j=1}^n\big ((Sv)_i-(Sw)_i\big )\nabla V_{ij}^\eta * v_j\bigg ) + {\text {div}}\bigg (\sum _{j=1}^n (Sw)_i\nabla V_{ij}^\eta * (v_j-w_j)\bigg ). \end{aligned}$$

Multiplying these equations by $(Sv)_i-(Sw)_i$, summing from $i=1,\ldots ,n$, integrating over $\mathbb {R}^d$, and using the Cauchy-Schwarz inequality leads to

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|Sv-Sw|^2 \mathrm{d}x + \sigma \int \limits _{\mathbb {R}^d}|\nabla (Sv-Sw)|^2 \mathrm{d}x \\&\le \sum _{i,j=1}^n \big \Vert ((Sv)_i-(Sw)_i)\nabla V_{ij}^\eta * v_j\big \Vert _{L^2}\Vert \nabla ((Sv)_i-(Sw)_i)\Vert _{L^2} \\&\quad + \sum _{i,j=1}^n \big \Vert (Sw)_i\nabla V_{ij}^\eta * (v_j-w_j)\big \Vert _{L^2} \Vert \nabla ((Sv)_i-(Sw)_i)\Vert _{L^2}. \end{aligned}$$

We deduce from Young’s convolution inequality that

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|Sv-Sw|^2 \mathrm{d}x + \frac{\sigma }{2}\int \limits _{\mathbb {R}^d}|\nabla (Sv-Sw)|^2 \mathrm{d}x \\&\quad \le C(\sigma )\sum _{i,j=1}^n\big \Vert ((Sv)_i-(Sw)_i)\nabla V_{ij}^\eta * v_j\big \Vert _{L^2}^2 + C(\sigma )\sum _{i,j=1}^n \big \Vert (Sw)_i\nabla V_{ij}^\eta * (v_j-w_j)\big \Vert _{L^2}^2 \\&\quad \le C(\sigma )\sum _{i,j=1}^n\Vert \nabla V_{ij}^\eta * v_j\Vert _{L^\infty }^2 \Vert (Sv)_i-(Sw)_i\Vert _{L^2}^2 \\&\qquad + C(\sigma ) \max _{i=1,\ldots ,n}\Vert (Sw)_i\Vert _{L^\infty }^2\sum _{i,j=1}^n \Vert \nabla V_{ij}^\eta * (v_j-w_j)\Vert _{L^2}^2 \\&\quad \le C(\sigma )\sum _{i,j=1}^n\Vert \nabla V_{ij}^\eta \Vert _{L^1}^2\Vert v_j\Vert _{L^\infty }^2 \Vert Sv-Sw\Vert _{L^2}^2 \\&\qquad + C(\sigma )\Vert Sw\Vert _{L^\infty }^2\sum _{i,j=1}^n \Vert \nabla V_{ij}^\eta \Vert _{L^1}^2\Vert v_j-w_j\Vert _{L^2}^2. \end{aligned}$$

By definition of the metric on Y, we have shown that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert Sv-Sw\Vert _{L^2}^2 \le C_1(\sigma ,\eta ,M)\Vert Sv-Sw\Vert _{L^2}^2 + C_2(\sigma ,\eta ,M)\Vert v-w\Vert _{L^2}^2. \end{aligned}$$

The constants $C_1$ and $C_2$ depend on M (and hence on $u^0$) in such a way that they become larger if $\Vert u^0\Vert _{H^s}$ is large but they stay bounded for small values of $\Vert u^0\Vert _{H^s}$. Thus, because of $v(0)=w(0)$, Gronwall’s inequality gives

$$\begin{aligned} \Vert Sv(t)-Sw(t)\Vert _{L^2}^2 \le C_2(\sigma , \eta ,M)te^{C_1(\sigma , \eta , M)t}\sup _{0<s<t}\Vert v(s)-w(s)\Vert _{L^2}^2, \end{aligned}$$

and the definition of the metric leads to

$$\begin{aligned} {\text {dist}}(Sv,Sw)^2 \le C_2(\sigma , \eta ,M)te^{C_1(\sigma , \eta , M)t}{\text {dist}}(v,w)^2. \end{aligned}$$

Then, choosing $T^*>0$ such that $C_2(\sigma ,\eta ,M)T^*e^{C_1(\sigma ,\eta ,M)T^*}\le 1/2$ shows that $S:Y\rightarrow Y$ is a contraction. Again, $T^*$ depends on $u^0$ but it is bounded from below for small values of $\Vert u^0\Vert _{H^s}$. Thus, we can apply the Banach fixed-point theorem, finishing the proof. $\square$

Lemma 5

(A priori estimates) Let assumption (6) hold. For the local solution $u_\eta$ to problem (4), the uniform estimate (7) holds. In particular, the solution $u_\eta$ can be extended to a global one.

Proof

We proceed similarly as in the proof of Lemma 4. We choose $\alpha$ of order $|\alpha |\le s$, apply the operator $D^\alpha$ on both sides of (4), multiply the resulting equation by $D^\alpha u_{\eta ,i}$, and integrate over $\mathbb {R}^d$. By the Cauchy–Schwarz inequality, the Moser-type calculus inequality, and Young’s convolution inequality and writing $u_i$ instead of $u_{\eta ,i}$, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|D^\alpha u|^2 \mathrm{d}x + \sigma \int \limits _{\mathbb {R}^d}|\nabla D^\alpha u|^2 \mathrm{d}x \\&\quad \le \sum _{i,j=1}^n\big \Vert D^\alpha (u_iV_{ij}^\eta * \nabla u_j)\big \Vert _{L^2}\Vert \nabla D^\alpha u_i\Vert _{L^2} \\&\quad \le C_M\sum _{i,j=1}^n\Big (\Vert u_i\Vert _{L^\infty }\Vert D^s(V_{ij}^\eta * \nabla u_j)\Vert _{L^2}+ \Vert D^s u_i\Vert _{L^2}\Vert V_{ij}^\eta * \nabla u_j\Vert _{L^\infty }\Big ) \Vert \nabla D^\alpha u_i\Vert _{L^2} \\&\quad \le C_M\sum _{i,j=1}^n \Big (C\Vert u\Vert _{H^s}\Vert V_{ij}^\eta \Vert _{L^1}\Vert D^s \nabla u\Vert _{L^2} + \Vert D^s u\Vert _{L^2}\Vert V_{ij}^\eta \Vert _{L^1}\Vert \nabla u\Vert _{L^\infty }\Big ) \Vert \nabla D^\alpha u\Vert _{L^2} \\&\quad \le C^*\sum _{i,j=1}^nA_{ij}\Vert u\Vert _{H^s}\Vert \nabla u\Vert _{H^{s}}\Vert \nabla D^\alpha u\Vert _{L^2}, \end{aligned}$$

where $C_M$ is the constant from Lemma 15, $C^*>1$ depends on $C_M$ and the constant of the embedding $H^s(\mathbb {R}^d)\hookrightarrow L^\infty (\mathbb {R}^d)$, and we have used $\Vert V_{ij}^\eta \Vert _{L^1}=A_{ij}$. Summation of all $|\alpha |\le s$ leads to

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert u\Vert _{H^s}^2 + \sigma \Vert \nabla u\Vert _{H^s}^2 \le C^*\sum _{i,j=1}^n A_{ij}\Vert u\Vert _{H^s}\Vert \nabla u\Vert _{H^{s}}^2, \end{aligned}$$

which can be written as

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert u\Vert _{H^s}^2 + 2\bigg (\sigma - C^*\sum _{i,j=1}^nA_{ij}\Vert u\Vert _{H^s}\bigg ) \Vert \nabla u\Vert _{H^{s}}^2 \le 0. \end{aligned}$$

(11)

This inequality holds for all $t\in [0,T]$, where $T<T^*$. By Lemma 17, applied to $f(t)=\Vert u(t)\Vert _{H^s}^2$, $g(t)=\Vert \nabla u(t)\Vert _{H^s}$, $a=\sigma$, and $b=C^*\sum _{i,j=1}^nA_{ij}$, we find that $\Vert u(t)\Vert _{H^s}^2 \le (a/b)^2$ for $t\in [0,T]$. Here, we use Assumption (6). We deduce that $(d/\mathrm{d}t)\Vert u\Vert _{H^s}^2\le 0$ and consequently $\Vert u(t)\Vert _{H^s}\le \Vert u^0\Vert _{H^s}$ for $t\in [0,T]$.

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 $\Vert u(T)\Vert _{H^s}$ is not larger as $\Vert u^0\Vert _{H^s}$. Note that $T^*$ becomes smaller only when the initial datum is larger in the $H^s$ norm. Hence, u(t) exists for $t\in [T,2T]$ and inequality (11) holds. As before, we conclude from Lemma 17 that $\Vert u(t)\Vert _{H^s}\le \Vert u^0\Vert _{H^s}$ for $t\in [T,2T]$. This argument can be continued, obtaining a global solution satisfying $\Vert u(t)\Vert _{H^s}\le \Vert u^0\Vert _{H^s}$ for all $t>0$. Then, under the stronger assumption (6),

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert u\Vert _{H^s}^2 \le -2\bigg (\sigma - C^*\sum _{i,j=1}^nA_{ij}\Vert u^0\Vert _{H^s}\bigg )\Vert \nabla u\Vert _{H^{s}}^2 \le -\gamma \Vert \nabla u\Vert _{H^{s}}^2, \end{aligned}$$

which leads to (7), finishing the proof. $\square$

Lemma 6

(Uniqueness of solutions) Let assumption (6) hold. Then the solution to problem (4) is unique in the class of functions $u\in L^\infty (0,\infty ;H^s(\mathbb {R}^d;\mathbb {R}^n))\cap L^2(0,\infty ;H^{s+1}(\mathbb {R}^d;$ $\mathbb {R}^n))$.

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 $i=1,\ldots ,n$, and integrate over $\mathbb {R}^d$. Then, for all $0<t<T$ and some $T>0$,

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|u-v|^2 \mathrm{d}x + \sigma \int \limits _{\mathbb {R}^d}|\nabla (u-v)|^2 \mathrm{d}x \\&\quad \le \sum _{i,j=1}^n\big (\Vert u_i-v_i\Vert _{L^2}\Vert V_{ij}^\eta * \nabla u_j\Vert _{L^\infty } + \Vert v_i\Vert _{L^\infty }\Vert V_{ij}^\eta * \nabla (u_j-v_j)\Vert _{L^2}\big ) \Vert \nabla (u_i-v_i)\Vert _{L^2} \\&\quad \le \sum _{i,j=1}^n\big (\Vert u-v\Vert _{L^2}\Vert V_{ij}^\eta \Vert _{L^1}\Vert \nabla u\Vert _{L^\infty } + \Vert v\Vert _{L^\infty }\Vert V_{ij}^\eta \Vert _{L^1}\Vert \nabla (u-v)\Vert _{L^2}\big )\Vert \nabla (u-v)\Vert _{L^2} \\&\quad \le C\sum _{i,j=1}^nA_{ij}\big (\Vert u\Vert _{H^{s+1}}\Vert u-v\Vert _{L^2}\Vert \nabla (u-v)\Vert _{L^2} + \Vert v\Vert _{H^s}\Vert \nabla (u-v)\Vert _{L^2}^2\big ). \end{aligned}$$

By assumption, $\sum _{i,j=1}^nA_{ij}\Vert v\Vert _{H^s} \le \sigma -\gamma$. Thus, using the Cauchy–Schwarz inequality, it follows that

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|u-v|^2 \mathrm{d}x + \sigma \int \limits _{\mathbb {R}^d}|\nabla (u-v)|^2 \mathrm{d}x \\&\quad \le C(\varepsilon )\bigg (\sum _{i,j=1}^n A_{ij}\Vert u\Vert _{H^{s+1}}\bigg )^2 \Vert u-v\Vert _{L^2}^2 + \varepsilon n^2\Vert \nabla (u-v)\Vert _{L^2}^2 + (\sigma -\gamma )\Vert \nabla (u-v)\Vert _{L^2}^2 \\&\quad \le C(\varepsilon )\Vert u(t)\Vert _{H^{s+1}}^2\Vert u-v\Vert _{L^2}^2 + \sigma \Vert \nabla (u-v)\Vert _{L^2}^2, \end{aligned}$$

if we choose $\varepsilon \le \gamma /n^2$. Observe that the norm $\Vert u\Vert _{L^2(0,\infty ;H^{s+1})}$ is bounded. This allows us to apply the Gronwall inequality, and together with the fact that $\Vert (u-v)(0)\Vert _{L^2}=0$, we infer that $\Vert (u-v)(t)\Vert _{L^2}=0$, concluding the proof. $\square$

3 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 $u_\eta$ be the solution to (4). We prove that a subsequence of $(u_\eta )$ converges to a solution to problem (1). In view of the uniform estimate (7), there exists a subsequence of $(u_\eta )$, which is not relabeled, such that, as $\eta \rightarrow 0$,

$$\begin{aligned} u_\eta \rightharpoonup u\quad \text{ weakly } \text{ in } L^2(0,T;H^{s+1}(\mathbb {R}^d)). \end{aligned}$$

(12)

We show that u is a weak solution to problem (1). First, we claim that

$$\begin{aligned} V_{ij}^\eta * \nabla u_{\eta ,j} \rightharpoonup a_{ij}\nabla u_j \quad \text{ weakly } \text{ in } L^2(0,T;L^2(\mathbb {R}^d)). \end{aligned}$$

To prove this statement, we observe that $V_{ij}^\eta * \psi \rightarrow a_{ij}\psi$ strongly in $L^2(0,T;L^2(\mathbb {R}^d))$ for any $\phi \in L^2(0,T;L^2(\mathbb {R}^d))$ [32, Theorem 9.10] and $\nabla u_{\eta ,j}\rightharpoonup \nabla u_j$ weakly in $L^2(0,T;L^2(\mathbb {R}^d))$. Therefore, for all $\psi \in C_0^\infty (\mathbb {R}^d\times (0,T);\mathbb {R}^n)$, as $\eta \rightarrow 0$,

$$\begin{aligned}&\bigg |\int \limits _0^T\int \limits _{\mathbb {R}^d}\big (V_{ij}^\eta * \nabla u_{\eta ,j} - a_{ij}\nabla u_j) \cdot \psi \mathrm{d}x\mathrm{d}t\bigg | \\&\quad = \bigg |\int \limits _0^T\int \limits _{\mathbb {R}^d}\bigg (\int \limits _{\mathbb {R}^d}V_{ij}^\eta (x-y)\nabla u_{\eta ,j}(y,t)\mathrm{d}y \bigg )\cdot \psi (x,t)\mathrm{d}x\mathrm{d}t \\&\qquad - \int \limits _0^T\int \limits _{\mathbb {R}^d}a_{ij}\nabla u_{j}(y,t)\cdot \psi (y,t)\mathrm{d}y\mathrm{d}t\bigg | \\&\quad \le \bigg |\int \limits _0^T\int \limits _{\mathbb {R}^d}\bigg (\int \limits _{\mathbb {R}^d}V_{ij}^\eta (x-y)\psi (x,t)\mathrm{d}x - a_{ij}\psi (y,t)\bigg )\cdot \nabla u_{\eta ,j}(y,t)\mathrm{d}y\mathrm{d}t\bigg | \\&\qquad + |a_{ij}|\bigg |\int \limits _0^T\int \limits _{\mathbb {R}^d}(\nabla u_{\eta ,j}-\nabla u_j)(y,t)\cdot \psi (y,t)\mathrm{d}y\mathrm{d}t\bigg | \\&\quad \le \Vert V_{ij}^\eta *\psi - a_{ij}\psi \Vert _{L^2(0,T;L^2(\mathbb {R}^d))} \Vert \nabla u_{\eta ,j}\Vert _{L^2(0,T;L^2(\mathbb {R}^d))} \\&\qquad + |a_{ij}|\bigg |\int \limits _0^T\int \limits _{\mathbb {R}^d}(\nabla u_{\eta ,j}-\nabla u_j)(y,t)\cdot \psi (y,t)\mathrm{d}y\mathrm{d}t\bigg | \rightarrow 0, \end{aligned}$$

which proves the claim. Estimate (7) and the embedding $H^s(\mathbb {R}^d)\hookrightarrow L^\infty (\mathbb {R}^d)$ show that

$$\begin{aligned} \big \Vert u_{\eta ,i}V_{ij}^\eta * \nabla u_{\eta ,j}\big \Vert _{L^2(0,T;L^2)} \le \Vert u_{\eta ,i}\Vert _{L^\infty (0,T;L^\infty )}\Vert V_{ij}^\eta \Vert _{L^1} \Vert \nabla u_{\eta ,j}\Vert _{L^2(0,T;L^2)} \le C \end{aligned}$$

(13)

and consequently, for any $T>0$,

$$\begin{aligned} \Vert \partial _t u_{\eta ,i}\Vert _{L^2(0,T;H^{-1})} \le \sigma _i\Vert \nabla u_{\eta ,i}\Vert _{L^2(0,T;L^2)} + \sum _{j=1}^n\big \Vert u_{\eta ,i}V_{ij}^\eta * \nabla u_{\eta ,j}\big \Vert _{L^2(0,T;L^2)} \le C. \end{aligned}$$

The weak formulation of (4) reads as

$$\begin{aligned} \int \limits _0^T\langle \partial _t u_{\eta ,i},\phi \rangle \zeta (t)\mathrm{d}t = \int \limits _0^T\int \limits _{\mathbb {R}^d}\bigg (\sigma _i\nabla u_{\eta ,i} + \sum _{j=1}^n u_{\eta ,i} \nabla V_{ij}^\eta * \nabla u_{\eta ,j}\bigg )\cdot \nabla \phi \mathrm{d}x\zeta (t)\mathrm{d}t, \end{aligned}$$

(14)

where $\phi \in C_0^\infty (\mathbb {R}^d)$ with ${\text {supp}}(\phi )\subset B_R(0)$ and $\zeta \in C^\infty ([0,T])$. Since the ball $B_R(0)$ is bounded and the embedding $H^1(B_R(0))\hookrightarrow L^2(B_R(0))$ is compact, the Aubin–Lions lemma [35] gives the existence of a subsequence of $(u_\eta )$, which is not relabeled, such that $u_\eta \rightarrow u$ strongly in $L^2(0,T;L^2(B_R(0)))$ as $\eta \rightarrow 0$, and the limit coincides with the weak limit in (12). We deduce that

$$\begin{aligned} u_{\eta ,i}V_{ij}^\eta * \nabla u_{\eta ,j} \rightharpoonup a_{ij}u_i\nabla u_j \quad \text{ weakly } \text{ in } L^1(0,T;L^1(B_R(0))). \end{aligned}$$

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 $u_i-v_i$, summing from $i=1,\ldots ,n$, integrating over $\mathbb {R}^d$, and using the Cauchy–Schwarz inequality leads to

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|u-v|^2 \mathrm{d}x + \sigma \int \limits _{\mathbb {R}^d}|\nabla (u-v)|^2 \mathrm{d}x \\&\quad \le \sum _{i,j=1}^n A_{ij}\Vert u\Vert _{H^{s+1}}\Vert u-v\Vert _{L^2}\Vert \nabla (u-v)\Vert _{L^2} + \sum _{i,j}A_{ij}\Vert v\Vert _{H^s}\Vert u-v\Vert _{L^2}^2. \end{aligned}$$

In view of estimate (8), this becomes

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|u-v|^2 \mathrm{d}x + \frac{\sigma }{2}\int \limits _{\mathbb {R}^d}|\nabla (u-v)|^2 \mathrm{d}x \le C(t)\Vert u-v\Vert _{L^2}^2, \end{aligned}$$

and the constant $C(t)>0$ is integrable (as it depends on $\Vert u(t)\Vert _{H^{s+1}}$). Gronwall’s inequality then implies that $(u-v)(t)=0$ for $t>0$. $\square$

Lemma 8

(Error estimate) Let the assumptions of Proposition 2 hold. Let u be the solution to (1) and $u_\eta$ be the solution to (4). Then the error estimate (9) holds.

Proof

We take the difference of Eqs. (4) and (1),

$$\begin{aligned}&\partial _t(u_{\eta ,i}-u_i) - \sigma _i\Delta (u_{\eta ,i}-u_i) = {\text {div}}\bigg (\sum _{j=1}^n u_{\eta ,i}V_{ij}^\eta * \nabla u_{\eta ,j} - \sum _{i=1}^n a_{ij}u_i\nabla u_j\bigg ) \\&\quad = {\text {div}}\sum _{j=1}^n\Big ((u_{\eta ,i}-u_i)V_{ij}^\eta * \nabla u_{\eta ,j} + u_i(V_{ij}^\eta * \nabla u_{\eta ,j} - a_{ij}\nabla u_{\eta ,j}) + a_{ij}u_i\nabla (u_{\eta ,j}-u_j)\Big ). \end{aligned}$$

Multiplying this equation by $u_{\eta ,i}-u_i$, summing from $i=1,\ldots ,n$, integrating over $\mathbb {R}^d$, using the Cauchy–Schwarz inequality, and the estimate $|a_{ij}|\le A_{ij}$, we find that

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|u_\eta -u|^2 \mathrm{d}x + \sigma \int \limits _{\mathbb {R}^d}|\nabla (u_\eta -u)|^2 \mathrm{d}x \\&\quad \le \sum _{i,j=1}^n\Vert u_{\eta ,i}-u_i\Vert _{L^2}\Vert V_{ij}^\eta * \nabla u_{\eta ,j}\Vert _{L^\infty }\Vert \nabla (u_{\eta ,i}-u_i)\Vert _{L^2} \\&\qquad + \sum _{i,j=1}^n \Vert u_i\Vert _{L^\infty }\Vert V_{ij}^\eta * \nabla u_{\eta ,j} - a_{ij}\nabla u_{\eta ,j}\Vert _{L^2}\Vert \nabla (u_{\eta ,j}-u_j)\Vert _{L^2} \\&\qquad + \sum _{i,j=1}^n A_{ij}\Vert u_i\Vert _{L^\infty } \Vert \nabla (u_{\eta ,j}-u_j)\Vert _{L^2}\Vert \nabla (u_{\eta ,i}-u_i)\Vert _{L^2} \\&\quad = I_ 1 + I_2 + I_3. \end{aligned}$$

We estimate the right-hand side term by term. First, by the continuous embedding $H^{s}(\mathbb {R}^d)\hookrightarrow L^\infty (\mathbb {R}^d)$,

$$\begin{aligned} |I_1|&\le \sum _{i,j=1}^n\Vert u_{\eta ,i}-u_i\Vert _{L^2}\Vert V_{ij}^\eta \Vert _{L^1} \Vert \nabla u_{\eta ,j}\Vert _{L^\infty }\Vert \nabla (u_{\eta ,i}-u_i)\Vert _{L^2} \\&\le C(\gamma )\bigg (\sum _{i,j=1}^n A_{ij}\Vert \nabla u_{\eta }\Vert _{H^{s}}\bigg )^2 \Vert u_\eta -u\Vert _{L^2}^2 + \frac{\gamma }{4}\Vert \nabla (u_\eta -u)\Vert _{L^2}^2. \end{aligned}$$

To estimate $I_2$, let $g\in L^2(\mathbb {R}^d;\mathbb {R}^n)$. Since ${\text {supp}}V_{ij}^\eta \subset B_\eta (0)$, the fundamental theorem of calculus shows that

$$\begin{aligned} \bigg |\int \limits _{\mathbb {R}^d}&\big (V_{ij}^\eta * \nabla u_{\eta ,j} - a_{ij}\nabla u_{\eta ,j}\big )(x)\cdot g(x)\mathrm{d}x\bigg | \\&= \bigg |\int \limits _{\mathbb {R}^d}\int \limits _{B_\eta (0)} V_{ij}^\eta (y)\sum _{k=1}^d \bigg (\frac{\partial u_{\eta ,j}}{\partial x_k}(x-y) - \frac{\partial u_{\eta ,j}}{\partial x_k}(x)\bigg ) g_k(x) \mathrm{d}y\mathrm{d}x\bigg | \\&= \bigg |\int \limits _{\mathbb {R}^d}\int \limits _{B_\eta (0)} V_{ij}^\eta (y)\bigg (\int \limits _0^1 \sum _{k,\ell =1}^d\frac{\partial ^2 u_{\eta ,j}}{\partial x_k\partial x_\ell }(x-ry)y_\ell dr\bigg )g_k(x) \mathrm{d}y\mathrm{d}x\bigg | \\&\le \eta \int \limits _0^1\int \limits _{B_\eta (0)} |V_{ij}^\eta (y)|\int \limits _{\mathbb {R}^d} |D^2 u_{\eta ,j}(x-ry)|\,|g(x)|\mathrm{d}x\mathrm{d}y\mathrm{d}r \\&\le \eta \int \limits _0^1\int \limits _{B_\eta (0)}|V_{ij}^\eta (y)|\, \Vert D^2 u_{\eta ,j}(\cdot -ry)\Vert _{L^2}\Vert g\Vert _{L^2}\mathrm{d}y\mathrm{d}r \\&\le \eta \Vert V_{ij}^\eta \Vert _{L^1}\Vert D^2 u_{\eta ,j}\Vert _{L^2}\Vert g\Vert _{L^2} = \eta A_{ij}\Vert D^2 u_{\eta ,j}\Vert _{L^2}\Vert g\Vert _{L^2}. \end{aligned}$$

This shows that

$$\begin{aligned} \Vert V_{ij}^\eta * \nabla u_{\eta ,j} - a_{ij}\nabla u_{\eta ,j}\Vert _{L^2} \le \eta C\Vert D^2 u_{\eta ,j}\Vert _{L^2} \le \eta C \end{aligned}$$

(15)

and consequently,

$$\begin{aligned} |I_2|&\le C(\gamma )\sum _{i,j=1}^n\Vert V_{ij}^\eta * \nabla u_{\eta ,j} - a_{ij}\nabla u_{\eta ,j}\Vert _{L^2}^2 + \frac{\gamma }{4}\Vert \nabla (u_\eta -u)\Vert _{L^2}^2 \\&\le C(\gamma )\eta ^2 + \frac{\gamma }{4}\Vert \nabla (u_\eta -u)\Vert _{L^2}^2. \end{aligned}$$

Finally, by Assumption (6),

$$\begin{aligned} |I_3| \le \bigg (\sum _{i,j=1}^nA_{ij}\Vert u\Vert _{H^{s}}\bigg ) \Vert \nabla (u_\eta -u)\Vert _{L^2}^2 \le (\sigma -\gamma )\Vert \nabla (u_\eta -u)\Vert _{L^2}^2. \end{aligned}$$

Therefore,

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int \limits _{\mathbb {R}^d}|u_\eta -u|^2 \mathrm{d}x + \frac{\gamma }{2}\int \limits _{\mathbb {R}^d}|\nabla (u_\eta -u)|^2 \mathrm{d}x \\&\quad \le C(\gamma )\bigg (\sum _{i,j=1}^nA_{ij}\Vert \nabla u_{\eta }\Vert _{H^{s}}\bigg )^2 \Vert u_\eta -u\Vert _{L^2}^2 + C(\gamma )\eta ^2 \\&\quad \le C(\gamma )(\sigma -\gamma )^2\Vert u_\eta -u\Vert _{L^2}^2 + C(\gamma )\eta ^2, \end{aligned}$$

and Gronwall’s lemma gives the conclusion. $\square$

4 Existence of solutions to the stochastic systems

Let the initial data $\xi _1,\ldots ,\xi _n$ be square integrable. We prove the solvability of the stochastic ordinary differential systems (2), (3), and (5).

Lemma 9

(Solvability of the stochastic many-particle system) For any fixed $\eta >0$, problem (2) has a unique strong solution $X_{\eta ,i}^{k,N}$ that is ${\mathcal {F}}_t$-adapted.

Proof

By assumption, the gradient $\nabla V_{ij}^\eta$ is bounded and Lipschitz continuous. Then [30, Theorem 5.2.1] or [31, Theorem 3.1.1] show that there exists a (up to $\mathbb {P}$-indistinguishability) unique strong solution to (2). $\square$

Lemma 10

(Solvability of the nonlocal stochastic system) Let $u_\eta$ be a solution to the nonlocal diffusion system (4) satisfying $|\nabla u_\eta |\in L^\infty (0,\infty ;W^{1,\infty }(\mathbb {R}^d;\mathbb {R}^n))$. Then problem (3) has a unique strong solution $\bar{X}_{\eta ,i}^k$ with probability density function $u_{\eta ,i}$.

Remark 11

We have shown in Proposition 2 that if $u^0\in H^s(\mathbb {R}^d;\mathbb {R}^n)$ and the smallness condition (6) holds, there exists a unique solution $u_\eta \in L^\infty (0,\infty ;H^s(\mathbb {R}^d;\mathbb {R}^n))$. The regularity for $u_\eta$, required in Lemma 10, is fulfilled if $s>d/2+2$, thanks to the embedding $u_{\eta ,i}\in L^\infty (0,\infty ;H^s(\mathbb {R}^d)) \hookrightarrow L^\infty (0,\infty ;W^{2,\infty }(\mathbb {R}^d))$ for $i=1,\ldots ,n$. $\square$

Proof of Lemma 10

We proceed as in the proof of Lemma 3.2 of [8]. Let $u_{\eta ,i}\in L^\infty (0,T;H^s(\mathbb {R}^d))\cap L^2(0,T;H^{s+1}(\mathbb {R}^d))$ be the unique solution to (4) satisfying $u_{\eta }(0)=u^0$ in $\mathbb {R}^d$, where $u^0_i$ is the density of $\xi ^k_i$. By assumption, $\nabla V_{ij}^\eta * u_{\eta ,j} = V_{ij}^\eta * \nabla u_{\eta ,j}$ is bounded and Lipschitz continuous. Therefore,

$$\begin{aligned} d\bar{X}_{\eta ,i}^k = -\sum _{j=1}^n(\nabla V_{ij}^\eta * u_{\eta ,j})(\bar{X}_{\eta ,i}(t),t)\mathrm{d}t + \sqrt{2\sigma _i}dW_i^k(t), \quad \bar{X}_{\eta ,i}^k(0)=\xi _i^k, \end{aligned}$$

has a unique strong solution $\bar{X}_{\eta ,i}^k$. By [6], the stochastic process $\bar{X}_{\eta ,i}^k$ is absolutely continuous with respect to the Lebesgue measure, thus there exists the probability density function $\bar{u}_{\eta ,i}$ of $\bar{X}_{\eta ,i}^k$, which is Lipschitz continuous. Let $\phi _i$ be a smooth test function. Then Itô’s lemma implies that

$$\begin{aligned}&\phi _i(\bar{X}_{\eta ,i}^k(t),t) - \phi _i(\xi _i^k,0) \\&\quad = \int \limits _0^t\partial _s\phi _i(\bar{X}_{\eta ,i}^k(s),s)\mathrm{d}s - \sum _{j=1}^n\int \limits _0^t(V_{ij}^\eta * \nabla u_{\eta ,j})(\bar{X}_{\eta ,i}^k(s),s) \cdot \nabla \phi _i(\bar{X}_{\eta ,i}^k(s),s)\mathrm{d}s \\&\qquad + \sigma _i\int \limits _0^t\Delta \phi _i(\bar{X}_{\eta ,i}^k(s),s)\mathrm{d}s + \sqrt{2\sigma _i}\int \limits _0^t\nabla \phi _i(\bar{X}_{\eta ,i}^k(s),s)dW_i^k(s). \end{aligned}$$

Applying the expectation

$$\begin{aligned} \mathbb {E}\big (\phi _i(\bar{X}_{\eta ,i}^k(t),t)\big ) = \int \limits _{\mathbb {R}^d}\phi _i(x,t)\bar{u}_{\eta ,i}(x,t)\mathrm{d}x \end{aligned}$$

to the previous expression yields

$$\begin{aligned} \int \limits _{\mathbb {R}^d}\phi _i(x,t)\bar{u}_{\eta ,i}(x,t)\mathrm{d}x&= \int \limits _{\mathbb {R}^d}\phi _i(x,0)\bar{u}_{\eta ,i}(x,0)\mathrm{d}x \\&\quad + \int \limits _0^t\int \limits _{\mathbb {R}^d}\big (\partial _s\phi _i(x,s) + \sigma _i\Delta \phi _i(x,s)\big ) \bar{u}_{i,\eta }(x,s)\mathrm{d}x\mathrm{d}s \\&\quad - \sum _{j=1}^n\int \limits _0^t\int \limits _{\mathbb {R}^d}\nabla \phi _i(x,s)\cdot (V_{ij}^\eta * \nabla u_{\eta ,j})(x,s) \bar{u}_{\eta ,i}(x,s)\mathrm{d}x\mathrm{d}s. \end{aligned}$$

This is the weak formulation of the linear problem

$$\begin{aligned} \partial _t \bar{u}_{\eta ,i} - \sigma _i\Delta \bar{u}_{\eta ,i} = {\text {div}}\bigg (\sum _{j=1}^n \bar{u}_{\eta ,i}V_{ij}^\eta * \nabla u_{\eta ,j}\bigg ), \quad u_{\eta ,i}(0)=u^0. \end{aligned}$$

The solution $\bar{u}_{\eta ,i}$ of this linear problem is Lipschitz continuous. Thus, by regularity theory [41, Theorem 1.3.11], it holds that $\bar{u}_{\eta ,i}\in L^\infty (0,T;H^s(\mathbb {R}^d))\cap L^2(0,T;H^{s+1}(\mathbb {R}^d))$. But since $u_{\eta ,i}$ is the unique solution to problem (4) in that solution class, this implies that $\bar{u}_{\eta ,i}=u_{\eta ,i}$. This finishes the proof. $\square$

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 $|\nabla u|\in L^\infty (0,\infty ;W^{1,\infty }(\mathbb {R}^d;\mathbb {R}^n))$. Then there exists a unique strong solution $\widehat{X}_i^k$ with probability density function $u_i$.

5 Proof of Theorem 3

First, we show an estimate for the difference $X_{\eta ,i}^{k,N}-\bar{X}_{\eta ,i}^k$.

Lemma 13

Let the assumptions of Theorem 3 hold. Then, for any $t>0$,

$$\begin{aligned} {\sup _{k=1,\ldots ,N}}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |X_{\eta ,i}^{k,N}(s) - \bar{X}_{\eta ,i}^k(s)\big |^2\bigg ) \le \frac{C(t)}{N^{1-C(t)\varepsilon }}, \end{aligned}$$

where the constant C(t) depends on t, n, $\Vert D^2 V_{ij}\Vert _{L^\infty }$, and the initial datum $u^0$.

Proof

We set for $k=1,\ldots ,N$,

$$\begin{aligned} {S^k_t = \sum _{i=1}^n\sup _{0<s<t} \big |X_{\eta ,i}^{k,N}(s) - \bar{X}_{\eta ,i}^k(s)\big |^2.} \end{aligned}$$

The difference of Eqs. (2) and (3), satisfied by $X_{\eta ,i}^{k,N}$ and $\bar{X}_{\eta ,i}^k$, respectively, equals

$$\begin{aligned}&X_{\eta ,i}^{k,N}(t) - \bar{X}_{\eta ,i}^k(t) \\&\quad = -\int \limits _0^t \sum _{j=1}^n\frac{1}{N}\sum _{\ell =1}^N\Big ( \nabla V_{ij}^\eta \big (X_{\eta ,i}^{k,N}(s) - X_{\eta ,j}^{\ell ,N}(s)\big ) - (\nabla V_{ij}^\eta * u_{\eta ,j})(\bar{X}_{\eta ,i}^k(s),s)\Big )\mathrm{d}s \end{aligned}$$

and thus,

$$\begin{aligned} \sum _{i=1}^n \big |X_{\eta ,i}^{k,N}(s) - \bar{X}_{\eta ,i}^k(s)\big |^2&\le \sum _{i=1}^n\bigg (\int \limits _0^t\frac{1}{N}\sum _{j=1}^n\bigg |\sum _{\ell =1}^N\Big ( \nabla V_{ij}^\eta \big (X_{\eta ,i}^{k,N}(s)-X_{\eta ,j}^{\ell ,N}(s)\big ) \\&\quad - (\nabla V_{ij}^\eta * u_{\eta ,j})(\bar{X}_{\eta ,i}^k(s),s)\Big )\bigg |\mathrm{d}s\bigg )^2. \end{aligned}$$

Taking the supremum in (0, t) and the expectation and using the Cauchy-Schwarz inequality with respect to t yields

$$\begin{aligned} \sup _{k=1,\ldots ,N}\mathbb {E}(S^k_t)&\le \sum _{i=1}^n\frac{t}{N^2} \int \limits _0^t\sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{j=1}^n \bigg |\sum _{\ell =1}^N\Big ( \nabla V_{ij}^\eta \big (X_{\eta ,i}^{k,N}(s)-X_{\eta ,j}^{\ell ,N}(s)\big ) \\&\quad - (\nabla V_{ij}^\eta * u_{\eta ,j})(\bar{X}_{\eta ,i}^k(s),s)\Big )\bigg |\mathrm{d}s\bigg )^2 \\&\quad \le C\sum _{i=1}^n\frac{t}{N^2}\int \limits _0^t\bigg \{\sup _{k=1,\ldots ,N} \mathbb {E}\sum _{j=1}^n \\&\qquad \times \bigg |\sum _{\ell =1}^N\Big ( \nabla V_{ij}^\eta \big (X_{\eta ,i}^{k,N}(s)-X_{\eta ,j}^{\ell ,N}(s)\big ) - \nabla V_{ij}^\eta \big (X_{\eta ,i}^{k,N}(s)-\bar{X}_{\eta ,j}^{\ell }(s)\big ) \Big )\bigg |^2 \\&\qquad + \sup _{k=1,\ldots ,N}\mathbb {E}\sum _{j=1}^n\bigg |\sum _{\ell =1}^N \Big (\nabla V_{ij}^\eta \big (X_{\eta ,i}^{k,N}(s)-\bar{X}_{\eta ,j}^{\ell }(s)\big ) - \nabla V_{ij}^\eta \big (\bar{X}_{\eta ,i}^{k}(s)-\bar{X}_{\eta ,j}^{\ell }(s)\big )\Big ) \bigg |^2 \\&\qquad + \sup _{k=1,\ldots ,N}\mathbb {E}\sum _{j=1}^n\bigg |\sum _{\ell =1}^N \Big (\nabla V_{ij}^\eta \big (\bar{X}_{\eta ,i}^{k}(s)-\bar{X}_{\eta ,j}^{\ell }(s)\big ) - (\nabla V_{ij}^\eta * u_{\eta ,j})(\bar{X}_{\eta ,i}^k(s),s)\Big ) \bigg |^2\bigg \}\mathrm{d}s \\&\quad =: J_1 + J_2 + J_3. \end{aligned}$$

We estimate the terms $J_1$, $J_2$, and $J_3$ separately.

Let $L_{ij}^\eta$ be the Lipschitz constant of $\nabla V_{ij}^\eta$. Because of $V_{ij}^\eta (z) = \eta ^{-d}V_{ij}(|z|/\eta )$, we obtain $L_{ij}^\eta = \eta ^{-d-2}\Vert D^2 V_{ij}\Vert _{L^\infty }$. Hence,

$$\begin{aligned} |J_1|&\le \sum _{i=1}^n\frac{t}{N^2}\int \limits _0^t\mathbb {E}\bigg (\sum _{j=1}^n(L_{ij}^\eta )^2 2N \sum _{\ell =1}^N\big |X_{\eta ,j}^{\ell ,N}(s)-\bar{X}_{\eta ,j}^\ell (s)\big |^2\bigg )\mathrm{d}s\\&\le 2t\sum _{i,j=1}^n(L_{ij}^\eta )^2\int \limits _0^t\frac{1}{N}\sum _{\ell =1}^N\mathbb {E}\bigg ( \big |X_{\eta ,j}^{\ell ,N}(s)-\bar{X}_{\eta ,j}^\ell (s)\big |^2\bigg )\mathrm{d}s \\&\le 2t\sum _{i,j=1}^n(L_{ij}^\eta )^2\int \limits _0^t\sup _{k=1,\ldots ,N}\mathbb {E}( S^k_s)\mathrm{d}s \\&\le \frac{2tn}{\eta ^{2d+4}}\sup _{i,j=1,\ldots ,n}\Vert D^2 V_{ij}\Vert _{L^\infty }^2 \int \limits _0^t\sup _{k=1,\ldots ,N}\mathbb {E}(S^k_s)\mathrm{d}s. \end{aligned}$$

Furthermore, by similar arguments,

$$\begin{aligned} |J_2|\le & {} \sum _{i=1}^n\frac{t}{N^2}\int \limits _0^t \sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{j=1}^n(L_{ij}^\eta )^2 \bigg (\sum _{\ell =1}^N \big |X_{\eta ,i}^{k,N}(s)-\bar{X}_{\eta ,j}^k(s)\big |\bigg )^2\bigg )\mathrm{d}s \\\le & {} 2t\sum _{i,j=1}^n(L_{ij}^\eta )^2\int \limits _0^t\sup _{k=1,\ldots ,N}\mathbb {E}\bigg ( \big |X_{\eta ,i}^{k,N}(s)-\bar{X}_{\eta ,j}^k(s)\big |^2\bigg )\mathrm{d}s \\\le & {} \frac{2tn}{\eta ^{2d+4}}\sup _{i,j=1,\ldots ,n}\Vert D^2V_{ij}\Vert _{L^\infty }^2 \int \limits _0^t \sup _{k=1,\ldots ,N}\mathbb {E}(S^k_s)\mathrm{d}s. \end{aligned}$$

For the third term, we set

$$\begin{aligned} Z_{i,j}^{k,\ell }(s) := \nabla V_{ij}^\eta \big (\bar{X}_{\eta ,i}^{k}(s)-\bar{X}_{\eta ,j}^{\ell }(s)\big ) - (\nabla V_{ij}^\eta * u_{\eta ,j})(\bar{X}_{\eta ,i}^k(s),s), \end{aligned}$$

and write the square as a product of two sums $Z_{i,j}^{k,1},\ldots ,Z_{i,j}^{k,N}$:

$$\begin{aligned} |J_3|&= \sum _{i=1}^n\frac{t}{N^2}\int \limits _0^t \sup _{k=1,\ldots ,N}\mathbb {E}\sum _{j=1}^n\bigg |\sum _{\ell =1}^N Z_{i,j}^{k,\ell }(s)\bigg |^2 \mathrm{d}s \\&= \sum _{i,j=1}^n\frac{t}{N^2}\int \limits _0^t\sup _{k=1,\ldots ,N}\mathbb {E}\bigg ( \sum _{\ell =1}^N Z_{i,j}^{k,\ell }(s)\cdot \sum _{m=1}^N Z_{i,j}^{k,m}(s)\bigg ) \mathrm{d}s \\&\le \sum _{i,j=1}^n\frac{t}{N^2}\sum _{\ell ,m=1}^N\int \limits _0^t\sup _{k=1,\ldots ,N} \mathbb {E}\bigg (Z_{i,j}^{k,\ell }(s)\cdot Z_{i,j}^{k,m}(s)\bigg )\mathrm{d}s \\&= \sum _{i,j=1}^n\frac{t}{N^2}\sum _{\ell =1}^N\int \limits _0^t\sup _{k=1,\ldots ,N} \mathbb {E}\big |Z_{i,j}^{k,\ell }(s)\big |^2 \mathrm{d}s \\&\quad + \sum _{i,j=1}^n\frac{t}{N^2}\sum _{\ell \ne m}\int \limits _0^t\sup _{k=1,\ldots ,N} {\mathbb {E}\big (Z_{i,j}^{k,\ell }(s) Z_{i,j}^{k,m}(s)\big )}\mathrm{d}s. \end{aligned}$$

We show that the last term vanishes. Indeed, since $\bar{X}^k_{\eta ,i}$, $\bar{X}^\ell _{\eta ,j}$, and $\bar{X}^m_{\eta ,j}$ are independent by construction (since we consider N independent copies) with distribution functions $u_{\eta ,i}$, $u_{\eta ,j}$, and $u_{\eta ,j}$, respectively, the joint distribution equals $u_{\eta ,i}\otimes u_{\eta ,j} \otimes u_{\eta ,j}$. This gives for $\ell \ne m$ and $i\ne j$ or, when $i=j$, $k\ne \ell , k \ne m$,

$$\begin{aligned} \mathbb {E}\big (Z_{i,j}^{k,\ell }(s) Z_{i,j}^{k,m}(s)\big )&= \int \limits _{\mathbb {R}^d}\int \limits _{\mathbb {R}^d}\int \limits _{\mathbb {R}^d}\nabla V_{ij}^\eta (\xi -x)\nabla V_{ij}^{\eta }(\xi -y) u_{\eta ,i}(\xi )u_{\eta ,j}(x)u_{\eta ,j}(y)\mathrm{d}x\mathrm{d}\xi \mathrm{d}y \\&\quad + \int \limits _{\mathbb {R}^d}\big (\nabla V_{ij}^{\eta }*u_{\eta ,j}(\xi )\big )^2 u_{\eta ,i}(\xi )d\xi \\&\quad - 2\int \limits _{\mathbb {R}^d}\int \limits _{\mathbb {R}^d}\nabla V_{ij}^\eta (\xi -x) \big (\nabla V_{ij}^{\eta }*u_{\eta ,j}(\xi )\big )u_{\eta ,i}(\xi )u_{\eta ,j}(x)\mathrm{d}x\mathrm{d}\xi = 0. \end{aligned}$$

If $\ell \ne m$, $i=j$, and $k=\ell$ or $k=m$, it holds by a similar calculation that $\mathbb {E}(Z_{i,j}^{k,\ell }(s) Z_{i,j}^{k,m}(s))=0$, and thus, $\mathbb {E}(Z_{i,j}^{k,\ell }(s) Z_{i,j}^{k,m}(s))=0$ for all $\ell \ne m$. Also the expectation of $Z_{i,j}^{k,\ell }$ vanishes since

$$\begin{aligned} \mathbb {E}\big (Z_{i,j}^{k,\ell }(s)\big )&= \int \limits _{\mathbb {R}^d}\int \limits _{\mathbb {R}^d}\big (\nabla V_{ij}^\eta (\xi -x) - (\nabla V_{ij}^\eta * u_{\eta ,j})(\xi ,s)\big )u_{\eta ,i}(\xi ,s)u_{\eta ,j}(x,s)\mathrm{d}x\mathrm{d}\xi \\&= \int \limits _{\mathbb {R}^d} u_{\eta ,i}(\xi ,s)\bigg (\int \limits _{\mathbb {R}^d} \nabla V_{ij}^\eta (\xi -x)u_{\eta ,j}(x,s) \mathrm{d}x - (\nabla V_{ij}^\eta * u_{\eta ,j})(\xi ,s)\bigg ) \mathrm{d}\xi = 0. \end{aligned}$$

We have shown that for $\ell =1,\ldots ,N$, $Z_{i,j}^{k,\ell }$ are uncorrelated. Therefore, using the estimates $|\nabla V_{ij}^\eta |\le C\eta ^{-d-1}$, $\Vert \nabla V_{ij}^\eta \Vert _{L^2}\le C\eta ^{-1}$, and consequently

$$\begin{aligned} \Vert \nabla V_{ij}^\eta * u_{\eta ,j}\Vert _{L^\infty } \le \Vert \nabla V_{ij}^\eta \Vert _{L^2}\Vert u_{\eta ,j}\Vert _{L^2} \le C\eta ^{-1}, \end{aligned}$$

we deduce that

$$\begin{aligned} |J_3| = \sum _{i,j=1}^n\frac{t}{N^2}\sum _{\ell =1}^N\int \limits _0^t\sup _{k=1,\ldots ,N} \mathbb {E}\big |Z_{i,j}^{k,\ell }(s)\big |^2 \mathrm{d}s \le \frac{n^2}{N}\frac{Ct^2}{\eta ^{d+2}}. \end{aligned}$$

Summarizing these estimations, we conclude that

$$\begin{aligned} {\sup _{k=1,\ldots ,N}}\mathbb {E}(S^k_t) \le 2\frac{tn}{\eta ^{2d+4}}\sup _{i,j=1,\ldots ,n}\Vert D^2 V_{ij}\Vert _{L^\infty }^2 \int \limits _0^t{\sup _{k=1,\ldots ,N}}\mathbb {E}(S^k_s)\mathrm{d}s + \frac{n^2}{N}\frac{Ct^2}{\eta ^{d+2}}, \end{aligned}$$

and, by Gronwall’s inequality,

$$\begin{aligned} {\sup _{k=1,\ldots ,N}}\mathbb {E}(S^k_t) \le \frac{Ct^2}{N\eta ^{d+2}}\exp \bigg (\frac{Ct}{\eta ^{2d+4}}\bigg ), \quad t>0. \end{aligned}$$

For fixed $\varepsilon \in (0,1)$ and $\eta \in (0,1)$, we choose $N\in \mathbb {N}$ such that $\varepsilon \log N\ge \eta ^{-2d-4}$. Using $\eta ^{-2d-2}\le \eta ^{-2d-4} \le \exp (C\eta ^{-2d-4})$ for $C\ge 1$, we obtain

$$\begin{aligned} \sup _{k=1,\ldots ,N}\mathbb {E}(S^k_t)&\le \frac{C}{N\eta ^{d+2}} \exp \bigg (\frac{Ct}{\eta ^{2d+4}}\bigg ) \le \frac{C}{N}\exp \bigg (\frac{C(1+t)}{\eta ^{2d+4}}\bigg ) \\&\le \frac{C}{N}\exp \big (C\varepsilon (1+t)\log N\big ) \le CN^{-1+C\varepsilon (1+t)}. \end{aligned}$$

This proves the result. $\square$

Next, we prove an estimate for the difference $\bar{X}_{\eta ,i}^k-\widehat{X}_i^k$.

Lemma 14

Let the assumptions of Theorem 3 hold. Then for $t>0$,

$$\begin{aligned} \mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t}\big |\bar{X}_{\eta ,i}^k(s) - \widehat{X}_i^k(s)\big |\bigg ) \le C(t)\eta , \quad \forall k=1,\ldots ,N. \end{aligned}$$

Proof

Since $\bar{X}_{\eta ,i}^k$ and $\widehat{X}_{\eta ,i}^k$ 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 by $\bar{X}_{\eta ,i}$ and $\widehat{X}_i$, respectively, and setting

$$\begin{aligned} U_t = \sum _{i=1}^n\sup _{0<s<t} \big |\bar{X}_{\eta ,i}(s)-\widehat{X}_i(s)\big |, \end{aligned}$$

if follows that

$$\begin{aligned} \mathbb {E}(U_t)&\le \sum _{i=1}^n\int \limits _0^t\mathbb {E}\bigg (\sum _{j=1}^n \Big |a_{ij}\nabla u_j(\widehat{X}_i(s),s) - (V_{ij}^\eta * \nabla u_{\eta ,j}) (\bar{X}_{\eta ,i}(s),s)\Big |\bigg )\mathrm{d}s \\&\le \sum _{i=1}^n\int \limits _0^t\mathbb {E}\bigg (\sum _{j=1}^n \Big |a_{ij}\nabla u_j(\widehat{X}_i(s),s) - (V_{ij}^\eta * \nabla u_{j}) (\widehat{X}_{i}(s),s)\Big |\bigg )\mathrm{d}s \\&\quad + \sum _{i=1}^n\int \limits _0^t\mathbb {E}\bigg (\sum _{j=1}^n \Big |(V_{ij}^\eta * \nabla u_{j})(\widehat{X}_{i}(s),s) - (V_{ij}^\eta * \nabla u_{\eta ,j})(\widehat{X}_{i}(s),s)\Big |\bigg )\mathrm{d}s \\&\quad + \sum _{i=1}^n\int \limits _0^t\mathbb {E}\bigg (\sum _{j=1}^n \Big |(V_{ij}^\eta * \nabla u_{\eta ,j})(\widehat{X}_{i}(s),s) - (V_{ij}^\eta * \nabla u_{\eta ,j})(\bar{X}_{\eta ,i}(s),s)\Big |\bigg )\mathrm{d}s \\&=: K_1 + K_2 + K_3. \end{aligned}$$

Using (15), the first two terms on the right-hand side are estimated according to

$$\begin{aligned} K_1&\le \sum _{i,j=1}^n\int \limits _0^t\int \limits _{\mathbb {R}^d}\big |\big (a_{ij}\nabla u_j(x,s) - (V_{ij}^\eta * \nabla u_j)(x,s)\big )u_i(x,s)\big |\mathrm{d}x\mathrm{d}s \\&\le n^2\max _{i,j=1,\ldots ,n}\big \Vert a_{ij}\nabla u_j-V_{ij}^\eta * \nabla u_j \big \Vert _{L^2(0,t;L^2)}\Vert u_i\Vert _{L^2(0,t;L^2)} \\&\le C(n)\eta \Vert D^2 u\Vert _{L^2(0,t;L^2)} \le C\eta , \\ K_2&\le \sum _{i,j=1}^n\int \limits _0^t\int \limits _{\mathbb {R}^d}\big |V_{ij}^\eta * \nabla (u_j-u_{\eta ,j}) u_i\big | \mathrm{d}x\mathrm{d}s \\&\le n^2\max _{i,j=1,\ldots ,n} \Vert V_{ij}^\eta \Vert _{L^1}\Vert \nabla (u_j-u_{\eta ,j})\Vert _{L^2(0,t;L^2)} \Vert u_i\Vert _{L^2(0,t;L^2)} \le C\eta , \end{aligned}$$

where we have used Lemma 16 (ii) and the error estimate from Lemma 13. Finally, the term $K_3$ can be controlled by

$$\begin{aligned} K_3&\le \sum _{i,j=1}^n \Vert \nabla (V_{ij,}^\eta * \nabla u_{\eta ,j})\Vert _{L^\infty } \int \limits _0^t\mathbb {E}(U_s)\mathrm{d}s \\&\le n^2\max _{i,j=1,\ldots ,n}\Vert V_{ij}^\eta \Vert _{L^1}\Vert D^2 u_\eta \Vert _{L^\infty } \int \limits _0^t\mathbb {E}(U_s)ds \le C\int \limits _0^t\mathbb {E}(U_s)\mathrm{d}s. \end{aligned}$$

We need the assumption $s>d/2+2$ for the continuous embedding $H^s(\mathbb {R}^d)\hookrightarrow W^{2,\infty }(\mathbb {R}^d)$, which allows us to estimate $D^2 u_\eta$ in $L^\infty (\mathbb {R}^d)$. This shows that

$$\begin{aligned} \mathbb {E}(U_t) \le C\eta + C\int \limits _0^t\mathbb {E}(U_s)\mathrm{d}s, \end{aligned}$$

and Gronwall’s inequality yields $\mathbb {E}(U_t)\le C(t)\eta$ for $t>0$. The statement of the lemma follows after taking the supremum over $t>0$. $\square$

Lemmas 13 and 14 imply Theorem 3. Indeed, it follows that

$$\begin{aligned}&\sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |X_{\eta ,i}^{k,N}(s) - \widehat{X}_i^k(s)\big |\bigg ) \\&\quad \le \sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |X_{\eta ,i}^{k,N}(s) - \bar{X}_{\eta ,i}^k(s)\big |\bigg ) \\&\qquad + \sup _{k=1,\ldots ,N}\mathbb {E}\bigg (\sum _{i=1}^n\sup _{0<s<t} \big |\bar{X}_{\eta ,i}^k(s) - \widehat{X}_i^k(s)\big |\bigg ) \\&\quad \le CN^{(-1+C(t)\varepsilon )/2} + C(t)\eta \le C(t)\eta , \end{aligned}$$

since the choice $\log N\ge 1/(\varepsilon \eta ^{2d+4})$ is equivalent to

$$\begin{aligned} N^{(-1+C(t)\varepsilon )/2} \le \exp \bigg (\frac{1}{2\varepsilon }(-1+C(t)\varepsilon )\eta ^{-2d-4}\bigg ), \end{aligned}$$

and the right-hand side is smaller than $\eta$ possibly times a constant C(t).

Appendix A. Some auxiliary results

We recall some auxiliary results.

Lemma 15

(Moser-type calculus inequality; Prop. 2.1 in [25]) Let f, $g\in H^s(\mathbb {R}^d)$ with $s>d/2+1$ and let $\alpha \in \mathbb {N}^d$ be a multi-index of order $|\alpha |\le s$. Then, for some constant $C_M>0$ only depending on s and d,

$$\begin{aligned} \Vert D^\alpha (fg)\Vert _{L^2} \le C_M\big (\Vert f\Vert _{L^\infty }\Vert D^s g\Vert _{L^2} + \Vert g\Vert _{L^\infty }\Vert D^s f\Vert _{L^2}\big ). \end{aligned}$$

Lemma 16

(Young’s convolution inequality; formula (7) on page 107 in [23]) (i) Let $1\le p, q\le \infty$ satisfying $1/p+1/q=1/r+1$ and $f\in L^p(\mathbb {R}^d)$, $g\in L^q(\mathbb {R}^d)$. Then

$$\begin{aligned} \Vert f*g\Vert _{L^r} \le \Vert f\Vert _{L^p}\Vert g\Vert _{L^q}. \end{aligned}$$

(ii) Let p, q, $r\ge 1$ satisfying $1/p+1/q+1/r=2$ and $f\in L^p(\mathbb {R}^d)$, $g\in L^q(\mathbb {R}^d)$, $h\in L^r(\mathbb {R}^d)$. Then, for some constant $C>0$ only depending on p, q, r, and d,

$$\begin{aligned} \bigg |\int \limits _{\mathbb {R}^d}(f * g)h \mathrm{d}x\bigg | \le C\Vert f\Vert _{L^p}\Vert g\Vert _{L^q}\Vert h\Vert _{L^r}. \end{aligned}$$

Lemma 17

(Gronwall-type inequality) Let a, $b>0$, $g\in C^0([0,T])$ with $g(t)\ge 0$ for $t\in [0,T]$ and $f:[0,T]\rightarrow [0,\infty )$ be absolutely continuous such that

$$\begin{aligned} f'(t) \le -g(t)\big (a-b\sqrt{f(t)}\big )\quad \text{ for } t>0 \end{aligned}$$

and $0<f(0) \le (a/b)^2$. Then $f(t) \le (a/b)^2$ for all $t\in [0,T]$. If $g:[0,T]\rightarrow [0,\infty )$ is any function, we still have the strict inequality: $f(t) < (a/b)^2$ for all $t\in [0,T]$.

We present a proof of this lemma since we could not find a reference in the literature.

Proof

First, let $f(0) < (a/b)^2$. We claim that $f(t)<(a/b)^2$ for $t\in [0,T]$. Assume that there exists $t_0\in [0,T]$ such that $f(t_0)\ge (a/b)^2$. By continuity, there exists $t_1\in [0,t_0]$ such that $f(t_1)<(a/b)^2$ and $f'(t_1) > 0$. This leads to the contradiction $0 < f'(t_1) \le -g(t_1)(a-b\sqrt{f(t_1)}) \le 0$, proving the claim.

Since $f(t)<(a/b)^2$ for $t\in [0,T]$, the differential inequality can be written as

$$\begin{aligned} \frac{f'(t)}{a-b\sqrt{f(t)}} \le -g(t), \quad t\in [0,T]. \end{aligned}$$

(16)

Introduce

$$\begin{aligned} F(s) = \int \limits _0^s\frac{d\sigma }{a-b\sqrt{\sigma }} = \int \limits _0^{\sqrt{s}}\frac{2\tau \mathrm{d}\tau }{a-b\tau } = -\frac{2}{b^2}\big (b\sqrt{s} + a\log (a-b\sqrt{s}) - a\ln a\big ). \end{aligned}$$

Then integrating (16) over (0, t) leads to

$$\begin{aligned} F(f(t)) - F(f(0)) \le -\int \limits _0^t g(\tau )\mathrm{d}\tau , \end{aligned}$$

which, after a computation, is equivalent to

$$\begin{aligned} a - b\sqrt{f(t)} \ge \big (a-b\sqrt{f(0)}\big )\exp \bigg (\frac{b}{a}\sqrt{f(0)} - \frac{b}{a}\sqrt{f(t)} + \frac{b^2}{2a}\int \limits _0^t g(\tau )\mathrm{d}\tau \bigg ). \end{aligned}$$

Finally, we choose a sequence of initial data $f_0^\delta < (a/b)^2$ such that $f_0^\delta \rightarrow f_0\le (a/b)^2$ as $\delta \rightarrow 0$. To each $f_0^\delta$, we associate a function $f^\delta$ satisfying the differential inequality. The proof shows that $f^\delta (t)<(a/b)^2$. In the limit $\delta \rightarrow 0$, this reduces to $f(t)\le (a/b)^2$, where $f(t)=\lim _{\delta \rightarrow 0}f^\delta (t)$ for $t\in [0,T]$. $\square$