Propagation of chaos for the Keller-Segel equation over bounded domains
Razvan C. Fetecau Hui Huang Weiran Sun

Propagation of Chaos]Propagation of chaos for the Keller-Segel equation over bounded domains
[
    Etsuji Yamamoto^1
    December 30, 2023
=====================


In this paper we rigorously justify the propagation of chaos for the parabolic-elliptic Keller-Segel equation  over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller-Segel equation for bounded weak solutions.  


Keywords:
	Coupling method, chemotaxis, Newtonian potential, mean-field limit, uniqueness, Wasserstein metric.




§ INTRODUCTION

In biology, chemotaxis describes the phenomenon that cells and organisms direct their movement in response to chemical gradients. This mechanism enables bacteria such as Escherichia coli (E. coli)  to swim toward the highest concentration of food  or flee from poisons. It also facilitates the movement of sperm towards the egg during fertilization.




Mathematically, one of the most classical models for studying chemotaxis is the Keller-Segel (KS)-type equations, introduced in <cit.> to describe aggregation of the slime mold amoebae. 

There exists a vast literature on the well-posedness and mean-field limits of Keller-Segel type equations  posed over the whole space ^d (see for example <cit.>). Realistic physical settings in biological experiments involve however bounded domains (for example, cells such as bacteria or slime mold are cultured in Petri dishes which have physical boundaries). This motivates the work in the present paper, which is devoted to a Keller-Segel type equation over bounded domains.



Our specific goal in this paper is to justify the mean-field limit leading to the parabolic-elliptic Keller-Segel model in a bounded convex domain D. Specifically, let ρ_t be the macroscopic density and F be the interaction force among particles. Then the KS equation under investigation has the form

    ∂_tρ_t =νρ_t-∇·[ρ_t F∗ρ_t]  , 
          x∈ D ,    t>0 ,
    ρ_t(x)|_t=0=ρ_0(x) ,
         x∈ D  , 
    ⟨ν∇ρ_t-ρ_t F∗ρ_t,n⟩=0  ,
         x ∈∂ D  ,

where ·· denotes the dot product in ^d (d≥ 2), n is the normal direction to ∂ D, and the convolution is defined as

    F ∗ρ_t (x) = ∫_D F(x - y) ρ_t (y)  ,
        for any x ∈ D .

The particular F considered in this paper is the negative gradient of the Newtonian potential given by

    F(x) = -∇Φ(x)
    =-C_*x/|x|^d ,
      ∀  x∈ℝ^d\{0} ,    d≥2 ,
 
with C_*=Γ(d/2)/2π^d/2 and

    Φ(x)
    =1/2πln|x|  ,     if  d=2 ,
    
    -C_d /|x|^d-2 ,      if  d≥3 .

Here, C_d=1/d(d-2)α_d with α_d=π^d/2/Γ(d/2+1) denoting the volume of d-dimensional unit ball.  





Equation (<ref>) is also commonly written as a system of equations for the density ρ_t and the concentration c = - Φ∗ρ_t of the chemo-attractant. The main mechanism driving the solutions of  (<ref>) is the competition between the diffusion and the nonlocal aggregation. We refer the reader to the review paper <cit.>  or Chapter 5 in <cit.> for more discussions of this model.

The main purpose of this paper is to justify the propagation of chaos for the KS system (<ref>). To do this, we model the dynamics of the particles by using equations on three levels: a continuity equation on the macroscopic level (<ref>),  a stochastic equation on the mesoscopic level  and a stochastic interacting particle system on the microscopic level. 




The methodology in this paper is to approach the PDE model (<ref>) from its underlying microscopic stochastic particle system. Specifically, let (Ω,ℱ,(ℱ_t)_t≥0,) be a probability space endowed with a filtration (ℱ_t)_t≥0 and {B_t^i}_i=1^N be N independent d-dimensional Brownian motions on this probability space. The system of interacting particles underlying the KS equation (<ref>) is given by:

    X_t^i=X^i_0+1/N-1∑_j≠ i^N ∫_0^tF(X_s^i-X_s^j) +√(2ν)B_t^i-R_t^i ,   i=1,⋯, N,    t>0 ,
    
    
    R_t^i=∫_0^tn(X_s^i)   d|R^i|_s ,   |R^i|_t=∫_0^t1_∂ D(X_s^i)   d|R^i|_s  , 
        
    X^i_t |_t = 0 = X^i_0 ,

where {X^i_t}_i=1^N are the trajectories of N particles and F given by (<ref>) models pairwise interaction between the particles. We assume that the initial data {X^i_0}_i=1^N are independently, identically distributed (i.i.d.) random variables with a common probability density function ρ_0(x) and √(2ν) is a constant.
Also, n(x) denotes the outward normal to ∂ D at the point x∈∂ D and R_t^i is a reflecting process associated to X_t^i  with a bounded total variation. Moreover, |R|_t^i is the total variation of R_t^i on [0,t], namely

    |R|_t^i=sup∑_k|R_t_k^i-R_t_k-1^i| ,

where the supremum is taken over all partitions such that 0=t_0<t_1<⋯<t_n=t.

Solving the stochastic differential equation (SDE) (<ref>) is known as the Skorokhod problem. Skorokhod introduced this problem in 1961 <cit.>, where he considered a SDE with a reflecting diffusion process on D=(0,∞). The multi-dimensional version of Skorokhod's problem was solved by Tanaka <cit.>, where the domain D was assumed to be convex. In <cit.> Lions and Sznitman extended the result to a general domain satisfying certain admissibility conditions, where the admissibility roughly means that the domain can be approximated by smooth domains in a certain sense. Later, Saisho <cit.> removed the admissibility condition of the domain by applying the techniques used in <cit.>.


The analysis of the limiting process of interacting particle systems as N →∞ is usually called the mean-field limit. 
There is extensive literature <cit.> on mean-field limits for various models when the spatial domain is the whole space ^d. For second order systems,  mean-field limits with globally Lipschitz forces in the whole space ^d were obtained by Braun and Hepp <cit.> and Dobrushin <cit.>. Later, in <cit.>,  these results were extended to particle systems with locally Lipschitz interacting forces under certain moment control assumptions. The whole space case with a singular force was treated in <cit.>. 
In particular, the mean-field limits for the  KS system (<ref>) in the whole space were shown in <cit.>. 


Meanwhile, there is much less work on mean-field limits for systems over bounded domains. In <cit.>, the author derived rigorously the mean-field limit for particle systems with reflecting boundary conditions and bounded Lipschitz interaction forces. 

In Carrillo et al. <cit.>, the authors considered a model in general domains (which is not necessarily convex ) with a λ-geodesically convex potential. 

Very recently Choi and Salem <cit.> proved the mean-field limit for the case where the particle system has a particular type of bounded discontinuous interacting force. 

The intermediate model between the microscopic model (<ref>) and the macroscopic equation (<ref>) is the following stochastic equation for the mean-field self-consistent stochastic process  (Y_t)_t≥0:

    {[                Y_t=Y_0+∫_0^t∫_DF(Y_s-y)ρ_s(y) +√(2ν)B_t-R̃_t ,    t>0 ,; [3pt]
    R̃_t=∫_0^tn(Y_s)   d|R̃|_s ,   |R̃|_t=∫_0^t1_∂ D(Y_s)   d|R̃|_s ,;                                                                         ].

where (ρ_t)_t≥0 is the marginal density of (Y_t)_t≥0 for any t≥ 0 and Y_0 has density ρ_0(x). The solution of (<ref>) is considered in the strong sense when the Brownian motion B_t is prescribed.  By Itô's formula <cit.>, it can be shown that the probability density function ρ_t of Y_t satisfies the KS equation (<ref>) in the weak sense (see Definition <ref> below).

The goal of the present paper is to justify the mean-field limit from the particle system (<ref>) with the Newtonian force (<ref>) to the mean-field KS equation (<ref>) (see Theorem <ref>).  
More precisely, we show an explicit convergence rate from the weak solution of the mean-field equation (<ref>) to the empirical measure associated with the particle system (<ref>). The main idea is to introduce an approximate mean-field dynamics {Y_t^i}_i=1^N satisfying (<ref>) and show that it well approximates the interacting particles {X_t^i}_i=1^N with a regularized F for large N's. Our main contribution is to extend the propagation of chaos posed on the whole space with a singular force <cit.> or posed on bounded domains with bounded forces <cit.> to bounded convex domains with a singular force. We also show a universal upper bound of the expectation of the collision time in ^2  (see Theorem <ref>).


Before proving the propagation of chaos,  we need to establish the well-posedness of solutions for the KS equation (<ref>) and the stochastic equation (<ref>). 
There is a huge literature on the well-posedness of the KS equation (see for example <cit.> for the existence and to <cit.> for the uniqueness).  There are also results for variants of the KS equation set on bounded domains with no-flux boundary conditions (see for example <cit.>).
In this paper, we use a coupling method to justify the well-posedness of equations (<ref>) and (<ref>). This coupling method has been used to show the well-posedness for the KS equation in the whole space in <cit.>. Its main idea is to make use of the link between the KS equation (<ref>) and the stochastic equation (<ref>) driven by a Brownian motion. Our main contribution in this part is to extend such coupling method from ^d to the bounded domain case.











The layout of the paper is as follows: in Section <ref> we explain in detail the settings of our equations and the precise statements of the main theorems. In Section <ref>, we prove some technical lemmas that are used in the proofs. In Section <ref>, we establish existence, uniqueness, and stability of the KS equation (<ref>) and the stochastic equation (<ref>). Finally, we show the finite-time collision and propagation of chaos in Section <ref>. 































§ SETTINGS AND MAIN RESULTS
 







In this section we explain the basic setting and main results in this paper.
Our main approach in investigating the mean-field limit with a singular potential is to apply a regularization mechanism <cit.>.  To this end, let J(x) be a blob function such that

    J ∈ C^∞(ℝ^d)  ,
      
    J ≥ 0  ,
       J(x)⊂B(0, 1) ,
      ∫_B(0, 1)J(x) =1.

For any > 0, define J_ε(x)=1/ε^dJ(x/ε) and

    F_ε(x)=J_ε∗ F(x)  .

The regularized version of (<ref>) has the form

    X_t^i,ε=X^i_0+1/N-1∑_j≠ i^N ∫_0^tF_ε(X_s^i,ε-X_s^j,ε) +√(2ν)B_t^i,ε-R_t^i,ε ,   i=1,⋯, N,    t>0 , 
    
    R_t^i,ε=∫_0^tn(X_t^i,ε)   d |R^i,ε|_s ,   |R^i,ε|_t=∫_0^t1_∂ D(X_s^i,ε)   d|R^i,ε|_s  ,

Here we assume that the initial data {X^i_0}_i=1^N are i.i.d. random variables with a common probability density function ρ_0(x).

The global well-posedness of (<ref>) is shown in <cit.>, which states 

	Let N≥ 2. For any initial data (X_0^i)_i=1,⋯,N∈D^N and for any T>0, there exists at least one weak solution (X_t^i,ε,R_t^i,ε,B_t^i,ε)_i=1,⋯,N to system (<ref>) on the time interval [0,T].



	Note that the regularized system (<ref>) is the same as the original system (<ref>) before the collision time τ defined in Theorem <ref>.



Next, we give the definition of weak solutions to the KS equation (<ref>).
 (Weak solutions)
	Assume D⊂^d is a bounded convex domain with  boundary D ∈ C^1.
	Let T > 0 and 0≤ρ_0(x)∈  L^∞(D). A probability density function ρ_t(x)∈ L^∞(0, T; L^∞(D)) is a weak solution to (<ref>) if it holds that
	
    ∫_Dρ_t(x)ϕ(x)    =∫_Dρ_0(x)ϕ(x) +ν∫_0^t∫_DΔϕ(x)ρ_s(x) 
           
    	+∫_0^t∫_Dρ_s(x)F∗ρ_s(x)·∇ϕ(x)   ,

	for any test function ϕ∈C^∞(D̅) satisfying ⟨∇ϕ,n ⟩=0 on ∂ D.



In our first main theorems we establish the well-posedness of (<ref>) and (<ref>) using the coupling method.
(Existence)
	Let Y_0 be a random variable with density ρ_0∈ L^∞(D). Suppose the singular interaction force F is given by (<ref>). Then there exists T > 0 such that equation (<ref>) has a strong solution (Y_t,R̃_t) and equation (<ref>) with initial data ρ_0 has a weak solution ρ_t∈ L^∞(0, T; L^∞(D)). Moreover, ρ_t is the density of Y_t up to time T. 
 

One main tool we use in establishing the above well-posedeness result is the Wasserstein distance. For the convenience of the reader, we give a brief introduction on the topology of the p-Wasserstein space.  We refer readers to <cit.> for further background.  Let us denote all probability measures on D by 𝒫(D), then for p≥ 1, consider the space

    𝒫_p(D)={f∈𝒫(D): ∫_D|x|^p   df(x)<∞},

on which we will define our distances. 


	(p-Wasserstein  distance) Let p∈ℕ and f, g∈𝒫_p(D). The p-Wasserstein  distance between f and g is defined by
		
    W_p(f,g)
    		= inf_π∈Λ(f, g){∫_D× D|x-y |^p (x,y)}^1/p
    		= inf_X∼ f,Y∼ g{[|X-Y|^p]}^1/p.

		where Λ(f, g) is the set of joint probability measures on D× D with marginals f and g respectively and (X,Y) are all possible couples of random variables with f and g as their respective distributions.  


(∞-Wasserstein distance) Let f, g∈𝒫_∞(D), then the  ∞-Wasserstein distance between f and g is defined by
		
    W_∞(f,g):=inf_X∼ f,Y∼ g|X-Y|=inf_π∈Λ(f,   g)π_(x,y)∈ D× D|x-y|  ,
	
		where random variables X,  Y and joint distribution Λ(f, g) are used in Definition <ref>. Here
		
    |X-Y|:=inf{λ≥ 0:  (|X-Y|>λ)=0} ,

		and
		
    π_(x,y)∈ D× D |x - y|:=inf{λ≥ 0:  π({(x,y)∈ D× D: |x-y|> λ})=0} .


The definitions of -ess sup and π will be used in the proof of Lemma <ref>. As a slight abuse of notation, we also use W_p(ρ_1,ρ_2) to denote the Wasserstein distance  of two measures  f,g with densities ρ_1,ρ_2 respectively. In <cit.>, it has been shown that for any 1 ≤ p <∞, the space 𝒫_p(D) endowed with the metric W_p is a complete metric space.  

Using the Wasserstein distance, we will establish the Dobrushin's type stability for (<ref>) and (<ref>). The result is given by the following theorem.

(Stability and Uniqueness)
	(1) If {Y_t^i}_i=1,2 are two strong solutions to (<ref>) with initial data {Y_0^i}_i=1,2 and densities {ρ_t^i}_i=1,2 respectively. Then there exist some constants C_1, C_2 depending only on D, T, ρ^1_t_L^∞(0, T; L^∞(D)) and ρ^2_t_L^∞(0, T; L^∞(D)) such that 
	
    sup_t∈[0,T]|Y^1_t-Y^2_t| 
    	≤ 
    	C_1 max{|Y^1_0-Y^2_0|,     (|Y^1_0-Y^2_0|)^exp(-C_2T)} .

	
	(2) Let {ρ^i_t}_i=1,2∈ L^∞(0, T; L^∞(D)) be two weak solutions to (<ref>) with initial data {ρ_0^i}_i=1,2 respectively.  Then there exist some constants C_1, C_2 depending only on D, T, ρ^1_t_L^∞(0, T; L^∞(D)) and ρ^2_t_L^∞(0, T; L^∞(D)) such that 
	
    sup_t∈[0,T]W_∞(ρ_t^1,ρ_t^2) 
    	≤ 
    	C_1 max{W_∞(ρ_0^1, ρ_0^2),     (W_∞(ρ_0^1, ρ_0^2))^exp(-C_2T)}.

	Uniqueness then follows from the stability estimates.


	This Dobrushin's type stability has been obtained for the KS equation in whole space <cit.>. In a domain with boundaries, one has to deal with an extra term resulting from the boundary effect. The key observation is that due to the convexity of the domain, this extra term possesses a good sign (see in (<ref>) and (<ref>)), and thus it does not add essential difficulties to the stability estimates. Such observation has been used in earlier works <cit.>.
	
	



Our second main theorem is to show the propagation of chaos and justify the mean-field limit of the regularized particle system (<ref>). First, we recall the definition of a chaotic sequence <cit.>:
(ρ-chaotic)
	Let E be a Polish space and ρ a probability measure on E. Let {X^i}_i=1^N be a sequence of exchangeable random variables. Then {X^i}_i=1^N is ρ-chaotic if 
	
    μ^N
    	:=1/N∑_i=1^Nδ_X^i→ρ    when   N→∞ ,

	that is, for any ϕ∈ C_b(D), 
	
    ∫_D ϕ^N →∫_Dϕρ    in law .


A sufficient condition for the sequence {X^i}_i=1^N to be ρ-chaotic is 
<cit.>
	Let ρ∈𝒫_p(E) with p ≥ 1. Let μ^N be the empirical measure defined in (<ref>). If 
	
    [W_p(μ^N,ρ)]→0N→∞ ,

	then the sequence {X^i}_i=1^N is ρ-chaotic.


The result of propagation of chaos states
(Propagation of chaos)
	Let ρ_t ∈ L^∞(0, T; L^∞(D)) be a weak solution to equation (<ref>) with initial data ρ_0∈ L^∞(D). Assume that {X_0^i}_i=1^N are N i.i.d. random variables with the common density ρ_0 and {X_t^i,ε}_i=1^N satisfy the particle system (<ref>) with the initial data {X_0^i}_i=1^N. Let μ^X, _t be the associated empirical measure given by
	
    μ_t^X,ε:=1/N∑_i=1^Nδ_X_t^i,ε .

	Then there exist a cut-off parameter ε∼log^-1/d(N) and  constants C_1,C_2, N_0>0 depending only on ρ_0_L^∞(D), D and T such that
	
    sup_t∈[0,T][W_2(μ_t^X,ε,ρ_t)]
    	≤ C_1[ω(log^-1/d(N))]^exp(-C_2T)     for all N≥ N_0 ,

	where ω(x) is defined in (<ref>).



	In the theorem above, we assume that the initial data {X_0^i}_i=1^N are i.i.d. random variables with the common density ρ_0. This is a stronger assumption than {X_0^i}^N_i=1 being ρ_0-chaotic. Our theorem shows that under this stronger assumption on the initial data, the system remains chaotic for a finite time. 





§ PRELIMINARIES
 
In this section we collect the technical lemmas that are used in the proofs of the main theorems. Throughout this paper, we will denote any generic constants as C, which may change from line to line. 
The notation f_L^p(D) represents the usual L^p(D)-norm for any 1≤ p≤∞. We may suppress the dependence on D when there is no confusion.  

The first lemma contains some useful properties of the regularized force F_, which states
 <cit.> 
	Let F_ be the regularized force defined in (<ref>). Then
	

	(a)
	F_ε(-x)=-F_ε(x), F_ε(x)=F(x) for any |x|≥ε, and |F_ε(x)|≤ |F(x)| for any x ∈^d.
	
	(b)
	|∂^β F_ε(x)|≤ C_β ε^1-d-|β| for any x∈ℝ^d and |β| ≥ 0.
	
	
	



An immediate consequence of Lemma <ref> is
 
	For any function ρ∈  L^∞(D), there exists a universal constant C that only depends on D such that for all ε≥ε^'≥0, 
	

	(a)
	F_ε∗ρ_L^∞(D)≤ Cρ_L^∞(D).
	
	(b)
	F_ε∗ρ-F_ε'∗ρ_L^∞(D)≤ C ερ_L^∞(D).


	(a). By Lemma <ref> (a), we have
	
    F_ε∗ρ≤∫_D F_ (x - y) ρ_L^∞(D)≤∫_DC_∗/|x - y|^d-1ρ_L^∞(D)≤
    	C ρ_L^∞(D) .

	
	
	
	(b). By Lemma <ref> (a) again, 
	we have
	
    |F_ε∗ρ(x)-F_ε'∗ρ(x)|
    	   ≤∫_D|F_ε(x-y)-F_ε^'(x-y)|ρ(y) 
       ≤ Cρ_L^∞(D)∫_{y:|x-y|≤ε}/|x-y|^d-1≤ C ερ_L^∞(D) ,

	where C is a generic constant. 


Using the above two lemmas, we can prove a crucial Quasi-Log-Lipschitz estimate for F_. 
 
	Let X_1, X_2 be two random variables with densities ρ_1, ρ_2 ∈  L^∞(D) respectively (X_1  and X_2 may not be independent). Suppose r_0 is large enough such that D ⊆ B(0, r_0). Then for any 0 ≤ε' ≤ε < 1, there exists a constant C depending only on D, ρ_1_ L^∞ and ρ_2_ L^∞ such that
	
    F_ε∗ρ_1(X_1)-F_ε'∗ρ_2(X_2)≤ 
    	C ω()
    	+ Cω|X_1-X_2| ,

	where
	
    ω(r)
    	=
    	1  ,     if  r≥ 1 ,
    
    	r(1-log r)  ,     if 0<r< 1 .



	Splitting the difference into two parts such that
	
    F_ε∗ρ_1(X_1)-F_ε'∗ρ_2(X_2)
    	   = F_ε∗ρ_1(X_1) - F_ε∗ρ_2(X_2)
    	+ F_ε∗ρ_2(X_2)-F_ε'∗ρ_2(X_2)
       =:I_1+I_2  ,

	we will derive the desired bound by bounding I_1, I_2 separately. The bound of I_2 is readily obtained by Lemma <ref> (b), which shows  that there exists a constant C depending only on ρ_2_ L^∞ such that
	
    |I_2|≤ C ε .

	
	The estimate of I_1 can	be obtained by following the approach in <cit.>.
	To this end, we introduce the joint distribution of X_1 and X_2 as π:=ℒ(X_1,X_2). Then 
	
    |I_1|= |∬_D× D F_ε(X_1-x_1)-F_ε(X_2-x_2)π(_1,_2)|.

	Let ℓ be the constant defined by
	
    ℓ:
    	=|X_1-X_2|
    	+π_(x_1,x_2)∈ D× D |x_1 - x_2|  .
 
	By their definitions in (<ref>) and (<ref>), we have
	
    |X_1-X_2|
    	=π_(x_1,x_2)∈ D× D |x_1 - x_2|  .

	Splitting |I_1| into two parts, we have
	
    ∬_D× D F_ε(X_1-x_1)-F_ε(X_2-x_2)π(_1, _2)
       ≤∬_({|X_1-x_1|≤ 2ℓ}∩ D)× D F_ε(X_1-x_1)-F_ε(X_2-x_2)π(_1, _2)
          
    	+ ∬_ ({|X_1-x_1|> 2ℓ}∩ D)× D F_ε(X_1-x_1)-F_ε(X_2-x_2)π(_1, _2)
        =: I_11 + I_12 .

	Note that over the integration domain of I_11, we have
	
    |X_2-x_2|
    	   ≤ 
    	|X_1-x_1|+|X_2-X_1+x_1-x_2| 
       ≤ 
    	2ℓ+|X_1-X_2|
    	+ π_(x,y)∈ D× D |x_1- x_2|
    	= 3ℓ .

	Therefore, I_11 satisfies
	
    I_11   ≤∫_|X_1-x_1|≤ 2ℓC/|X_1-x_1|^d-1ρ_1(x_1)_1
    	+ ∫_|X_2-x_2|≤ 3ℓC/|X_2-x_2|^d-1ρ_2(x_2)_2 
       ≤ 
    	C(ρ_1_∞+ρ_2_L^∞(D)) ℓ ,

	where we have applied the bound |F_ε(x)|≤ |F(x)|.
	
	To bound |I_12|, we separate the two cases where 2 ℓ > 2 and 2 ℓ≤ 2. First, if 2ℓ > 2, then X_1 - x_1 > 2 ℓ > 2. By the definition of ℓ we have
	
    ℓ≥ |X_1-x_1|-|X_2-x_2| >2ℓ- |X_2-x_2| ,

	which implies |X_2-x_2|>ℓ>2ε.
	Therefore, if 2 ℓ >, then
	
    F_ε(X_1-x_1)
    	= F(X_1-x_1),   F_ε(X_2-x_2)=F(X_2-x_2)  ,

	and
	
    F_ε(X_1-x_1) - F_ε(X_2-x_2)
    	≤C|X_2-X_1+x_1-x_2|/|X_1-x_1+ξ(X_2-X_1+x_1-x_2)|^d    for some ξ∈ [0, 1] .

	Note that
	
    |X_1-x_1+ξ(X_2-X_1+x_1-x_2)|  
    	≥    |X_1-x_1|-|X_2-X_1+x_1-x_2|
    	
    ≥   |X_1-x_1|-ℓ≥ |X_1-x_1|- 1/2 |X_1-x_1|=1/2 |X_1-x_1| .

	Therefore,
	
    |I_12| 
    	   ≤∬_{2ℓ< |X_1-x_1| ≤ r_0}× D |X_2-X_1+x_1-x_2|
    	C/|X_1-x_1+ξ(X_2-X_1+x_1-x_2)|^dπ(_1,_2)  
       ≤ Cℓ∫_2ℓ <|X_1-x_1|≤ r_02^d/|X_1-x_1|^dρ_1(x_1)_1
    	≤ Cρ_1_L^∞ℓlog(r_0/2ℓ) ,
    	    if 2 ℓ > 2 .

	Thus it follows from (<ref>) and (<ref>) that
	
    |I_1|
    	≤ C ℓ (1-log(ℓ)) ,
    	  if ε<ℓ<1 .

	
	
	In the case where 2 ℓ≤ 2, we further divide I_12 as
	
    I_12   ≤∬_ ({|X_1-x_1|> 2}∩ D)× D F_ε(X_1-x_1)-F_ε(X_2-x_2)π(_1, _2)
          
    	+ ∬_ ({2 ℓ≤ |X_1-x_1| ≤ 2}∩ D)× DF_ε(X_1-x_1)-F_ε(X_2-x_2)π(_1, _2) 
    	:= I^(1)_12 + I^(2)_12 .

	The first term I^(1)_12 is bounded in a similar as in the case 2 ℓ > 2, which gives
	
    I^(1)_12≤ 
    	C ρ_1_L^∞log(r_0/2)  .

	By Lemma <ref> (b), the second term I^(2)_12 satisfies
	
    I^(2)_12   ≤∬_ (|X_1-x_1| ≤ 2}∩ D)× DX_1 - X_2 - x_1 + x_2^-dπ(_1, _2)
       ≤
    	C ℓ^-d∫_|X_1 - x_1| ≤ 2 ρ(x_1) _1
    	≤
    	C ℓ .

	Combining (<ref>) and (<ref>), we have
	
    I_12≤ C ω()  ,
    	    if 2 ℓ≤ 2 .

	Collecting estimates (<ref>) and (<ref>), we derive that 
	
    |I_1|
    	≤ 
    	C ω() ,
    	  if 0<ℓ≤ε .

	Combing this with (<ref>), we concludes that
	
    |I_1|
    	≤ 
    	C ω() +Cω(ℓ) ,
    	  if 0<ℓ<1 .

	where C depends only on D, ρ_1_L^∞(D) and ρ_2_L^∞(D). For ℓ≥ 1, it follows from Lemma <ref> (i) that |I_1|≤ C. We thereby conclude our proof.
	
	
	
	
	
	
	
	
	
	
	
	
	
	



There are two Gronwall-type inequalities used in the later proofs. The first one is
 <cit.> 
	Let f be a nonnegative scalar function satisfying
	
    f(t)
    	≤ 
    	f_0exp(C∫_0^t f(s)),   t≥ 0  .

	Then 
	
    f(t)
    	≤f_0/1-Cf_0t ,   t≥ 0 ,

	where C is a positive constant.




The second Gronwall-type inequality states
 
	Let {f_ε(t)}_ε>0 be a family of nonnegative continuous functions  satisfying
	
    f_ε^2(t)
    	≤ 
    	C∫_0^tf_ε(s)ω(f_ε(s))+Cω(ε) T   t∈[0,T] ,

	where C>0 is a constant, T is independent of , and ω(x) is defined in (<ref>). Suppose  is small enough such that Cω(ε) T < 1.
	Then there exist two constants C_T and 0 <_0 < 1/2 depending only on T and C  such that if ω(ε)≤_0, then
	
    sup_t∈[0,T]f_ε(t)
    	≤ 
    	C_Tω(ε)^1/2e^-CT<1 .



	Let R_ be the continuous function defined by
	
    R_ε(t)
    	:= Cω(ε) T
    	+∫_0^t Cf_ε(s) ω(f_ε(s)) .

	Since R_ε(0)<1 and R_ε(t) is continuous in time, there  exists T^∗>0 such that R_ε(t) <1 for all t∈[0,T^∗). We will show that such T^∗ can be extended to T by choosing  small enough.
	
	Indeed, for ω(ε)< 1/2, define
	
    F(s)
    	:= exp(1-exp(-Cs))ω(ε)^1/2exp(-Cs)  and  
    	T_ε
    	:= 1/Clog(1-1/2log(ω(ε)))  .

	Then F is an increasing function and the particular choice of T_ satisfies 
	
    F(T_) = 1  ,
    	    
    	T_≥1/Clog1 + 1/2log2 > 0  ,
    	    ω(ε) < 1/2  .

	Now we show that 0 < R_ε(t)<1 for all t∈[0,T_ε). We prove this by contradiction. Assume there exists T_1<T_ε such that 
    R_ε(t) < 1
    	    for any t ∈ [0, T_1)  and  
    	R_ε(T_1) = 1  .

	Since ω(x) = x 1 - log x on (0, 1] and it is nondecreasing, by (<ref>) we have
	
    d R_ε(t)/
    	= Cf_ε(t)ω(f_ε(t))
    	≤ 
    	C(R_ε(t))^1/2ω((R_ε(t))^1/2)  ,

	which gives
	
    d/(R_ε(t))^1/2
    	= 1/2(R_ε(t))^-1/2 d R_ε(t)/≤C/2ω((R_ε(t))^1/2)≤ C(R_ε(t))^1/2(1-log((R_ε(t))^1/2)) .

	Solving this differential inequality on t ∈ [0, T_1], we get
	
    (R_ε(t))^1/2≤ 
    	e^1-(1-1/2log(Cω(ε)T_1))e^-CT_1≤exp(1-exp(-CT_1))ω(ε)^1/2exp(-CT_1) = F(T_1) .

	By the definitions and properties of F, T_1, T_, we have
	
    (R_ε(T_1))^1/2≤ F(T_1)
    	<    F(T_ε) = 1 ,

	which contradicts the assumption R_ε(T_1)=1. 
	Hence we have proved that  R_ε(t) < 1 for all t∈[0,T_ε) with T_ defined in (<ref>).  Notice that by the definition of T_, there exists _0 < 1/2 depending only on C and T such that T_ε>T for all ω(ε)≤_0. For such  we have 
	
    R_ε(t) <1     for all t∈ [0,T] .
 
	Then repeating the derivation of (<ref>), we obtain that
	
    f_ε(t)
    	≤ 
    	(R_ε(t))^1/2≤ F(T) 
    	≤ C_Tω(ε)^1/2e^-CT  t∈[0,T]  .






§ WELL-POSEDNESS OF THE NONLINEAR SDE AND PDE
 
In this section we establish the well-posedness of the nonlinear SDE (<ref>) and show that the marginal density of its solution is the unique weak solution to the KS equation (<ref>).


 §.§ Regularized system

The existence of solutions to (<ref>) and (<ref>) will be obtained through the convergence of approximate solutions to regularized versions of (<ref>) and (<ref>), where the singular force F is replaced by F_ defined in (<ref>). 




First, we show the well-posedness of the regularized PDE 

    ∂_tρ_t^ε=νρ_t^ε-∇·[ρ_t ^ε F_ε∗ρ_t^ε] ,     x∈ D ,    t>0 ,
    
    ρ_t^ε(x)|_t=0=ρ_0(x) ,    x∈ D  , 
    
    ⟨ν∇ρ_t^ε-ρ_t^ε F_ε∗ρ_t^ε,n⟩=0 ,    ∂ D .



	Suppose ρ_0 ∈ L^∞(D). Then there exists T > 0 independent of  such that equation (<ref>) has a weak solution ρ_t^ε∈ L^∞(0, T; L^∞(D)) with the bound
	
    sup_0≤ t≤ Tρ_t^ε_L^∞(D)≤ 
    	2 ρ_0_L^∞(D) .

	
	



	
	In order to show the existence of a weak solution to (<ref>), first we consider the regularized equation
	
    ∂_tρ_t^ε,δ
    	=νρ_t^ε,δ
    	- ∇·[ρ_t ^ε F_ε∗ρ_t^ε,δ]  ,     x∈ D ,    t>0 ,
    	
    ρ_t^ε,δ(x)|_t=0=ρ_0^δ(x)  ,     x∈ D  ,
    	
    ⟨ν∇ρ_t^ε,δ-ρ_t^ε,δ F_ε∗ρ_t^ε,δ,n⟩=0    ∂ D ,

	where ρ_0^δ∈ C^∞(D) satisfies
	
    ρ_0^δ_L^∞(D)≤ρ_0_L^∞(D) ,
    	    ρ_0^δ→ρ_0  L^1δ→ 0 .

	
	
	Then for any fixed , δ>0, there exists T_1 (which may depend on , δ) such that system (<ref>) has a smooth nonnegative solution ρ^, δ_t ∈ C([0, T_1); L^∞(D)) c.f. <cit.>. In order to remove the dependence of T_1 on , δ, we show (a priori) that there exist T, C > 0 which only depend on ρ_0_L^∞ and the domain D such that
	
    sup_t∈[0,T]ρ_t^ε,δ_L^∞(D)≤ C  .

	
	
	Indeed, for any p ≥ 2,  we have
	
    dρ_t^ε,δ_L^p(D)^p/   =p∫_D(ρ_t^ε,δ)^p-1(νΔρ_t^ε,δ-∇·(ρ_t^ε,δ F_ε∗ρ_t^ε,δ)) 
       =pν∫_∂ D(ρ_t^ε,δ)^p-1∇ρ_t^ε,δ· n   dS(x)-pν∫_D∇ρ_t^ε,δ·∇ ((ρ_t^ε,δ)^p-1) 
         
    	- p∫_∂ D(ρ_t^ε,δ)^p F_ε∗ρ_t^ε,δ· n   dS(x)+p∫_Dρ_t^ε,δ F_ε∗ρ_t^ε,δ·∇((ρ_t^ε,δ)^p-1) 
       =-p(p-1)ν∫_D(ρ_t^ε,δ)^p-2|∇ρ_t^ε,δ|^2 
    	+p∫_Dρ_t^ε,δ F_ε∗ρ_t^ε,δ·∇((ρ_t^ε,δ)^p-1)  .

	re-organize equation (<ref>) as
	
    dρ_t^ε,δ_L^p(D)^p/
    	+4ν(p-1)/p∇((ρ_t^ε,δ)^p/2)_L^2(D)^2
    	   =(p-1)∫_D F_ε∗ρ_t^ε,δ·∇((ρ_t^ε,δ)^p) 
       =2(p-1)∫_D (ρ_t^ε,δ)^p/2F_ε∗ρ_t^ε,δ·∇((ρ_t^ε,δ)^p/2)  .

	By (a) in Lemma <ref>, one has
	
    ∫_D (ρ_t^ε,δ)^p/2F_ε∗ρ_t^ε,δ·∇((ρ_t^ε,δ)^p/2) ≤ 
    	Cρ_t^ε,δ_L^∞(D)(ρ_t^ε,δ)^p/2_L^2(D)∇((ρ_t^ε,δ)^p/2)_L^2(D) .

	Applying Cauchy-Schwarz inequality in (<ref>), we obtain
	
    dρ_t^ε,δ_L^p(D)^p/
    	+4ν(p-1)/p∇((ρ_t^ε,δ)^p/2)_L^2(D)^2
    	≤2ν(p-1)/p∇((ρ_t^ε,δ)^p/2)_L^2(D)^2+Cpρ_t^ε,δ_L^∞(D)^2ρ_t^ε,δ_L^p(D)^p  ,

	
	
	
	
	where C is independent of ε, δ, and p. Gronwall's inequality then leads to
	
    ρ_t^ε,δ_L^p(D)≤ρ^δ_0_L^p(D)expC∫_0^tρ_s^ε_L^∞(D)^2     for any p ≥ 2 .

	Now we send p→∞ to get
	
    ρ_t^ε,δ_L^∞(D)≤ρ^δ_0_L^∞(D)exp(C∫_0^tρ_s^ε_L^∞(D)^2 )  .

	Finally, applying Lemma <ref> with f(t)=ρ_t^ε,δ_L^∞(D)^2 yields that
	
    ρ_t^ε,δ_L^∞(D)^2≤ρ_0^δ_L^∞(D)^2/1- 2Cρ_0^δ_L^∞(D)^2t≤ρ_0_L^∞(D)^2/1- 2Cρ_0_L^∞(D)^2t .

	Therefore, the uniform existence time T can be chose as
	
    T = 3/8 C ρ_0^2_L^∞(D) ,

	such that
	
    ρ^, δ_t_L^∞(D)≤ 
    	2 ρ_0_L^∞(D)    on [0, T) .

	
	
	
	
	
	Therefore there exists a ρ^_t ∈ L^∞(0, T; L^∞(D)) such that up to a subsequence, 
	
    ρ_t^ε,δ⇀^∗ρ_t^ε   L^∞(0, T; L^∞(D)) .

	
	Now we show that ρ_t^ε is a weak solution to (<ref>).  
	Notice that for any ϕ∈ C^∞(D) satisfying ⟨∇ϕ(x),n(x) ⟩=0 on ∂ D, we have
	
    ∫_Dρ_t^ε,δ(x)ϕ(x)     = ∫_Dρ_0^δ(x)ϕ(x) 
    	+ν∫_0^t∫_DΔϕ(x)ρ_s^ε,δ(x) 
          
    	+ ∫_0^t ∫_Dρ_s^ε,δ (x) F_ε∗ρ_s^ε,δ(x)·∇ϕ(x)  .

	Then by taking a continuous version of the left-hand side and (<ref>), we derive that 
	
    ∫_Dρ_t^ε,δ(x)ϕ(x)    →∫_Dρ_t^ε(x)ϕ(x)  ,
    	
    	
    	
    	
    	
    	    ∫_Dρ_0^δ(x)ϕ(x) →∫_Dρ_0(x)ϕ(x)  ,
    	
    ν∫_0^t∫_D   Δϕ(x)ρ_s^ε,δ(x) →ν∫_0^t∫_DΔϕ(x)ρ_s^ε(x)  ,

	as δ→ 0. For the last term on the right-hand side of (<ref>), we have
	
	
	
	
	
    |∫_0^t∫_Dρ_s^ε,δ(x)F_ε∗ρ_s^ε,δ(x)·∇ϕ(x) 
    	- ∫_0^t∫_Dρ_s^ε(x)F_ε∗ρ_s^ε(x)·∇ϕ(x) | 
    ≤   |∫_0^t∫_Dρ_s^ε,δ(x)F_ε∗ρ_s^ε,δ(x)·∇ϕ(x) 
    	- ∫_0^t∫_Dρ_s^ε,δ(x)F_ε∗ρ_s^ε(x)·∇ϕ(x) | 
        +|∫_0^t∫_Dρ_s^ε,δ(x)F_ε∗ρ_s^ε(x)·∇ϕ(x) - ∫_0^t∫_Dρ_s^ε(x)F_ε∗ρ_s^ε(x)·∇ϕ(x) | 
    ≤    
    	C∫_0^t∫_D |F_ε∗(ρ_s^ε,δ-ρ_s^ε)(x)| 
    	+ C |∫_0^t∫_D(ρ_s^ε,δ-ρ_s^ε)ϕ(x) | .

	By (<ref>), one has
	
    |∫_0^t∫_D(ρ_s^ε,δ-ρ_s^ε)ϕ(x) |→ 0 
    	  δ→0 .

	Notice that for any x∈ D, one has
	
    |F_ε∗(ρ_s^ε,δ-ρ_s^ε)(x)|≤∫_D F_ε(x-y)(ρ_s^ε,δ(y)-ρ_s^ε(y)) → 0
    	  δ→0 ,

	since F_ε∈ L^1(D). It follows from the dominated convergence theorem that
	
    ∫_0^t∫_D |F_ε∗(ρ_s^ε,δ-ρ_s^ε)(x)| → 0
    	  δ→0 .

	Therefore,
	
    ∫_0^t∫_Dρ_s^ε,δ(x)F_ε∗ρ_s^ε,δ(x)·∇ϕ(x) →∫_0^t∫_Dρ_s^ε(x)F_ε∗ρ_s^ε(x)·∇ϕ(x)   δ→ 0  .

	Combining the above limits, we have shown that ρ^_t is a weak solution to (<ref>) in the sense that
	
    ∫_Dρ_t^ε(x)ϕ(x)    =∫_Dρ_0(x)ϕ(x)+ν∫_0^t∫_DΔϕ(x)ρ_s^ε(x) 
           
    	+ ∫_0^t∫_Dρ_s^ε(x)F_ε∗ρ_s^ε(x)·∇ϕ(x)  .

	with the bound in (<ref>).












Next, we show that for any weak solution ρ_t^ε∈ L^∞(0, T; L^∞(D)) to (<ref>). One can construct a process Y_t^ε with ρ_t^ε being its density function, which satisfies regularized self-consistent SDE

    Y_t^ε
    =Y_0 + ∫_0^t∫_DF_ε(Y_s^ε-y)ρ_s^ε(y) 
    +√(2ν)B_t - R̃_t^ε,     t>0  ,
    
    R̃_t^ε
    = ∫_0^t n(Y_s^ε)   d |R̃^ε|_s  , 
      
    |R̃^ε|_t
    =∫_0^t1_∂ D(Y_s^ε)   d|R̃^ε|_s  ,

where we require Y_t^ε possessing a density function ρ_t^ε(x). The result states as follows












	If ρ_t^ε∈ L^∞(0, T; L^∞(D)) is a weak solution to (<ref>) with the initial data ρ_0∈ L^∞(D), then given any random variable Y_0 with the density ρ_0, there is a process Y_t^ε with the marginal density ρ_t^ε satisfying (<ref>) with the initial data Y_0.


	Consider the following linear PDE
	
    {[    ∂_tρ_t^ε=νρ_t^ε-∇·[ρ_t^ε V_ε [g_t]] ,    x∈ D ,    t>0 ,; ρ_t(x)|_t=0=ρ_0(x) , x∈ D ⟨ν∇ρ_t^ε-ρ_t V_ε [g_t] ,n⟩=0∂ D , ].

	where V _ε[g_t]=∫_DF_ε(x-y)g_t(y)dy, g_t(x)∈ L^∞(0, T; L^∞(D)) is a given function. According to Lemma <ref> (i), the term V_ε[g_t] is bounded, thus 
	the linear PDE (<ref>) has a unique weak solution in L^∞(0,T;L^1(D)). 
	
	
	
	
	
	
	Next let V_ε[ρ_t^ε]=∫_DF_ε(x-y)ρ_t^ε(y) dy, where ρ_t^ε∈ L^∞(0, T; L^∞(D)) is a weak solution to (<ref>).   According to <cit.>, the following SDE
	
    {[                  Y_t^ε=Y_0+∫_0^tV_ε[ρ_t^ε](Y_s^ε) +√(2ν)B_t-R̃_t^ε ,    t>0 ,; R̃_t^ε=∫_0^tn(Y_s^ε)   d|R̃^ε|_s ,   |R̃^ε|_t=∫_0^t1_∂ D(Y_s^ε)   d|R̃^ε|_s ,;                                                                               ].

	has a strong solution Y_t^ε with density ρ̅_t^ε since V_ε[ρ_t^ε] is smooth. 
	For ϕ∈ C^∞(D) satisfying ⟨∇ϕ(x),n(x) ⟩=0 for x∈∂ D, one applies Itô's formula:
	
    ϕ(Y_t^ε)   =ϕ(Y_0)+∫_0^t⟨∇ϕ(Y_s^ε),   d Y_s^ε⟩+1/2∫_0^t (d Y_s^ε)^T∇^2 ϕ(Y_s^ε)   d Y_s^ε
       =ϕ(Y_0)+∫_0^t⟨∇ϕ(Y_s^ε),V_ε[ρ_t^ε](Y_s^ε)⟩+√(2ν)∫_0^t⟨∇ϕ(Y_s^ε),   dB_s⟩
          -∫_0^t⟨∇ϕ(Y_s^ε),   dR̃_s^ε⟩ +ν∫_0^tΔϕ(Y_s^ε)  .

	Using the boundary condition, one has
	
    ∫_0^t⟨∇ϕ(Y_s^ε),   dR̃_s^ε⟩ =∫_0^t⟨∇ϕ(Y_s^ε),n(Y_s^ε)⟩  d|R̃^ε|_s=0 .

	Taking expectation on (<ref>), it follows from above that
	
    ∫_Dρ̅_t^ε(x)ϕ(x)    = ∫_Dρ_0(x)ϕ(x) 
    	+ν∫_0^t∫_DΔϕ(x)ρ̅_s^ε(x) 
    	+ ∫_0^t∫_Dρ̅_s^ε(x)V_ε[ρ_t^ε]·∇ϕ(x)  .

	This implies that ρ̅_t^ε is a weak solution to (<ref>) associated to V_ε[ρ_t^ε]. Notice that ρ_t^ε is also a weak solution to (<ref>) associated to V_ε[ρ_t^ε]. By the uniqueness of weak solution to (<ref>), we obtain ρ_t^ε=ρ̅_t^ε.  Hence we have proved that there exists a process Y_t^ε with density ρ_t^ε， which is a strong solution to the regularized mean-field dynamics (<ref>).
	
	
	
	




 §.§ The original system


Now we show the proof of Theorem <ref>, which asserts the existence of solutions to the SDE (<ref>) and the KS equation (<ref>).

 
	Note that the existence of weak solutions to the KS equation (<ref>) follows from applying the Itô's formula on SDE (<ref>). Thus we focus on proving the existence of a strong solution to the SDE (<ref>), which is obtained by first constructing approximate solutions to the regularized SDE (<ref>) and then passing to the limit. 
	
	
	
	
	To this end, we let ε>ε'>0 be arbitrary and F_, F_' be the corresponding regularized forces. 
	Denote  ρ_t^ε, ρ_t^ε' are two weak solutions to (<ref>) with the same initial condition ρ_0∈ L^∞(D). Then there exists T > 0 independent of , ' such that ρ_t^ε_L^∞(D) and ρ_t^ε'_L^∞(D) are uniformly bounded in ε and ε'  on [0, T].  According to Proposition <ref>, we can construct two processes Y^ε_t, Y^ε'_t satisfying (<ref>) with the same initial data Y_0 and Brownian motion B_t,  which have density functions ρ_t^ε and ρ_t^ε' respectively. 
	We shall show that there exist constants C, C_T, and ε_0 depending only on D, T and ρ_0_∞ such that if 0 < ' < ε< ε_0, then
	
    sup_t∈[0,T]𝒲_∞(ρ_t^ε,ρ_t^ε')
    	≤sup_t∈[0,T]|Y^ε_t-Y^ε'_t| ≤ C_Tω(ε)^1/2e^-CT .

	Indeed, by applying Itô's formula to Y^_t - Y^'_t, we have
	
    |Y_t^ε-Y_t^ε'|^2
    	   =2∫_0^t Y_s^ε-Y_s^ε',F_ε∗ρ_s^ε(Y_s^ε)-F_ε'∗ρ_s^ε'(Y_s^ε')
          
    	-2∫_0^t Y_s^ε-Y_s^ε',n(Y_s^ε) d|R̃^ε|_s-2∫_0^t
    	 Y_s^ε'-Y_s^ε,n(Y_s^ε') d |R̃^ε'|_s  .

	By the convexity of the domain D, one has
	
    (x-y)· n(x)≥ 0   x∈∂ D y∈D .

	Consequently, 
	
    Y_s^ε-Y_s^ε',n(Y_s^ε)≥ 0 ,
    	   d|R̃^ε|_s  ,

	and
	
    Y_s^ε'-Y_s^ε,n(Y_s^ε')≥ 0,
    	   d|R̃^ε'|_s  .

	Therefore, by Lemma <ref>, we have
	
    |Y_t^ε-Y_t^ε'|^2
    	   ≤ 
    	2∫_0^t⟨ Y_s^ε-Y_s^ε',F_ε∗ρ_s^ε(Y_s^ε)-F_ε'∗ρ_s^ε'(Y_s^ε')⟩
    	
    	
    	
    
    	   ≤ 2∫_0^t | Y_s^ε-Y_s^ε'| |F_ε∗ρ_s^ε(Y_s^ε)-F_ε'∗ρ_s^ε'(Y_s^ε')| 
       ≤ 2 ∫_0^t | Y_s^ε-Y_s^ε'| ( Cω(ε)+Cω(| Y_s^ε-Y_s^ε'| )) 
       ≤ Cω(ε) T+C∫_0^t | Y_s^ε-Y_s^ε'| ω(| Y_s^ε-Y_s^ε'| )  ,

	where C only depends on D, ρ_t^ε_∞,  and ρ_t^ε'_∞.
	Hence, by Lemma <ref>, there exist constants C_T, ε_0 depending only on C and T such that if ε< ε_0, then ω(ε)<_0 and the following holds
	
    sup_t∈[0,T]|Y^ε_t-Y^ε'_t| 
    	≤ 
    	C_Tω(ε)^1/2e^-CT,

	Therefore (<ref>) holds. Hence
	
	
	there exists a limiting stochastic process Y_t ∈ C([0,T],D) such that
	
    sup_t∈[0,T]|Y^ε_t-Y_t| →0
    	    ε→0  .

	Assume that Y_t has the marginal density ρ_t. Then by the bound
	
    sup_t∈[0,T]𝒲_∞(ρ_t^ε,ρ_t)≤sup_t∈[0,T]|Y^ε_t-Y_t|  ,

	we deduce that 
	
    sup_t∈[0,T]𝒲_∞(ρ_t^ε,ρ_t)→0 
    	    ε→0 ,

	and ρ_t∈ L^∞(0, T; L^∞(D)).
	
	
	We are left to show that there exists R_t such that (Y_t, R_t) is a strong solution to the SDE (<ref>). 
	
	By Lemma <ref> and (<ref>),
	
    |∫_0^t(F_ε∗ρ_s^ε(Y_s^ε)-F∗ρ_s(Y_s)) |
    	≤ 
    	C∫_0^tω(|Y^ε_s-Y_s|) + Cω(ε) t→0

	as → 0.
	
	
	
	
	
	
	
	
	
	
	We define the boundary process R̃_t as
	
    R̃_t
    	:= - Y_t-Y_0-∫_0^tF∗ρ_s(Y_s)-√(2ν) B_t  ,

	such that (Y_t, R_t) is the strong solution to the SDE (<ref>). The properties of R_t in (<ref>) are verified in the same way as in <cit.>, whose details are omitted here. 
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	


Similar to Proposition <ref>, we show that any weak solution ρ_t∈ L^∞(0, T; L^∞(D)) to (<ref>) can be represented as the family of time marginal density of a solution to (<ref>).

	If ρ_t∈ L^∞(0, T; L^∞(D)) is solution to (<ref>) with the initial data ρ_0∈ L^∞(D), then given any random variable Y_0 with the density ρ_0, there is a process Y_t with the marginal density ρ_t satisfying (<ref>) with the initial data Y_0.


	Proposition <ref> is a regularized version of this proposition, so the proof is almost the same. To this end, we change V_ε[g_t] into V [g_t]=∫_DF(x-y)g_t(y). The details are omitted here.




Using Proposition <ref>, we prove the stability estimates  in Theorem <ref>.


	
	
	
	First we prove stability of the nonlinear SDE (<ref>).
	Using similar arguments as in the proof of Theorem <ref>, we apply Itô's formula and get
	
    |Y_t^1-Y_t^2|^2   =|Y_0^1-Y_0^2|^2+2∫_0^t Y_s^1-Y_s^2,F∗ρ_s^1(Y_s^1)-F∗ρ_s^2(Y_s^2)
         
    	-2∫_0^t Y_s^1-Y_s^2,n(Y_s^1)  d|R̃^1|_s-2∫_0^t
    	 Y_s^2-Y_s^1,n(Y_s^2)  d|R̃^2|_s
       ≤ |Y_0^1-Y_0^2|^2+2∫_0^t Y_s^1-Y_s^2,F∗ρ_s^1(Y_s^1)-F∗ρ_s^2(Y_s^2) .

	Then it follows from Lemma <ref> that there exists a constant C depending only on ρ^1_L^∞(0,T; L^∞(D)) and ρ^2_L^∞(0,T; L^∞ (D))
	such that
	
    |Y_t^1-Y_t^2|^2≤ |Y_0^1-Y_0^2|^2+C∫_0^t| Y_s^1-Y_s^2|ω(| Y_s^1-Y_s^2| )  .

	
	The bound of |Y_t^1-Y_t^2| is derived by considering three possible cases:
	
	Case 1: There exists a small enough constant C_0(T) such that |Y^1_0-Y^2_0|≤ C_0(T). Here C_0(T) is determined by Lemma  <ref>. Actually, we can treat |Y_t^1-Y_t^2| as f_ε(t) in Lemma <ref> and |Y_0^1-Y_0^2|^2 as Cω(ε)T. Suppose that |Y^1_0-Y^2_0|<C_0(T)<1 sufficiently small. Then there exists two constants C_T and 0< _0<1/2 depending only on C and T such that if |Y_0^1-Y_0^2|^2<C_0(T)^2<_0, then
	
    |Y^1_t-Y^2_t|≤ C_T {|Y^1_0-Y^2_0|}^exp(-CT)<1   for  all  t∈ [0, T] .

	
	Case 2: If |Y^1_0-Y^2_0|≥ C_0(T) and  |Y^1_t-Y^2_t|≥ C_0(T) for any t∈ [0,T], then
	by the definition of ω(r),  one  obtains that there exists a constant C_1 depending only on C_0(T) such that 
	
    ω(|Y^1_t-Y^2_t|)≤ C_1 |Y^1_t-Y^2_t|  for  all  t∈ [0, T] .

	Thus it follows from (<ref>) that
	
    |Y^1_t-Y^2_t|^2 ≤ |Y^1_0-Y^2_0|^2+C_1C∫_0^t|Y^1_s-Y^2_s| |Y^1_s-Y^2_s| .

	Applying the Gronwall inequality, one has for any t∈ [0, T],
	
    |Y^1_t-Y^2_t|   ≤|Y^1_0-Y^2_0|(1+CC_1Te^CC_1T)^1/2 .

	
	Case 3: If |Y^1_0-Y^2_0|≥ C_0(T) and there exists a t_0∈ (0, T) such that |Y^1_t-Y^2_t|≥ C_0(T) for t∈ [0, t_0) and |Y^1_t_0-Y^2_t_0|= C_0(T).
	In this case, the estimate for the interval t∈ [0, t_0) is reduced to the Case 2 and one has
	
    |Y^1_t-Y^2_t|≤|Y^1_0-Y^2_0|(1+CC_1Te^CC_1T)^1/2.

	Over the interval [t_0,T], choosing t_0 as a new initial time and repeating the proof in Case 1 gives 
	
    |Y^1_t-Y^2_t|≤ C_T {|Y^1_t_0-Y^2_t_0|}^exp(-C(T-t_0))  for  all  t∈ [t_0, T] .

	By (<ref>) and the continuity of |Y^1_t-Y^2_t|, one has 
	
    |Y^1_t_0-Y^2_t_0|≤|Y^1_0-Y^2_0|(1+CC_1Te^CC_1T)^1/2 .

	Combining (<ref>), (<ref>) and (<ref>),  we obtain
	
    |Y^1_t-Y^2_t|
    	≤ 
    	C_2 max{|Y^1_0-Y^2_0|,   {|Y^1_0-Y^2_0|}^exp(-C_3T)} ,

	for any t∈ [0, T]. Combining (<ref>), (<ref>) and  (<ref>) finishes the proof of (<ref>). 
	
	
	
	
	Next we prove the stability of the KS equation (<ref>).
	Let ρ^1, ρ^2 ∈ L^∞(0, T; L^∞(D)) be two weak solutions to (<ref>) with the initial conditions ρ_0^1, ρ_0^2 ∈ L^∞(D) respectively. Take two initial random variables Y_0^1 and Y_0^2 such that
	
    𝒲_∞ (ρ_0^1, ρ_0^2)=|Y^1_0-Y^2_0|  .

	
	
	By Proposition <ref>, we build two self-consistent SDEs such that
	
    Y_t^i=Y_0^i+∫_0^t∫_DF(Y_s^i-y)ρ_s^i(y) +√(2ν)B_t-R̃_t^i ,    t>0, i=1,2,
    	
    R̃_t^i=∫_0^tn(Y_s^i)   d|R̃^i|_s ,
    	   
    	|R̃^i|_t=∫_0^t1_∂ D(Y_s^i)   d|R̃^i|_s  ,

	which have unique strong solutions (Y^i_t)_t≥0 with (ρ^i_t)_t≥0 being the marginal density of (Y^i_t)_t≥0. 
	Hence it follows from (<ref>) and (<ref>) that
	
    sup_t∈[0,T]𝒲_∞ (ρ_t^1,ρ_t^2)   ≤sup_t∈[0,T]|Y^1_t-Y^2_t|
       ≤ C_1max{|Y^1_0-Y^2_0|, {|Y^1_0-Y^2_0|}^exp(-C_2T)}
       = C_1 max{𝒲_∞ (ρ_0^1, ρ_0^2), {𝒲_∞ (ρ_0^1, ρ_0^2)}^exp(-C_2T)},

	which leads to the desired stability estimate (<ref>).



§ PROPAGATION OF CHAOS
 
In this section, we show an estimate of the collision time between particles and prove the propagation of chaos result stated in Theorem <ref>.



 §.§ Collision time between particles
 
It is known that in ^2, if 8πν<ρ_0_1=1, then solutions to the Keller-Segel equation (<ref>) concentrate within a finite time <cit.>. In this part, we show a similar bound for the expectation of the collision time for the interacting particle system (<ref>). 

	Assume d=2 and {X_0^i}_i=1^N be N i.i.d. random variables  with a common density ρ_0. Let {(X_t^i,R_t^i)}_i=1^N be weak solutions to (<ref>) with the initial data {X_0^i}_i=1^N. For any fixed T>0 and 8πν<1, we define
	
    A(t):=inf_s∈[0,t]min_i≠ j|X_s^i-X_s^j| ,
    	
    	
    	
    	    τ_ε
    	=   0         if ε≥ A(0)  ,
       sup{t∧ T: A(t)≥ε}       if ε< A(0)  ,

	where t ∧ T = min{t, T}. Let τ=lim_ε→ 0τ_ε. Then the expectation of the collision time satisfies
	
    (τ)≤2π[|X_0^1-X_0|^2]/1 -8πν ,
    	       X_0:=1/N∑_i=1^NX_0^i .

	

 Since system (<ref>) has a weak solution until the collision time τ 
	and F_ε(x)=F(x) for any |x|>ε, we know that X_t^i≡ X_t^i,ε and F_ε(x)=F(x) for t∈ [0,τ_ε], where {X_t^i,ε}_i=1^N is a global solution to the regularized particle system (<ref>). Let us denote 
    X_0:=1/N∑_i=1^NX_0^i .

	By the Itô's formula and (<ref>),  for t∈[0,τ_ε], one has
	
    |X^i,ε_t-X_0|^2
    	   = |X_0^i-X_0|^2+ 2∫_0^tX^i,ε_s-X_0,   1/N-1∑_j≠ i^N F(X^i,ε_s-X^j,ε_s)
          + 2∫_0^t X^i,ε_s-X_0,   √(2ν)  dB_s^i,ε
    	- 2∫_0^t X^i,ε_s-X_0,     dR_s^i,ε+4ν t 
       ≤ 	
    	|X_0^i-X_0|^2+ 2∫_0^tX^i,ε_s-X_0,   1/N-1∑_j≠ i^N F(X^i,ε_s-X^j,ε_s)
          + 2∫_0^t X^i,ε_s-X_0,   √(2ν)  dB_s^i,ε+4ν t  ,

	where we have used the convexity of the domain D such that
	
    ∫_0^t X^i,ε_s-X_0,   dR_s^i
    	= ∫_0^t X^i,ε_s-X_0,     n(X^i,ε_t)   d|R^i|_s ≥ 0 .
	
	Then summing (<ref>) for all i's, one has
	
    ∑_i=1^N|X^i,ε_t-X_0|^2
    	   ≤∑_i=1^N|X_0^i-X_0|^2+
    	2/N-1∑_  i, j =1  i ≠ j^N
    	∫_0^tX^i,ε_s -X_0,    F(X^i,ε_s-X^j,ε_s)
         +2√(2ν)∑_i=1^N∫_0^tX^i,ε_s-X_0,     dB_s^i,ε+4ν Nt .

	Note that 
    ∫^τ__0𝔼[|X^i,ε_t-X_0|^2] < ∞ ,
 
	since D is bounded, which implies that <cit.>
	
    𝔼[∫_0^tX^i,ε_t-X_0,    dB_t^i,ε]=0 .
	
	Also note that
	
    2/N-1∑_  i, j =1  i ≠ j^N X^i,ε_s - X_0, F(X^i,ε_s-X^j,ε_s)
    	= 1/N-1∑_  i, j 
    		=1  i ≠ j^N X^i,ε_s-X^j,ε_s,     F(X^i,ε_s-X^j,ε_s)
    	=-N/2π .

	Then we take expectation on (<ref>) and choose t=τ_ε. By the exchangeability of {(X^i,ε_t)_t≥0}_i=1^N, one has
	
    0≤𝔼[|X^i,ε_τ_ε-X_0|^2]
    	≤[|X_0^i-X_0|^2]+
    	4ν-1/2π𝔼[τ_ε]=[|X_0^1-X_0|^2]+
    	4ν-1/2π𝔼[τ_ε] .

	By the positivity of left hand and 8πν<1, we obtain
	
    𝔼[τ_ε]≤2π[|X_0^1-X_0|^2]/1 -8πν .

	Hence, the desired bound for τ holds by the monotone convergence theorem.





 §.§  Propagation of chaos and the proof of Theorem <ref>


In order to prove Theorem <ref>, we first show the following lemma:

	Let {X_t^i,ε}_i=1^N be N exchangeable random variables on D satisfying (<ref>). Let {Y_t^i,ε}_i=1^N be N i.i.d. random variables on D satisfying (<ref>) with the common density ρ_t^ε∈ L^∞(0,T;L^∞(D)). Then there exists a constant C>0 depending only on T, D and ρ^ε_L^∞(0,T;L^∞(D)) such that
	
    sup_i=1,⋯,N|1/N-1∑_j≠ i^NF_ε(X_t^i,ε-X_t^j,ε)-∫_DF_ε(Y_t^i,ε-y)ρ_t^ε(y)|≤  Cε^-dsup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|+ℋ_N ,

	where ℋ_N is given by
	
    ℋ_N:=sup_i=1,⋯,N|1/N-1∑_j≠ i^NF_ε(Y_t^i,ε-Y_t^j,ε)-∫_DF_ε(Y_t^i,ε-y)ρ_t^ε(y)| ,

	and it satisfies
	
    [|ℋ_N|^2]≤Cε^-2(d-1)/N-1 .



	For i=1, we split the error
	
    |1/N-1∑_j=2^NF_ε(X_t^1,ε-X_t^j,ε)-∫_DF_ε(Y_t^1,ε-y)ρ_t^ε(y)|
    ≤   |1/N-1∑_j=2^NF_ε(X_t^1,ε-X_t^j,ε)-1/N-1∑_j= 2^NF_ε(Y_t^1,ε-Y_t^j,ε)|
       +|1/N-1∑_j=2^NF_ε(Y_t^1,ε-Y_t^j,ε)-∫_DF_ε(Y_t^1,ε-y)ρ_t^ε(y)|
    	=:|I_1|+|I_2| .
	
	
	By Lemma <ref> (b) one has
	
    |I_1|≤1/N-1∑_j=2^N Cε^-d(|X_t^1,ε-Y_t^1,ε|+|X_t^j,ε-Y_t^j,ε|)=Cε^-d/N-1∑_j=1^N|X_t^j,ε-Y_t^j,ε|  .

	This shows
	
    |I_1|≤ Cε^-dsup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε| .

	Next, we rewrite I_2 as
	
    I_2=1/N-1∑_j=2^NZ_j ,

	with 
	
    Z_j:=F_ε(Y_t^1,ε-Y_t^j,ε)-∫_DF_ε(Y_t^1,ε-y)ρ_t^ε(y) ,
    	     j ≥ 2  .

	Since {Y_t^j,ε}_j=1^N are i.i.d. random variables, it holds that 
	
    [Z_jZ_k]=0  ,
    	     j ≠ k  .

	To see this, we consider Y_t^1,ε is given and denote 
	
    '[·]=[·| Y_t^1,ε] .

	Since Z_j and Z_k are independent for j ≠ k, it holds that '[Z_jZ_k]='[Z_j]'[Z_k]=0 when j≠  k. Hence
	
    [Z_jZ_k]=_1'[Z_jZ_k]=0 ,
    	     j ≠ k   ,

	where _1 means taking expectation on Y_t^1,ε.
	
	Hence, one concludes that
	
    [|I_2|^2]
    	   ≤1/N-1[|Z_2|^2] 
    	≤2/N-1[F_ε^2(Y_t^1,ε-Y_t^2,ε)+(∫_DF_ε(Y_t^1,ε-y)ρ_t^ε(y))^2]
    	
    	≤Cε^-2(d-1)/N-1 ,

	where we have used Lemma <ref> (b) and Lemma <ref> (a).
	Combining estimates (<ref>) and (<ref>), we have
	
    |1/N-1∑_j=2^NF_ε(X_t^1,ε-X_t^j,ε)-∫_DF_ε(Y_t^1,ε-y)ρ_t^ε(y)|
    	≤ Cε^-dsup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|+|I_2| ,

	with the bound (<ref>) for I_2.
	
	
	
	The same estimate holds for the case i=2,⋯,N.
	


To prove the propagation of chaos, we need to introduce an 
auxiliary  stochastic mean-field dynamics {Y_t^i}_i=1^N satisfying


    {[ Y_t^i=Y_0^i+∫_0^t∫_DF(Y_s^i-y)ρ_s(y) +√(2ν)B_t^i-R̃_t^i ,    t>0 ,   i=1,⋯,N,;  R̃_t^i=∫_0^tn(Y_s^i)   d|R̃^i|_s,   |R̃^i|_t=∫_0^t1_∂ D(Y_s^i)   d|R̃^i|_s ,;                                                                               ].

where (ρ_t)_t≥0 is the common marginal density of ({Y_t^i}_i=1^N)_t≥0 for any t≥ 0.  Here we set the initial date (Y_0^i)_i=1,⋯, N=(X_0^i)_i=1,⋯, N i.i.d. sharing the same density ρ_0(x). In other words here ({Y_t^i}_i=1^N)_t≥0 are N copies of the strong solutions to (<ref>).  Similarly let {Y_t^i,ε}_i=1^N be N copies of the strong solutions to the regularized mean-field dynamics (<ref>):

    Y_t^i,ε
    =Y_0^i + ∫_0^t∫_DF_ε(Y_s^i,ε-y)ρ_s^ε(y) 
    +√(2ν)B_t ^i- R̃_t^i,ε,     t>0  ,   i=1,⋯,N,
    
    R̃_t^i,ε
    = ∫_0^t n(Y_s^i,ε)   d |R̃^i,ε|_s  , 
      
    |R̃^i,ε|_t
    =∫_0^t1_∂ D(Y_s^i,ε)   d|R̃^i,ε|_s  .



Finally, we present the proof of Theroem <ref> regarding the propagation of chaos.

	It follows from Theorem <ref>, Theorem <ref> and Theorem <ref> that there exists a weak solution {X_t^i,ε}_i=1^N to (<ref>) and a unique pathwise solution {Y_t^i}_i=1^N to (<ref>) with marginal density ρ_t on the time interval [0,T] for some T > 0, such that ρ_t∈ L^∞(0, T; L^∞(D)). This implies that we are able to define solutions for those two equations on the same probability space with the same initial condition and Brownian motion. 
	Then it follows from the same argument as in the proof of (<ref>), we obtain that there exists constants C_T, ε_0 depending only on T, ρ^ε_L^∞(0, T; L^∞(D)) and ρ_L^∞(0, T; L^∞(D)) such that if ε<ε_0, then for any 1≤ i≤ N
	
    sup_t∈[0,T]|Y^i,ε_t-Y_t^i| ≤ C_Tω(ε)^1/2e^-CT.

	
	Since the  Brownian motions in (<ref>) and (<ref>) are prescribed, we can assume B_t^i=B_t^i,ε, where {B_t^i,ε}_i=1^N are used in (<ref>). 
	Using the similar argument as in the proof of Theorem <ref>, it follows from Itô's formula that
	
    |X_t^i,ε-Y_t^i,ε|^2   =2∫_0^t X_s^i,ε-Y_s^i,ε, 1/N-1∑_j≠ i^NF_ε(X_s^i,ε-X_s^j,ε)-∫_DF_ε(Y_s^i,ε-y)ρ_t^ε(y) 
         -2∫_0^t X_s^i,ε-Y_s^i,ε, n(X_s^i,ε)  d|R^i,ε|_s-2∫_0^t Y_s^i,ε-X_s^i,ε, n(Y_s^i,ε)  d|R̃^i,ε|_s 
       ≤ 2∫_0^t X_s^i,ε-Y_s^i,ε, 1/N-1∑_j≠ i^NF_ε(X_s^i,ε-X_s^j,ε)-∫_DF_ε(Y_s^i,ε-y)ρ_t^ε(y)  ,

	where in the second inequality we have used the convexity of the domain D. Applying Lemma <ref>, one finds
	
    |X_t^i,ε-Y_t^i,ε|^2   ≤ 2∫_0^t |X_s^i,ε-Y_s^i,ε| (Cε^-dsup_i=1,⋯,N|X_s^i,ε-Y_s^i,ε|+ℋ_N) 
       ≤ Cε^-d∫_0^tsup_i=1,⋯,N|X_s^i,ε-Y_s^i,ε|^2 +Cℋ_NT .

	Taking  the expectation on (<ref>) it gives
	
    [sup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|^2]
    	≤ 
    	Cε^-d∫_0^t[sup_i=1,⋯,N|X_s^i,ε-Y_s^i,ε|^2] +CTε^-(d-1)/√(N-1) ,

	where we have used 
	
    [|ℋ_N|]≤ ([|ℋ_N|^2])^1/2≤Cε^-(d-1)/√(N-1).

	Applying Gronwall's inequality, we have
	
    [sup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|]≤([sup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|^2])^1/2≤C_Tε^-d+1/2exp(C_Tε^-d)/(N-1)^1/4 ,

	for all t∈[0,T]. Let ε=(log N/8C_T)^-1/d, then one has
	
    [sup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|]   ≤([sup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|^2])^1/2
    	≤N^1/8log (N)/8C_T(N-1)^1/4≤ CN^-1/8log (N) .

	 
	
	
	
	
	Now recall the definition of the empirical measures
	
    μ_t^X,ε=1/N∑_i=1^Nδ_X_t^i,ε ,
    	    μ_t^Y,ε=1/N∑_i=1^Nδ_Y_t^i,ε ,
    	    μ_t^Y=1/N∑_i=1^Nδ_Y_t^i ,

	and notice that
	
    π_∗=1/N∑_i=1^Nδ_X_t^i,εδ_Y_t^i,ε∈Λ(μ_t^X,ε,μ_t^Y,ε) .

	Then by the definition of 𝒲_∞ distance one has 
	
    𝒲_∞(μ_t^X,ε,μ_t^Y,ε)≤π_∗_(x,y)∈ D× D|x-y|≤sup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|  .

	This leads to
	
    [𝒲_∞(μ_t^X,ε,μ_t^Y,ε)]≤[sup_i=1,⋯,N|X_t^i,ε-Y_t^i,ε|]≤ CN^-1/8log (N)

	and
	
    [𝒲_∞(μ_t^Y,ε,μ_t^Y)]≤[sup_i=1,⋯,N|Y_t^i,ε-Y_t^i|]≤ C_Tω(ε)^e^-CT≤ C[ω(log^-1/d(N))]^exp(-CT) .

	Furthermore, it follows from the the convergence estimate obtained in <cit.> that
	
    [𝒲_2^2(μ_t^Y,ρ_t)]≤ C{   N^-1/2+N^-q-2/q,        d<4q≠ 4 ,
       N^-1/2log(N+1)+N^-q-2/q,       d=4q≠ 4 ,
       N^-2/d+N^-q-2/q,        d>4q≠d/d-2 .
    .

	Combining inequalities (<ref>) and (<ref>), we arrive at 
	
    [𝒲_2(μ_t^X,ε,ρ_t)]≤[𝒲_2(μ_t^X,ε,μ_t^Y,ε)]+[𝒲_2(μ_t^Y,ε,μ_t^Y)]+[𝒲_2(μ_t^Y,ρ_t)]
    ≤   [𝒲_∞(μ_t^X,ε,μ_t^Y,ε)]+[𝒲_∞(μ_t^Y,ε,μ_t^Y)]+[𝒲_2(μ_t^Y,ρ_t)]
    ≤   CN^-1/8(log N)+C[ω(log^-1/d(N))]^exp(-CT)
       +C{   N^-1/4+N^-q-2/2q,        d<4q≠ 4,
       N^-1/4(log(N+1))^1/2+N^-q-2/2q,       d=4q≠ 4,
       N^-1/d+N^-q-2/2q,        d>4q≠d/d-2,
    . 
    ≤   C[ω(log^-1/d(N))]^exp(-CT) ,

	which completes the proof.


Acknowledgments:
This research was funded in part by Canada’s National Science
and Engineering Research Council (NSERC) through grants to RF
and WS. HH also wants to acknowledge support from National Nature Science Foundation of China (NSFC) (Grant No. 11771237).



amsxport







