Propagation of chaos for large Brownian particle system with Coulomb interaction

Propagation of chaos for large Brownian particle system with Coulomb interaction

Jian-Guo Liu^1*and Rong Yang^2*

*Correspondence:

[email protected];

[email protected]

1Departments of Physics and Mathematics, Duke University, Durham, NC 27708, USA

2College of Applied Sciences, Beijing University of Technology, Ping Le Yuan 100, Chaoyang District, Beijing 100124, People’s Republic of China

Abstract

We investigate a system ofNBrownian particles with the Coulomb interaction in any dimensiond≥2, and we assume that the initial data are independent and identically distributed with a common densityρ0satisfying

R^dρ0lnρ0dx<∞and

ρ0∈L^d+2d2(_R^d)∩L¹(_R^d,(1+ |x|²) dx). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. Ford=2, we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. WhenN→ ∞, 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.

Keywords: Noncollision among particles, Entropy and Fisher information estimates, Martingale problem, Uniqueness, de Finetti–Hewitt–Savage theorem

1 Background Let

,F,P

be a probability space, endowed with the standardd-dimensional Brownian motions associated with this space. In this article, we consider the particle system with the following form

dX_tⁱ= 1 N

N j=i

F(X_tⁱ−X_t^j) dt+√

2 dB_tⁱ, i=1,. . ., N, (1.1)

with the initial data{X₀ⁱ}^N_i=1, where{(X_tⁱ)_t≥0}^N_i=1are the trajectories ofNparticles (X_tⁱ∈R^d for anyt > 0), and{(Bⁱ_t)_t≥0}^N_i=1are a sequence of independentd-dimensional standard Brownian motions. The interparticles force is taken to be the Coulomb interaction, and it is described by the Newtonian potential,

F(x)= −∇Φ(x), Φ(x)=

⎧⎪

⎪⎨

⎪⎪

⎩ C_d

|x|^d−2 ifd≥3,

− 1

2πln|x| ifd=2,

(1.2)

whereC_d = 1 d(d−2)αd

, αd = π^d/2

Γ(d/2+1), i.e.,αdis the volume ofd-dimensional unit ball. We recastF(x) = ^C_|x^∗_|d^x, ∀x ∈ R^d\{0}, d ≥ 2, where C^∗ = ^Γ_2π^(d_d/2^/2). The first term

©The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

0123456789().,–: vol

on the right hand in (1.1) represents repulsive force on (X_tⁱ)_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_tⁱ)_t≥0and (X_t^j)_t≥0colliding with each other for some timet <∞, thenX_tⁱ =X_t^j,F(X_tⁱ−X_t^j)= ∞, and then the solution to (1.1) breaks up. Fortunately, we will prove that this will not happen. More precisely, when the initial dataX₀ⁱ = X₀^j almost surely (a.s.) for alli = 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) ford=2. To do this, we will show the following main result:

For any fixed timeT > 0, there exists a subsequence of the empirical measureμ^N :=

1 N

_N

i=1δ_Xtⁱ (μ^N areP(C([0, T];R^d))-valued random variables) converging in law to a deterministic probability measureμasN goes to infinity, whereP(C([0, T];R^d)) is the set of probability measures overC([0, T];R^d). Furthermore, the time marginal lawμthas a density functionρt which is the unique weak solution to the Poisson–Nernst–Planck equation (1.4) below.

In this paper, for k ≥ 1, we denote byPsym((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 = (X1,. . ., X_k) belongs toPsym((R^d)^k)). Whenf ∈Psym((R^d)^k) has a densityρ∈L¹((R^d)^k), we introduce the entropy and the Fisher information off:

H_k(f) := 1 k

R^kdρlnρdx and I_k(f) := 1 k

R^kd

|∇ρ|² ρ dx.

Sometimes, we also useH_k(ρ) andI_k(ρ) to presentH_k(f) andI_k(f), respectively. Iff has no density, we simply putH_k(f)= +∞andI_k(f)= +∞. Notice thatH_k(f^⊗k)=H1(f) andI_k(f^⊗k)=I₁(f).

We will split the proof of propagation of chaos into three steps. First, we denote f_t^N andρ_t^N as the joint time marginal distribution and density of (X_t¹,. . ., X_t^N)_0≤t≤T, respectively, L(μ^N) is the law of μ^N andΦ^N = _N¹ _N

i,j=1

i=j

Φ(x_i −x_j). In Lemma 3.1, when d = 2, using the uniform estimate of_t

0I_N(f_s^N) ds; whend ≥ 3, using the uni- form estimate of_t

0ρ_s^N,|∇Φ^N|²ds, we prove that the sequence{L(μ^N)}N≥2is tight in P

P(C([0, T];R^d))

. (It is well knownC([0, T];R^d) is a polish space andP(C([0, T];R^d)) is metrizable, and it is also a polish space, see the “Appendix”.) Therefore there exists a subsequence ofμ^N (without relabeling) and aP(C([0, T];R^d))-valued random measureμ such thatμ^N converges in law toμasNgoes 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 dataf₀in 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,{Bt}0≤t≤T), if a probability measure μ∈ P(C([0, T];R^d)) with time marginalμ0 at time t = 0 is endowed with

aμ-distributed canonical process (X_t)_0≤t≤T ∈ C([0, T];R^d), then let{Bt}0≤t≤T is the natural filtration generated by (Xt)0≤t≤T, i.e.,

Bt=σ{X_s, s≤t} (1.3)

andB = B(C([0, T];R^d)) (theσ-algebra of Borel sets overC([0, T];R^d)).μis called a solution to the

g, C_b²(R^d)

-self-consistent martingale problem with the initial distribution μ0(meaning thatX₀is distributed according toμ0), if for anyϕ ∈ C_b²(R^d), (X_t)_0≤t≤T induces the following process

Mt :=ϕ(Xt)−ϕ(X0)− _t

0

g(Xs,L(Xs)) ds for any t∈[0, T],

such that (Mt)_0≤t≤Tis a martingale with respect to (w.r.t.) the filtration{Bt}0≤t≤T, where g(x,L(X_s))=

C([0,T];R^d)∇ϕ(x)·F(x−y_s)μ(dy)+ ϕ(x) for any s∈[0, T].

Indeed, Lemma4.3gives a martingale estimate for theN-particle system and Lemma 4.2states a standard method of checking a process to be a martingale. Then Proposition 4.1shows thatμ(ω) is a solution to the above martingale problem for a.s.ω∈.

Third, denoting (μt(ω))_t≥0as the time marginal ofμ(ω). With the uniform estimates of entropy and the second moments for the particle system (1.1), Lemma3.2shows that (μt(ω))_t≥0has a density (ρt(ω))_t≥0a.s.. Using the fact thatμ(ω) is a.s. a solution to the self-consistent martingale problem in Definition1, Theorem5.2shows thatρ(ω) is the unique weak solution to the mean-field Poisson–Nernst–Planck (PNP) equations of single component:

⎧⎪

⎨

⎪⎩

∂tρ= ρ+ ∇ ·(ρ∇c), x∈R^d, t>0,

−c=ρ(t, x), ρ(t, x)t=0=ρ0(x),

(1.4)

i.e.,ρ(ω) is independent ofωand hence it is deterministic (so doesμ), 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 smoothFhas 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 desingularizeF byF_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 two- dimensional (2D) Navier–Stokes equation. In the vortex system, the interparticles force is given by F(x) = −∇^⊥Φ(x) ford = 2, where the operator∇^⊥ =

−_∂^∂_x2,_∂^∂_x

1

. In a series papers [18–20], 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 anyN ≥2 andT >0, if {(X_t^i,N)_t∈[0,T]}^N_i=1is the solution to the attractive model, then

P

∃s∈[0, T], ∃1≤i<j≤N:X_s^i,N =X_s^j,N >0,

i.e., the singularity is visited and the particle system is not clearly well defined. The sign of F is crucially used in Lemmas2.2and2.3to 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 theN-interacting particle sys- tem (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 theN 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 Definition1. 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 ofP(C([0, T];R^d)).

2 Global well posedness of theN-interacting particle system ind≥2 First, we give a definition of the strong solution to (1.1).

Definition 2 For any fixed T > 0, initial data {X₀ⁱ}^N_i=1 and given probability space ,F,P

endowed with a sequence of independentd-dimensional Brownian motions {(Bⁱ_t)_t≥0}^N_i=1, if there is a stochastic process{(X_tⁱ)_t∈[0,T]}^N_i=1adapted to (Ft)_t∈[0,T]such that {(X_tⁱ)_t∈[0,T]}^N_i=1satisfies (1.1) a.s. in the probability space (,F,(Ft)_t≥0,P) for allt∈[0, T], we say that{(X_tⁱ)_t∈[0,T]}^N_i=1is a global strong solution to (1.1).

Next, we state some results about the well posedness of theN-interacting particle system (1.1) and the entropy and regularity properties for the density of those particles.

Theorem 2.1 For any d ≥2, let N ≥2and T >0. Consider a sequence of independent d-dimensional Brownian motions {(Bⁱ_t)t≥0}^N_i=1and the independent and identically dis- tributed (i.i.d.) initial data{X₀ⁱ}^N_i=1with a common distribution f0satisfying H1(f0)<+∞

and a common densityρ0∈L^d+22d (R^d)∩L¹(R^d,(1+ |x|²) dx). Then

(i) There exists a unique global strong solution to(1.1)and thus a.s. X_tⁱ = X_t^j for all t∈[0, T], i=j.

(ii) Denote by (f_t^N)0≤t≤T the joint time marginal distribution function of (X_t¹,. . ., X_t^N)_0≤t≤Tand assumeρ0_Lr(R^d)<∞for some r>d≥2. Then f_t^N(X)has a density functionρ_t^N(X), and it is the unique weak solution to the following linear Fokker–Plank equation:

∂tρ^N = ρ^N +1

2∇ ·(ρ^N∇Φ^N), (2.1)

whereΦ^N = _N¹ _N

i,j=1

i=j

Φ(x_i−x_j).

(iii) Denote m2(ρ) :=

R^d|x|²ρdx. For all t >0, H_N(f_t^N)+

_t

0

I_N(f_s^N) ds≤H₁(f₀), ford≥2; (2.2) ρ_t^N,Φ^N

+ 1 2

_t

0

ρ_s^N,|∇Φ^N|²

ds≤(N−1)C(d)ρ0²

L^d+22d

ford≥3;

(2.3)

1≤i≤Nsup E[|X_tⁱ|²]≤

⎧⎪

⎪⎨

⎪⎪

⎩

m₂(ρ0)+

4+ 1 2π

t if d=2,

3m₂(ρ0)+3t

2C(d)ρ0²

L^d+22d +6td if d≥3,

(2.4)

where C(d)= _d(d−2)π¹ Γ(d)

Γ(^d₂)

²

d. (iv) For any d≥2and1<p<∞,

sup

t∈[0,T]

ρ_t^Np_Lp(RNd)+4(p−1)

p ∇((ρ^N)^p2)²_L2([0,T]×R^Nd)≤ ρ0^Np_Lp(Rd), (2.5) and there exists a constant C (depending only on T and the radius of the support of ρ^N) such that

∇ρ^N2_L2([0,T]×R^Nd)≤ 1

2ρ0^2N_L2(R^d); (2.6)

∂tρ^N2_L2(0,T;W−2,∞

loc (R^Nd))≤Cρ0^2N_L2 ford=2,3; (2.7)

∂tρ^N2_L2(0,T;W−1,∞

loc (R^Nd))≤C(ρ0^2N_L2 + ρ0^2N_Lq) for alld≥4

and someq>d. (2.8)

Additionally, the definition of weak solution to Eq. (2.1) is given as follows.

Definition 3 (Weak solution) Let the initial dataρ₀^N ∈L¹₊∩L^d+22d (R^Nd) andT > 0, we shall say thatρ^Nis a weak solution to (2.1) with the initial dataρ₀^N if it satisfies:

1. integrability and time regularity:

ρ^N ∈L^∞(0, T;L¹∩L^d+22d (R^Nd)), (ρ^N)^d+2d ∈L²(0, T;H¹(R^Nd))

∂tρ^N ∈L^k2(0, T;W_loc^−1,k1(R^Nd)) for some k₁, k₂≥1;

2. for allϕ∈C_c^∞(R^Nd), 0<t≤T, the following holds:

R^Ndρ^N(·, t)ϕdX=

R^Ndρ₀^NϕdX− 1 2

_t

0

R^Nd∇ϕ· ∇Φ^Nρ^NdXds +

_t

0

R^Ndρ^NϕdXds. (2.9)

Next, we will split into two subsections to prove Theorem2.1.

2.1 Noncollision among particles for the system (1.1)

Since the interacting forceFof (1.1) is singular, we regularizeFfirstly. 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 onΦ_ε.

Lemma 2.1 Suppose J(x)∈C²(R^d), supp J(x)⊂B(0,1), J(x)=J(|x|)and J(x)≥ 0. Let J_ε(x)= _ε¹_dJ(^x_ε)andΦ_ε(x)=J_ε∗Φ(x)for x∈R^d, F_ε(x)= −∇Φ_ε(x), then F_ε(x)∈C¹(R^d),

∇ ·F_ε(x)=J_ε(x)and

(i) F_ε(0) = 0and F_ε(x)= F(x)g

|x| ε

for any x = 0, where g(r)= _C¹∗

_r

0J(s)s^d−1ds, C^∗= Γ(d/2)

2π^d/2 , d≥2and g(r)=1for r≥1;

(ii) |F_ε(x)| ≤min_C|x|

ε^d ,|F(x)|

and|∇F_ε(x)| ≤ _ε^C_d;

(iii) For any bounded domain B and some1<q< _d^d₋₁,F_εL^q(B)is uniformly bounded inε;

(iv) when d ≥ 3, Φ_ε(x) = Φ(x)for any |x| ≥ ε > 0; when d = 2 and0 < ε ≤ 1, Φ_ε(x)=Φ(x)+Φ_ε(1)for any|x| ≥ε. And

Φε(ε)→ +∞, as ε→0⁺ for d≥2. (2.10)

Proof of(iv): Letr= |x|. By the proof of (i), one knows that r^d−1∂rΦε(r)= −

_r

0

J_ε(s)s^d−1ds= − ^r_ε

0

J(s)s^d−1ds= −C^∗g r

ε

. (2.11)

Then for anyr ≥ ε, we integrate the above equality and use the fact thatg(r) = 1 for r≥1,

−Φ_ε(r)= _∞

r ∂sΦ_ε(s) ds= −C^∗ _∞

r

g_s

ε

s^d−1 ds

= −C^∗ _∞

r

1

s^d−1ds= − C^∗

(d−2)r^d−2 for d≥3; (2.12) Φε(r)−Φε(1)=

_r

1 ∂sΦε(s) ds= −C^∗ _r

1

g_s

ε

s ds

= − 1 2π

_r

1

1

s ds= − 1

2π lnr for d=2,ε≤1. (2.13) In this article, we take a cutoff functionJ(x)≥0,J(x)∈C₀³(R^d),

J(x)=

C(1+cosπ|x|)² if |x| ≤1,

0 if |x|>1,

whereCis a constant such thatC|S^d−1|₁

0(1+cosπr)²r^d−1dr=1 and|S^d−1| =_Γ^2π_(d/2)^d/2 . Proof(i)of Theorem2.1: First, we consider the followingN-interacting particle system via the regularized force:

dX_t^i,ε= _N¹ _N

j=iF_ε(X_t^i,ε−X_t^j,ε) dt+√ 2 dBⁱ_t,

X_t^i,ε|t=0=X₀ⁱ, (2.14)

which has a unique global strong solution{(X_t^i,ε)_t≥0}^N_i=1. Define a random variable

A_ε(t) := inf

0≤s≤tmin

i=j |X_s^i,ε−X_s^j,ε|. FixT >0, define the stopping time

τε:=

0 ifε≥A_ε(0);

sup{t∧2T :A_ε(t)≥ε} ifε <A_ε(0). (2.15)

The key step is to prove that

ε→0limP(τε≤T)=0, (2.16)

We adapt the techniques of [6,24] to prove (2.16). Define a random process (Φ_t^ε,N)_0≤t≤2T as

Φ_t^ε,N := 1 N

N i,j=1

i=j

Φ(x_i−x_j)Φ_ε

X_t^i,_∧τ^ε_ε−X_t^j,ε_∧τε

. (2.17)

Then one has the following basic fact {τε≤T} ⊂

sup

t∈[0,T]Φ_t^ε,N ≥Φ_τ^ε,N_ε

, (2.18)

and the proof of (2.16) is divided into three steps as follows.

Step 1 We show that

Φ_t^ε,N =Φ₀^N+M_t∧τε− 2 N²

_t∧τε

0

N i=1

⎛

⎜⎜

⎜⎝ N j=1 j=i

F_ε(X_s^i,ε−X_s^j,ε)

⎞

⎟⎟

⎟⎠

2

ds, (2.19)

where

M_t∧τε := −

√2 N

N i,j=1 i=j

_t∧τε

0

F_ε(X_s^i,ε−X_s^j,ε)·(dB_sⁱ−dB^j_s);

Φ₀^N := 1 N

N i,j=1 i=j

Φ(X₀ⁱ−X₀^j) (2.20)

and we prove (Mt∧τε)0≤t≤Tis a martingale w.r.t. the filtration generated by the Brownian motions{(Bⁱ_t)_0≤t≤T}^N_i=1.

Using the Itô’s formula and the factΦε(x)= −J_ε(x)=0 on|x| ≥ε, one has Φ_ε

X_t^i,_∧τ^ε_ε−X_t^j,ε_∧τε

=Φ

X₀ⁱ −X₀^j

−√ 2

_t∧τε

0

F_ε(X_s^i,ε−X_s^j,ε)·(dB_sⁱ−dB^j_s)

− _t∧τε

0

F_ε(X_s^i,ε−Xs^j,ε)·

⎡

⎣1 N

N k=i

F_ε(X_s^i,ε−X_s^k,ε)− 1 N

N k=j

F_ε(Xs^j,ε−X_s^k,ε)

⎤

⎦ds.

(2.21) Summing (2.21) together, we obtain (2.19) and thus only need to show that (M_t∧τε)_0≤t≤T is a martingale. Using the fact |F_ε(x)| ≤ ^C_ε^|x|d in Lemma 2.1 and

(X_t^i,ε)_0≤t≤TN i=1 are exchangeable, one has

_T

0

E[|F_ε(X_t^i,ε−X_t^j,ε)|²] dt ≤ C_T

0 E[|X_t^i,ε−X_t^j,ε|²] dt ε^2d

≤ C_T

0 (E[|X_t^i,ε|²]+E[|X_t^j,ε|²]) dt ε^2d

= 2C_T

0 (E[|X_t^i,ε|²]) dt

ε^2d . (2.22)

If one can prove that _T

0 E|X_t^i,ε|²dt<C(ε, T), (2.23)

then by [17, Corollary 3.2.6],{_t∧τε

0 F_ε(X_s^i,ε−Xs^j,ε)·(dBⁱ_s−dB^js)}0≤t≤Tis a martingale w.r.t.

the filtration generated by the Brownian motions (B_tⁱ)_0≤t≤Tand (B^j_t)_0≤t≤T, and thenM_t∧τε is a martingale w.r.t. the filtration generated by the Brownian motions {(Bⁱ_t)_0≤t≤T}^N_i=1. Below we prove (2.23).

According to Eq. (2.14) and the fact (_N

i=1a_i)²≤N_N

i=1a²_i, one has E[|X_t^i,ε|²]=E[|X₀ⁱ+ 1

N _t

0

N j=i

F_ε(X_s^i,ε−X_s^j,ε) ds+√ 2Bⁱ_t|²]

≤3E[|X₀ⁱ|²]+ 3t N²E

⎡

⎢⎣ _t

0

⎛

⎝^N

j=i

F_ε(X_s^i,ε−X_s^j,ε)

⎞

⎠

2

ds

⎤

⎥⎦+6E[|Bⁱ_t|²]

≤3E[|X₀ⁱ|²]+3t NE

⎡

⎣ _t

0

N j=i

F_ε²(X_s^i,ε−Xs^j,ε) ds

⎤

⎦+6td.

Since{(X_t^i,ε)_t≥0}^N_i=1are exchangeable, one has E[|X_t^i,ε|²]≤3E[|X₀ⁱ|²]+ 3Ct

Nε^2d _t

0

N j=i

E[|X_s^i,ε|²]+E[|X_s^j,ε|²]

ds+6td

≤3E[|X₀ⁱ|²]+ 6Ct ε^2d

_t

0 E[|X_s^i,ε|²] ds+6td.

Hence by Gronwall’s lemma, one obtains (2.23).

Step 2 We prove that there exists a constantC (depending only onH1(ρ0),m2(ρ0), ρ0

L^d+22d ,d,TandN) such that for anyR>0 and small enoughε, P(τε≤T)≤ C

R +P

$

t∈inf[0,T]M_t∧τε >−R, sup

t∈[0,T]M_t∧τε ≥ 1

NΦε(ε)−R

%

(2.24) and split the proof into two cases.

Case1(d≥3): Using the factΦ_ε(x)> 0, ifτ_ε ≤T, thenΦ_τ^ε,N_ε ≥ _N¹Φ_ε(ε). Combining (2.18), one has

P(τε≤T)≤P

$ sup

t∈[0,T]Φ_t^ε,N ≥ 1 NΦε(ε)

%

=:I₁. (2.25)

From (2.19), one also has

0< Φ_t^ε,N ≤Φ₀^N +Mt∧τε. (2.26)

Directly from (2.25) and (2.26), one has I₁≤P

$ sup

t∈[0,T]M_t∧τε ≥ 1

NΦε(ε)−Φ₀^N

%

=P

$

t∈inf[0,T]M_t∧τε ≥ −Φ₀^N, sup

t∈[0,T]M_t∧τε ≥ 1

NΦε(ε)−Φ₀^N

%

(2.27) Then for anyR>0,

I₁≤P(−Φ₀^N ≤ −R)+P

$

t∈inf[0,T]M_t∧τε >−R, sup

t∈[0,T]M_t∧τε ≥ 1

NΦε(ε)−R

% . (2.28)

Using the Markov’s inequality to the first term of (2.28) and combining (2.25), one has P(τε≤T)≤ E[|Φ₀^N|]

R +P

$

t∈inf[0,T]M_t∧τε >−R, sup

t∈[0,T]M_t∧τε ≥ 1

NΦε(ε)−R

% . (2.29) Moreover,E[|Φ₀^N|] can be controlled by

E[|Φ₀^N|]= ρ₀^N,Φ^N =(N−1)

R^2dρ0(x)ρ0(y)Φ(x−y) dxdy

≤(N−1)C(d)ρ0²

L^d+22d

, (2.30)

where C(d) = _d(d−¹_2)π Γ(d)

Γ(^d₂)

²

d, and the last inequality comes from the Hardy–

Littlewood–Sobolev inequality. Plugging (2.30) into (2.29) gives (2.24) ford≥3.

Case2(d = 2): For small enoughε, using the fact thatΦ_ε(x)= −₂¹_πln|x| +Φ_ε(1)>

−_2π¹|x|for any|x| ≥εby (iv) in Lemma2.1, one has Φ_τ^ε,N_ε ≥ 1

NΦ_ε(ε)− 1 π

N i=1

|X_τ^i,ε_ε| ≥ 1

NΦ_ε(ε)− 1 π

N i=1

sup

t∈[0,T]|X_t^i,ε| if τ_ε≤T. (2.31) Combining (2.18), one has

P(τε≤T)≤P

$ sup

t∈[0,T]Φ_t^ε,N ≥ 1

NΦε(ε)− 1 π

N i=1

sup

t∈[0,T]|X_t^i,ε|

%

=:I₂. (2.32) From (2.19) and the fact thatΦε(x)>−_2π¹|x|for any|x| ≥εand small enoughε, one also has

− 1 π

N i=1

&&

&X_t^i,ε_∧τε&&&< Φ_t^ε,N ≤Φ₀^N +M_t∧τε. (2.33) DenoteY :=Φ₀^N +_π¹_N

i=1sup_t∈[0,T]|X_t^i,ε|. From (2.33), one has I₂ ≤P

$ sup

t∈[0,T]

M_t∧τε≥ 1

NΦ_ε(ε)−Φ₀^N− 1 π

N

i=1

sup

t∈[0,T]

|X_t^i,ε|

%

=P

$

t∈[0,T]inf M_t∧τε≥ −Φ0^N− inf

t∈[0,T]

1

π

N

i=1

&&

&X_t∧τ^i,ε_ε&&&

, sup

t∈[0,T]M_t∧τε ≥ 1

NΦε(ε)−Y

% .

(2.34) Combining (2.34) and (2.32), for anyR>0, one has

P(τε≤T)

≤P

$

t∈[0,T]inf M_t∧τε ≥ −Φ0^N− inf

t∈[0,T]

1

π

N

i=1

&&

&X_t∧τ^i,ε_ε&&&

, sup

t∈[0,T]

M_t∧τε≥ 1

NΦε(ε)−Y

%

≤P

$

t∈inf[0,T]M_t∧τε ≥ −Y, sup

t∈[0,T]M_t∧τε ≥ 1

NΦε(ε)−Y

%

≤P(−Y ≤ −R)+P

$

t∈[0,Tinf ]M_t∧τε>−R, sup

t∈[0,T]

M_t∧τε ≥ 1

NΦ_ε(ε)−R

%

. (2.35)

The first term of (2.35) is given by the Markov’s inequality P(−Y ≤ −R)≤ E'&&&Φ₀^N + _π¹_N

i=1sup_t∈[0,T]|X_t^i,ε|&&&( R

≤ E[|Φ₀^N|]+_π¹_N

i=1E'

sup_t∈[0,T]|X_t^i,ε|(

R . (2.36)

Therefore we need to evaluateE[sup_t∈[0,T]|X_t^i,ε|].

E )

t∈[0,T]sup |X_t^i,ε|

*

≤

E )

t∈[0,T]sup |X_t^i,ε|²

*¹₂

. (2.37)

By the Itô formula, one has d|X_t^i,ε|²=2X_t^i,ε·

⎛

⎝1 N

N j=i

F_ε(X_t^i,ε−X_t^j,ε) dt+√ 2 dB_tⁱ

⎞

⎠+4td. (2.38)

Since x·F_ε(x) = x ·F(x)g(^|x|_ε) = _2π¹g(^|x|_ε) ≤ _2π¹ by Lemma 2.1, then _N

i=12X_t^i,ε · 1

N

_N

j=iF_ε(X_t^i,ε−X_t^j,ε)

= _N¹ _N

i,j=1

i=j

Φ(xi −xj)(X_t^i,ε −X_t^j,ε)·F_ε(X_t^i,ε −X_t^j,ε) ≤ ^N₂⁻_π¹. Summing (2.38) and integrating in time, one has

N i=1

|X_t^i,ε|²≤ N i=1

|X₀ⁱ|²+

4+ 1 2π

Nt+2√ 2

N i=1

_t

0

X_s^i,ε·dBⁱ_s. (2.39) Taking expectation, one has

E )

sup

r∈[0,t]|X_r^i,ε|²

*

≤E )

sup

r∈[0,t]

$ _N

i=1

|X_r^i,ε|²

%*

≤ N i=1

E[|X₀ⁱ|²]+

4+ 1 2π

Nt

+2√ 2E

) sup

r∈[0,t]

&&

&&

&

N i=1

_r

0

X_s^i,ε·dB_sⁱ

&&

&&

&

*

≤ N i=1

E[|X₀ⁱ|²]+

4+ 1 2π

Nt

+2√ 2

⎛

⎝E

⎡

⎣sup

r∈[0,t]

&&

&&

&

N i=1

_r

0

X_s^i,ε·dBⁱ_s

&&

&&

&

2⎤

⎦

⎞

⎠

12

. (2.40)

From (2.23),_N

i=1_t

0X_s^i,ε·dB_sⁱ

0≤t≤T is a martingale w.r.t. the filtration generated by the Brownian motions{(Bⁱ_t)_t≥0}^N_i=1. Then using the Doob’s inequality for martingale [10, see p. 203, Theorem 7.31], one has

E

⎡

⎣sup

r∈[0,t]

&&

&&

&

N i=1

_r

0

X_s^i,ε·dBⁱ_s

&&

&&

&

2⎤

⎦≤4E

⎡

⎣&&

&&

&

N i=1

_t

0

X_s^i,ε·dB_sⁱ

&&

&&

&

2⎤

⎦. (2.41)

Combining the exchangeability of{(X_t^i,ε)_t≥0}^N_i=1, we obtain that E

⎡

⎣&&

&&

&

N i=1

_t

0

X_s^i,ε·dBⁱ_s

&&

&&

&

2⎤

⎦=E ) _N

i=1

_t

0 |X_s^i,ε|²ds

*

=NE+ _t

0 |X_s^i,ε|²ds ,

≤NE ) _t

0

sup

r∈[0,s]|X_r^i,ε|²ds

*

. (2.42)