Propagation of chaos for large Brownian particle system with Coulomb interaction

We investigate a system of N Brownian particles with the Coulomb interaction in any dimension d≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document}, and we assume that the initial data are independent and identically distributed with a common density ρ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0$$\end{document} satisfying ∫Rdρ0lnρ0dx<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{\mathbb {R}^{d}}\rho _0\ln \rho _0\,\hbox {d}x<\infty $$\end{document} and ρ0∈L2dd+2(Rd)∩L1(Rd,(1+|x|2)dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0\in L^{\frac{2d}{d+2}} (\mathbb {R}^{d}) \cap L^1(\mathbb {R}^{d}, (1+|x|^2)\,\hbox {d}x)$$\end{document}. We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document}, we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightarrow \infty $$\end{document}, the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson–Nernst–Planck equation of single component.


Background
Let , F, P be a probability space, endowed with the standard d-dimensional Brownian motions associated with this space. In this article, we consider the particle system with the following form are a sequence of independent d-dimensional standard Brownian motions. The interparticles force is taken to be the Coulomb interaction, and it is described by the Newtonian potential, where , i.e., α d is the volume of d-dimensional unit ball. We recast F (x) = C * x |x| d , ∀x ∈ R d \{0}, d ≥ 2, where C * = Γ (d/2) 2π d/2 . The first term on the right hand in (1.1) represents repulsive force on (X i t ) t≥0 by all other particles. The interacting particle system (1.1) is a typical physical model and appears in many applications. For example, in semiconductor, (i) the electrons interact with each other through the Coulomb repulsive force; (ii) the electrons interact with background, and it is modeled by Brownian motions; (iii) the mass of electron is very light and the inertia can be neglected, and the overdamped system of particles is used in (1.1).
Notice that if there exists two particles (X i t ) t≥0 and (X j t ) t≥0 colliding with each other for some time t < ∞, then X i t = X j t , F (X i t − X j t ) = ∞, and then the solution to (1.1) breaks up. Fortunately, we will prove that this will not happen. More precisely, when the initial data X i 0 = X j 0 almost surely (a.s.) for all i = j, we will show that there exists a unique global strong solution to system (1.1) and hence there is no collision a.s. among particles in (1.1).
The second object of this paper is to provide a rigorous theory on propagation of chaos for the above system (1.1) for d = 2. To do this, we will show the following main result: For any fixed time T > 0, there exists a subsequence of the empirical measure μ N : In this paper, for k ≥ 1, we denote by P sym ((R d ) k ) the set of symmetric probability measures on (R d ) k (the law of any exchangeable (R d ) k -valued random variable X = (X 1 , . . . , X k ) belongs to P sym ((R d ) k )). When f ∈ P sym ((R d ) k ) has a density ρ ∈ L 1 ((R d ) k ), we introduce the entropy and the Fisher information of f : Sometimes, we also use H k (ρ) and I k (ρ) to present H k (f ) and I k (f ), respectively. If f has no density, we simply put H k (f ) = +∞ and I k (f ) = +∞. Notice that H k (f ⊗k ) = H 1 (f ) and I k (f ⊗k ) = I 1 (f ). We will split the proof of propagation of chaos into three steps. First, we denote f N t and ρ N t as the joint time marginal distribution and density of (X 1 t , . . . , X N t ) 0≤t≤T , respectively, L(μ N ) is the law of μ N and Φ N = 1 is metrizable, and it is also a polish space, see the "Appendix".) Therefore there exists a subsequence of μ N (without relabeling) and a P(C([0, T ]; R d ))-valued random measure μ such that μ N converges in law to μ as N goes to infinity.
Second, for d = 2 and a.s. ω ∈ , we prove that μ(ω) is exactly a solution to the following self-consistent martingale problem with the initial data f 0 in a new probability space C([0, T ]; R d ), B, μ(ω) . This definition is the same as the Stroock-Varadhan [17], and it is a variant of the definition of nonlinear martingale problem in [11, p. 40].

Definition 1
In the probability space (C([0, T ]; R d ), B, μ, {B t } 0≤t≤T ), if a probability measure μ ∈ P(C([0, T ]; R d )) with time marginal μ 0 at time t = 0 is endowed with a μ-distributed canonical process (X t ) 0≤t≤T ∈ C([0, T ]; R d ), then let {B t } 0≤t≤T is the natural filtration generated by (X t ) 0≤t≤T , i.e., B t = σ {X s , s ≤ t} (1.3) and B = B(C([0, T ]; R d )) (the σ -algebra of Borel sets over C([0, T ]; R d )). μ is called a solution to the g, C 2 b (R d ) -self-consistent martingale problem with the initial distribution μ 0 (meaning that X 0 is distributed according to μ 0 ), if for any ϕ ∈ C 2 b (R d ), (X t ) 0≤t≤T induces the following process Indeed, Lemma 4.3 gives a martingale estimate for the N -particle system and Lemma 4.2 states a standard method of checking a process to be a martingale. Then Proposition 4.1 shows that μ(ω) is a solution to the above martingale problem for a.s. ω ∈ .
Third, denoting (μ t (ω)) t≥0 as the time marginal of μ(ω). With the uniform estimates of entropy and the second moments for the particle system (1.1), Lemma 3.2 shows that (μ t (ω)) t≥0 has a density (ρ t (ω)) t≥0 a.s.. Using the fact that μ(ω) is a.s. a solution to the self-consistent martingale problem in Definition 1, Theorem 5.2 shows that ρ(ω) is the unique weak solution to the mean-field Poisson-Nernst-Planck (PNP) equations of single component: (1.4) i.e., ρ(ω) is independent of ω and hence it is deterministic (so does μ), which finishes the proof of propagation of chaos.
The concept of propagation of chaos was originated by Kac [8]. The propagation of chaos for (1.1) with the smooth F has been rigorously proved by McKean in 1970s with a coupling method, and the mean-field equation is a class of nonlinear parabolic equations [16]. For singular interacting kernel, a cutoff parameter is usually introduced to desingularize F by F ε , and the coupling method sometimes still can be used to prove the propagation of chaos, c.f. [13].
The problem for the Newtonian potential without cutoff parameter is a challenging problem, which is the content of this paper. In this case, the coupling method can no longer be used and we adapt the nonlinear martingale problem method developed by Stroock-Varadhan [17]. Model (1.1) is closely related to the vortex system for the twodimensional (2D) Navier-Stokes equation. In the vortex system, the interparticles force is given by F (x) = −∇ ⊥ Φ(x) for d = 2, where the operator ∇ ⊥ = − ∂ ∂x 2 , ∂ ∂x 1 . In a series papers [18][19][20], Osada showed that the particles a.s. never encounter, so that the singularity of kernel a.s. never visited. He also studied the propagation of chaos for the Navier-Stokes equation with the random vortex method without regularized parameters. In a recent important work of Fournier et al. [4], the authors significantly improved Osada's result: (i) They proved the propagation of chaos for the 2D viscous vortex model with any positive viscosity coefficient; (ii) the convergence holds in a strong sense, called entropic.
Instead of repulsive force, if the attractive force is used (in this case, the sign of F is changed), then the mean-field equation is the Keller-Segel equation. Much of analysis used in this paper failed due to the change of sign. In fact, recently, there is a deep result proved by Fournier and Jourdain [5,Proposition 4]: For any N ≥ 2 and T > 0, if is the solution to the attractive model, then i.e., the singularity is visited and the particle system is not clearly well defined. The sign of F is crucially used in Lemmas 2.2 and 2.3 to achieve the uniform estimates. For a related work, Godinho and Quininao proved propagation of chaos for the subcritical Keller-Segel equations [6]. Some of their frameworks and techniques will be adapted to this paper. This paper is organized as follows. The well posedness of the N -interacting particle system (1.1) and the uniform estimates for the joint density of those particles are established in Sect. 2. In Sect. 3, we show the tightness of the empirical measures of the trajectories of the N particles. In Sect. 4, we prove that the limiting point of the empirical measures is a.s. solution to the self-consistent martingale problem in Definition 1. In Sect. 5, we provide a simple proof of the uniqueness of weak solution to the PNP equation (1.4), and then, we prove the propagation of chaos results. Finally, in the "Appendix" we provide a metrization of P(C([0, T ]; R d )).
2 Global well posedness of the N-interacting particle system in d ≥ 2 First, we give a definition of the strong solution to (1.1).

Definition 2
For any fixed T > 0, initial data {X i 0 } N i=1 and given probability space , F, P endowed with a sequence of independent d-dimensional Brownian motions is a global strong solution to (1.1).
Next, we state some results about the well posedness of the N -interacting particle system (1.1) and the entropy and regularity properties for the density of those particles.

5)
and there exists a constant C (depending only on T and the radius of the support of ρ N ) such that and some q > d. (2.8) Additionally, the definition of weak solution to Eq. (2.1) is given as follows.

Definition 3 (Weak solution)
Let the initial data ρ N 0 ∈ L 1 + ∩ L 2d d+2 (R Nd ) and T > 0, we shall say that ρ N is a weak solution to (2.1) with the initial data ρ N 0 if it satisfies: 1. integrability and time regularity: 2. for all ϕ ∈ C ∞ c (R Nd ), 0 < t ≤ T , the following holds: Next, we will split into two subsections to prove Theorem 2.1.

Noncollision among particles for the system (1.1)
Since the interacting force F of (1.1) is singular, we regularize F firstly. We directly recall below a lemma stated in [13, Lemma 2.1.], which collects some useful properties of the regularization. In addition, we add (iv) for a estimate on Φ ε .
Proof of (iv): Let r = |x|. By the proof of (i), one knows that (2.11) Then for any r ≥ ε, we integrate the above equality and use the fact that g(r) = 1 for r ≥ 1, In this article, we take a cutoff function J (x) ≥ 0, J (x) ∈ C 3 0 (R d ), Proof (i) of Theorem 2.1: First, we consider the following N -interacting particle system via the regularized force: which has a unique global strong solution Fix T > 0, define the stopping time The key step is to prove that lim ε→0 P(τ ε ≤ T ) = 0, (2.16) We adapt the techniques of [6,24] to prove (2.16). Define a random process (Φ ε,N t ) 0≤t≤2T as Then one has the following basic fact (2.18) and the proof of (2.16) is divided into three steps as follows.
Step 1 We show that and we prove (M t∧τ ε ) 0≤t≤T is a martingale w.r.t. the filtration generated by the Brownian Using the Itô's formula and the fact Φ ε (x) = −J ε (x) = 0 on |x| ≥ ε, one has . (2.22) If one can prove that is a martingale w.r.t. the filtration generated by the Brownian motions (B i t ) 0≤t≤T and (B j t ) 0≤t≤T , and then M t∧τ ε is a martingale w.r.t. the filtration generated by the Brownian motions According to Eq. (2.14) and the fact ( Hence by Gronwall's lemma, one obtains (2.23).
Step 2 We prove that there exists a constant C (depending only on H 1 (ρ 0 ), m 2 (ρ 0 ), , d, T and N ) such that for any R > 0 and small enough ε, and split the proof into two cases.
Directly from (2.25) and (2.26), one has Then for any R > 0, Using the Markov's inequality to the first term of (2.28) and combining (2.25), one has , and the last inequality comes from the Hardy- (2.32) From (2.19) and the fact that Φ ε (x) > − 1 2π |x| for any |x| ≥ ε and small enough ε, one also has (2.34) Combining (2.34) and (2.32), for any R > 0, one has (2.35) The first term of (2.35) is given by the Markov's inequality (2.37) By the Itô formula, one has
Since τ ε is decreasing with respect to ε, (2.16) implies that In other words, for a.s. ω ∈ , there exists a ε 0 (ω) such that if ε ≤ ε 0 (ω), Since T is arbitrary, the global existence and uniqueness of strong solution to the system (1.1) can be achieved immediately.

A uniform priori estimates for the density of N-interacting particle system
First, we start from the regularized system of (1.1) to achieve the uniform estimates of entropy and the second moments. Notice that the sign of F is crucially used in this section. For example, we used the positivity of J ε to prove (2.55), (2.56) and (2.70).
be the unique strong solution to (2.14) and (f N,ε t ) t≥0 be its joint time marginal distribution with density (ρ N,ε t ) t≥0 . We have the uniform estimates for entropy: We also have the second moment estimates: , applying the Itô formula one deduces that Taking expectation, one has We compute the entropy: where ρ (2),N,ε s is the second marginal density. Since . Then combining the positivity of J ε , (2.55) is obtained.
Next, multiplying (2.58) with Φ N,ε (x) and integrating, one has s t≥0 is a martingale and taking expectation of (2.39), one has are exchangeable, one obtains the second moment estimates for two dimension.
For d ≥ 3, since Using the identity:   Starting from the regularized system of (1.1), we also have a uniform priori regularity estimates.
be the unique strong solution to (2.14) and (ρ N,ε t ) t≥0 be its joint time marginal density. We have the uniform regularity estimates: For any d ≥ 2 and

70)
and there exists a constant C (depending only on T and the radius of the support of ρ N,ε ) such that L q ) for all d ≥ 4 and some q > d.

(2.73)
Proof For any p > 1, multiplying (2.58) with p(ρ N,ε ) p−1 and integrating, one has By the positivity of J ε , we have For any B R ⊂ R Nd , multiplying (2.58) with test function ϕ(x) ∈ C ∞ 0 (B R ) and integrating in space, one has L q ) for all d ≥ 4 and some q > d, (2.85) which finishes the proof of (2.72) and (2.73).
Next, we finish the rest proof of Theorem 2.1.
Proof (ii) of Theorem 2.1: Using the uniform estimates for the joint distribution of strong solution to (2.14), we split into three steps to study the joint distribution of strong solution to (1.1).
Step 1 We show that ρ N,ε t is relatively compact. Combining (2.55) and (2.57), then there exists a constant C independent of ε such that (2.88) Step 2 We show that ρ N obtained above is the unique weak solution to (2.1). For any ϕ ∈ C ∞ c (R Nd ), ρ N,ε t (X) satisfies the following equation: and then Combining the regularity of ρ N from Lemma 2.3, we obtain that ρ N is exactly a weak solution to (2.1). Supposeρ N is another weak solution to (2.1) with the same initial data. One has Step 3 Finally, we prove ρ N t (X) is the density of f N t (X). By (2.16), one has

Lemma 3.1 For any N
Proof For d = 2, we directly cite the proof of Lemma 5.2 in [4].
For d ≥ 3, in order to prove (i), it means that for fixed η > 0, T > 0, one should find a compact subset K η, Considering the particle system (1.1), for any 0 ≤ s < t ≤ T , one has A direct computation shows the time regularity of the Brownian motion term: The estimate for the drift term is given by Using the identity: and combining (2.3), one has Hence by the Markov's inequality, combining (3.2) and (3.7), for any η > 0, one can find a constant R η > 0 (depending only on d and ρ 0 Since E[|X 1,N 0 | 2 ] < ∞, then one can find a constant a η > 0 (depending only on m 2 (ρ 0 )) such that Now we construct the following set which is a compact subset of C([0, T ]; R d ) by Ascoli's theorem. Combining (3.4), (3.8) and (3.9), one has sup N ≥2 which finishes the proof of (i). (ii) follows from the exchangeability of [23,Proposition 2.2] or [15,Lemma 4.5].
From the tightness of {L(μ N )} in P P(C([0, T ]; R d )) by Lemma 3.1, one has that there exists a subsequence of μ N ∈ P(C([0, T ]; R d )) (without relabeling) and a random measure μ ∈ P(C([0, T ]; R d )) such that μ N → μ in law as N → ∞. (3.11) Next, we prove that the limited measure-valued process μ has a density a.s..

Lemma 3.2 For any N
Combining (2.4) and the exchangeability of (3.14) Similarly with (2.87), one has the following uniformly integrable property of ρ (j),N in And then using the Dunford-Pettis theorem, there exists a subsequence of ρ (j),N t (without relabeling) and ρ j t ∈ L 1 (R dj ) such that ρ (j),N t ρ j t in L 1 (R dj ) weakly as N → ∞, (3.16) and ρ j t satisfies (3.12).
Step 3 Now we prove that μ t has a density ρ t a.s. for any time t ≥ 0.
For t > 0, let π t = L(μ t ) ∈ P P(R d ) and define the projection π j t = P(R d ) g ⊗j π t (dg) ∈ P(R dj ) for any j ≥ 1 in the following sense Step 2, we know that f (j),N t narrowly converges to π j t as N → ∞ for all j ≥ 1. Then combining the uniform estimates (2.4) and applying Theorem 4.1 in [4] (a refined version of the de Finetti-Hewitt-Savage theorem), μ t has a density denoted by ρ t a.s. such that where the last inequality of (3.27) comes from (2.2) and (2.4).

The self-consistent martingale problem
As a preparatory work, recalling directly from the definition of time marginal law and the probability measure on the path space for a stochastic process, we have the following lemma.

Lemma 4.1
Let (X t ) 0≤t≤T ∈ C([0, T ]; R d ) be a stochastic process, μ ∈ P(C([0, T ]; R d )) be the law of (X t ), and μ t (x) be the time marginal law of (X t ) on the space R d . Then for any ψ ∈ C b (R d ) and t ∈ [0, T ], The following lemma gives a standard method of checking a stochastic process to be a solution to the martingale problem in Definition 1, and it is stated in [3, p. 174] without a proof. For completeness, we give a detail proof below.  0≤t≤T is a solution to the (g, C 2 b (R d ))-self-consistent martingale problem with the initial distribution μ 0 in Definition 1, i.e., let M t = ϕ(X t ) − ϕ(X 0 ) − t 0 g(X r , L(X r )) dr, then (M t ) 0≤t≤T is a martingale w.r.t. the filtration {B t } 0≤t≤T , and then where the first and second equalities come from Theorem B.2. b) and e) in [17], respectively.
(ii) By the definition of martingale, in order to prove (M t ) 0≤t≤T is a martingale w.r.t. the filtration {B t } 0≤t≤T , one need to show that for any 0 < s < t ≤ T , Without loss of generality, we assume that is an increase sequence and has the following inequality In other words, we have which is a contradiction to (4.4). By the fact that any bounded Borel measurable function can be approximated by a sequence of bounded continuous functions and using the dominated convergence theorem, one knows that (4.1) holds for any h 1 , . . . , h n ∈ B(R d ) is equivalent for any From Lemma 4.2, for solving the martingale problem in Definition 1, we just need to prove (4.1). Therefore we construct a functional ψ on C( We also define a functional on P(C([0, T ]; R d )) below, for any Q ∈ P(C([0, T ]; R d )), then we have the following martingale estimate lemma.
Proof By the definition of K ψ (Q), simple computation shows that (4.14) Using the Itô formula, for any Plugging (4.15) into (4.14), one has Then one has and then where C depends only on T , . Plugging (4.18) into (4.17), one can achieve (4.13) immediately.

Lemma 4.4 Let E be a polish space. Assume a sequence of P(E)-valued random variables
μ N converge in law to a random measure μ. For any ψ(x, y) ∈ C b (E × E) and Q ∈ P(E), Proof For any Q ∈ P(E), ψ(x, y) ∈ C b (E × E) and ϕ ∈ C b (R), define a functional Γ : Here, the space P(E) is endowed with a metric induced by the narrowly convergence, and it is a Polish space too. Note that ϕ ∈ C b (P(E)) if and only if a sequence μ N (∈ P(E)) narrowly converge to μ as N → ∞ ⇒ ϕ(μ N ) converges to ϕ(μ) as N → ∞. For any sequence Q N (∈ P(E)) narrowly converge to Q, by [1, p. 23, Theorem 2.8], the following convergence result holds, (4.20) holds.
Since the sequence μ N converges in law to μ, then which gives (4.19).
Following the spirit of [4], one has here ψ ε and K ψ ε are defined by (4.11) and (4.12). It is obvious that ψ ε ∈ C b (C([0, T ]; R d ) × C([0, T ]; R d )) for any fixed ε > 0. Then combining (3.11) and using Lemma 4.4, we obtain that Define Combining the fact |F ε (x)| ≤ |F (x)|, F ε (x) = F (x) for |x| ≥ ε by Lemma 2.1 and Lemma 4.1, there exists a constant C (depending only on d, Since μ s has a density ρ s a.s. by Lemma 3.2, then (4.25) By Lemma 4.3, there exists a constant C (depending only on From (4.14), one has where C is a constant depending only on When d = 2, similarly with the proof of Lemma 3.3. in [4], one obtains that where 0 < q < 2 and C is a constant depending only on q. Plugging (4.29) and (4.30) into (4.28) and (4.25), respectively, one has and where C is a constant depending only on q and T . Using (3.13) and (3.14), there exists a constant C (depending only on T , H 1 (f 0 ) and m 2 (ρ 0 )) such that Combining (4.26), (4.31) and (4.33), there exists a constant C (depending only on q, T , H 1 (f 0 ) and m 2 (ρ 0 )) such that Plugging (4.34) into (4.32), there exists a constant C (depending only on q, T , H 1 (f 0 ) and m 2 (ρ 0 )) such that Let ε goes to 0, one obtains that which means (4.21) holds.

The refined hyper-contractivity and uniqueness for the mean-field
Poisson-Nernst-Planck equations 1.4 In this subsection, we prove the uniqueness of weak solution to 1.4 by the standard semigroup method, see [12]. We use the following definition of weak solution to (1.4).
2. For all ϕ ∈ C ∞ 0 (R d ) and 0 < t ≤ T , the following holds, Remark 5.1 Notice that the regularity of ρ(t, x) is enough to make sense of each term in (5.3). By the Hardy-Littlewood-Sobolev inequality, one has .
Proof Follows the spirit of [12], we outline the proof briefly.
Step 3 Finally, since t 1 is a constant only depending on C(d, q, T, m 2 (ρ 0 ), H 1 (ρ 0 )) or C(d, q, T, ρ 0 L d 2 +γ ), taking t 1 as a new initial time, repeating the above process, we have that model (1.4) has a unique weak solution in t ∈ [t 1 , 2t 1 ]. One can continue this process and obtain a unique global solution in [0, T ).

Propagation of chaos result
First, for d = 2, we show that the limited measure-valued random variable μ satisfies that: For any ϕ ∈ C 2 b (R d ) and t ∈ [0, T ], the time marginal measure μ t ∈ P(R d ) a.s. solves the following equation ∇ϕ(x) · F (x − y) μ s (dx)μ s (dy)ds + . Denote μ as a limiting point of a subsequence of μ N . Then by Proposition 5.1, one knows that μ t satisfies (5.9) a.s.. And Lemma 3.2 shows that (μ t ) t≥0 has a density (ρ t ) t≥0 a.s. and ρ t takes the initial density ρ 0 at time t = 0. Recalling equation (5.9), we deduce that ∇ϕ(x) · x − y |x − y| 2 ρ s (x)ρ s (y) dxdyds i.e., ρ t a.s. is a weak solution to (1.4) with the initial data ρ 0 . Finally, by the uniqueness of weak solution to (1.4) from Theorem 5.1, ρ t is deterministic, which completes the proof of Theorem 5.2 immediately.
Finally, we make a remark on the possible using stochastic PDE method.