We have …rst taken N ! 1, then M ! 1

1 The PDE associated with local interaction

This section is more di¢ cult than previous ones, hence we sketch its content.

First, we reformulate the model with contact interaction in a form which is suitable for the investigation of its macroscopic limit, 1.1.

Then, since the macrocopic limit problem is very di¢ cult and innovative, we change the problem and freeze the scaling parameter in the interaction potential, 1.2.

With this trick we may apply Mean Field theory and deduce a PDE, parametrized by the freezed parameter, say M .

Then we send M to in…nity, to …nd a new PDE, called Porous Media equation, which in a sense corresponds to in…nitely many particles and in…nitely rescaled potential, 1.3.

However, we have not done the limit together, with M = N , as it should be. We have

…rst taken N ! 1, then M ! 1. Is it the same? We do not know the answer. We present a few simulations that indicate that the two results are at least numerically very close, for certain classes of problems, 1.4.

Encouraged by the numerical results, we describe in detail a partial result in the direc- tion of showing that the two limits are the same. It holds true for a modi…ed version of the rescaling, called intermediate or moderate interaction theory, 2-3.

1.1 Reformulation of the SDE with local interaction

Given the function g (r) 0, r 0, of the previous lecture, let G (r) 0, r 0, be a primitive of g (r) and let

V (x) = G (jxj) :

Example 1 g (r) = (1 r) 1_{r 1}, r 0, G (r) = (1 r)²=2 1_{r 1}. We consider V as a "potential". Writing V⁰ for rV , we have

V⁰(x) = g (jxj) x jxj: Hence we may write equation

dX_t^i;N = N^1=d XN j=1

g N^1=d X_t^i;N X_t^j;N X_t^i;N X_t^j;N X_t^i;N X_t^j;N

dt + dB_tⁱ

in the form

dX_t^i;N = N^1=d XN j=1

V⁰ N^1=d X_t^i;N X_t^j;N dt + dB_tⁱ: Setting further

V_N(x) = N V N^1=dx

we have

V_N⁰ (x) = N N^1=dV⁰ N^1=dx hence we may write the previous SDE as

dX_t^i;N = 1 N

XN j=1

V_N⁰ X_t^i;N X_t^j;N dt + dB_tⁱ:

With these new notations, recall the result:

Corollary 2 The correct rescaled dynamics to investigate the macroscopic limit of a sys- tem with local interaction V is

dX_t^i;N = 1 N

XN j=1

V_N⁰ X_t^i;N X_t^j;N dt + dB_tⁱ: (1)

1.2 The Mean Field equation with a concentrated potential

Let us decuple the number of particles from the scaling of the potential and consider the equation

dX_t^i;N;M = 1 N

XN j=1

V_M⁰ X_t^i;N;M X_t^j;N;M dt + dB_tⁱ (2) where now we have two parameters: N is the number of particles, M is a parameter which de…nes the rescaling

VM(x) = M V M^1=dx :

Obviously the true contact-interaction problem is to take M = N and send N ! 1.

But we already know the result of sending N ! 1 if we keep M …xed (it is the Mean Field theory): the empirical measure S_t^N;M = _N¹ PN

i=1 X_t^i;N;M weakly converges to the measure-valued solution _tof the Mean Field equation

@ _t

@t =

2

2 ^t+ div _t V_M⁰ _t :

Assume (we need it below) that _t has a density ut(x). More precisely, both _tand ut(x) depend on M ; so let us write u^M_t (x). This function is a weak solution of the PDE

@u^M_t

@t =

2

2 u^M_t + div u^M_t V_M⁰ u^M_t : (3) We ask two questions: does u^M_t converge to the solution u_t of some PDE? Is this PDE the correct one for the limit of the empirical measure S_t^N = S_t^N;N? We may answer the

…rst question, but the second one is open.

1.3 Taking the limit in the Mean Field PDE

Let us …rst investigate the limit, as M ! 1, of solutions u^M of equation (3). We do not want to be rigorous here (it is possible to state a rigorous result) and thus assume that u^M converges, in a suitable sense, to a function u. We want to …nd the equation satis…ed by u. We have, for every test function 2 C0¹,

u^M_t ; = u^M₀ ; +

2

2 Z t

0

u^M; ds Z t

0

V_M⁰ u^M; u^Mr ds

hence, under relatively weak convergence properties, the …rst three terms converge, hence we have

hut; i = hu0; i +

2

2 Z t

0 hu; i ds lim

M !1

Z t 0

V_M⁰ u^M; u^Mr ds:

In addition, if we prove that V_M⁰ u^M ! v in a suitable sense, we get hu^t; i = hu⁰; i +

2

2 Z t

0 hu; i ds Z t

0 hv; ur i ds:

This is the weak form of the PDE

@u

@t =

2

2 u + div (uv) :

We have only to identify v. We have, for every test function 2 C0¹, V_M⁰ u^M; = u^M; V_M⁰ ( ) : Moreover,

V_M⁰ ( ) (x) = Z

V_M⁰ (y x) (y) dy = r Z

V_M(y x) (y) dy:

Assume now that V_M is proportional to a sequence of molli…ers: this is true by the formula VM(x) = M V M^1=dx if V is proportional to a smooth probability density

V (x) = ²₁f (x)

where ²₁ > 0 and f is a pdf. In this case, since is smooth and compact support, RV_M(y x) (y) dy ! ²1 (x), hence (V_M⁰ ( ) ) (x) ! ²1r (x) and …nally

V_M⁰ u^M; ! ²1hu; r i = ²1hru; i : We have found

v = ²₁ru and therefore the limit PDE

@u

@t =

2

2 u +

2 1

2 u²: (4)

This equation is known in the literature as Porous Media equation.

Remark 3 Notice that the equation does not depend on the details of the interaction po- tential V , only on

2 1 =

Z

V (x) dx:

This looks strange, from the viewpoint of the particle system.

The result guessed so far is correct from the viewpoint of the PDEs: it is a theorem that the solution u^M of the Mean Field equation converges to the solution u of the Porous Media equation.

1.4 Open problem and a few simulations

Let us repeat the open problem. We have a few objects: the particle system with contact interaction (1) and its empirical measure S_t^N; the particle system with mean …eld interac- tion V_M⁰ (2) and its empirical measure S_t^N;M; the solution u^M_t of the mean …eld PDE (3);

the solution ut of the porous media PDE (4). We know that

N !1lim S_t^N;M = u^M_t lim

M !1u^M_t = u_t in suitable senses. Do we have also

N !1lim S_t^N = ut?

The answer is not known. In this section we give some evidence of the fact that the answer, although theoretically uncertain, numerically is very close to be positive, in the case of repulsive interaction with integrable potential V .

The aim of the following simulation is to see, at the particle level, what happens if we consider N particles with potential VM(x), for di¤erent choices of M . We consider the case, in d = 1,

V⁰(x) = 1_{jxj 1}(1 jxj) x jxj and we rescale it as above:

V_M⁰ (x) = M M^1=dV⁰ M^1=dx

= M²V⁰(M x) :

The following functions de…ne part of V⁰ with a generic rescaling, where in the sequel we shall take R = M ¹, = M².

norma=function(x) abs(x) H=function(r) (sign(r)+1)/2

g=function(r,R,alp) alp*H(1-r/R)*(1-r/R)

The code is now

M1=1; M2=100

N=200; Nloc=10; n=100000; dt=0.00001; sig=1; h=sqrt(dt); T=20; L=5; L0 = 1

e= matrix(1/N,N,1) alp1=M1^2; R1=1/M1 alp2=M2^2; R2=1/M2

X1=matrix(0,N,n); X2=matrix(0,N,n);

X1[,1]=rnorm(N,0,L0); X2[,1]=X1[,1];

plot(c(-L,L), c(0,0.6), type="n") for(t in 1:(n-1)){

DX1.0= matrix(X1[,t],N,N); DX1=DX1.0-t(DX1.0) DX2.0= matrix(X2[,t],N,N); DX2=DX2.0-t(DX2.0)

Kx1=g(norma(DX1),R1,alp1)*DX1/(norma(DX1)+ 0.000001) Kx2=g(norma(DX2),R2,alp2)*DX2/(norma(DX2)+ 0.000001) X1[,t+1]=X1[,t] + dt*Kx1%*%e + h*sig*rnorm(N)

X2[,t+1]=X2[,t] + dt*Kx2%*%e + h*sig*rnorm(N) if(t%%T==0 )

{polygon(c(-L,L,L,-L),c(0,0,0.6,0.6), col="white", border=NA) lines(density(X1[,t+1],bw=sd(X1[,t+1])/4),col="green")

lines(density(X2[,t+1],bw=sd(X2[,t+1])/4),col="red") }

}

Remark 4 The previous code, thanks to the help of participants, has been improved with respect to previous ones, by taking a matrix-approach to the computation of interactions between particles. This speeds-up considerably the computation.

Remark 5 Taking values like 2-3 instead of 4-5 in bw=sd(X2[,t+1])/4) smooths-out the pro…le but one then suspects that fatness is due to smoothing and not to di¤ usion.

The result seems to be that M does not matter: the di¤usion is essentially the same.

The result is similar for large densities:

M1=1; M2=100

N=200; Nloc=10; n=100000; dt=0.00001; sig=1; h=sqrt(dt); T=20; L=5; L0 = 0.2

e= matrix(1/N,N,1) alp1=M1^2; R1=1/M1 alp2=M2^2; R2=1/M2

X1=matrix(0,N,n); X2=matrix(0,N,n);

X1[,1]=rnorm(N,0,L0); X2[,1]=X1[,1];

plot(c(-L,L), c(0,2), type="n") for(t in 1:(n-1)){

DX1.0= matrix(X1[,t],N,N); DX1=DX1.0-t(DX1.0) DX2.0= matrix(X2[,t],N,N); DX2=DX2.0-t(DX2.0)

Kx1=g(norma(DX1),R1,alp1)*DX1/(norma(DX1)+ 0.000001) Kx2=g(norma(DX2),R2,alp2)*DX2/(norma(DX2)+ 0.000001) X1[,t+1]=X1[,t] + dt*Kx1%*%e + h*sig*rnorm(N)

X2[,t+1]=X2[,t] + dt*Kx2%*%e + h*sig*rnorm(N) if(t%%T==0 )

{polygon(c(-L,L,L,-L),c(0,0,2,2), col="white", border=NA) lines(density(X1[,t+1],bw=sd(X1[,t+1])/4),col="green") lines(density(X2[,t+1],bw=sd(X2[,t+1])/4),col="red") }

}

Replacing the potential with the following one

V⁰(x) = 10 1_{jxj 1}(1 jxj)¹⁰ x jxj does not change the picture:

norma=function(x) abs(x) H=function(r) (sign(r)+1)/2

g=function(r,R,alp) 10*alp*H(1-r/R)*(1-r/R)^10

Obviously we should go to M = N , but it seems that nothing changes. The conclusion seems to be that taking the limits N ! 1 …rst and then M ! 1 gives a very similar result as imposing M = N and taking N ! 1.

Some geometric intuition, not described here, seems to con…rm this result for interme- diate values of the density. When the density is very large, the case M = N could have

a stronger di¤usion, but this is not biologically relevant. When the density is very small the di¤usion of the Mean Field equation should prevail, but the Brownian di¤usion in that case becomes dominant.

Overall, we think that the Porous Media equation is a very good model for contact repulsive interaction, when V is integrable.

Exercise 6 Compare the simulation of the Porous Media PDE (4) with the particle system with true contact interaction (1).

For this reason, let us examine in detail the following case.

2 Intermediate interaction: preparation

2.1 Intermediate regime

The open problem above, namely whether the limit of the contact-interaction case is equal to taking …rst N ! 1 and then M ! 1, has a full rigorous (and positive) solution if we rescale the interaction potential in a weaker way. Karl Oelschäger identi…ed this intermediate problem, between mean …eld and contact interactions, called intermediate (or moderate) regime, where it is possible to prove that S_t^N converges to the solution of the porous media equation. It is the case when

VN(x) = N V N ^=dx

with 2 (0; 0), for a suitable ₀< 1. For = 0 we have mean …eld interaction; for = 1 we have contact interaction. The regime 2 (0; 0) is thus more local than mean …eld but not so realistic as contact interaction.

The restriction to the intermediate regime will be used to prove tightness. But also the problem of passage to the limit is very di¢ cult, as explained below. In this case, it is not the restriction on to play a role but an assumption on the structure of V as the convolution of two kernels, assumption again identi…ed by Oelschäger. Let us discuss …rst the issue of convergence.

2.2 Taking the limit As in a previous section we have:

Lemma 7 The empirical measure S_t^N satis…es the identity d S_t^N; _t = S_t^N;@ _t

@t dt S_t^N; r t V_N⁰ S_t^N dt + dM_t^;N+

2

2 S_t^N; _t dt

for all 2 C_b^1:2 [0; T ] R^d , where

M_t^;N = 1 N

XN i=1

Z t

0 r s X_s^i;N dB_sⁱ:

In order to understand the di¢ culty to take the limit in this identity, assume for a second that S_t^N weakly converges to some measure _t. We can easily pass to the limit in the terms D

S_t^N;^@_@t^tE

and S_t^N; _t ; the martingale term M_t^;N goes to zero in mean square (the proof is easy; we shall see below similar computations). But the limit of the nonlinear term

S_t^N; r t V_N⁰ S_t^N is much more di¢ cult.

Recall, from general facts of analysis, that weak convergence of functions is not su¢ cient to pass to the limit in nonlinear terms: the sequence fn(x) = sin (nx), x 2 [0; 1] converges weakly to zero, but f_n² does not. Hence the sole property of weak convergence of the measures S_t^N cannot be su¢ cient.

Another general fact we know from analysis is that if S_t^N weakly converge to _t and f_N is a sequence of functions which converges uniformly to f , then

N !1lim f_N; S_t^N = hf; ti But uniform convergence is essential.

Example 8 If _n = _xn, = _x, x_n ! x, then n * . If f_n converges uniformly to f , we see that hfⁿ; ni = fⁿ(xn) converges to f (x). But weaker convergences do not imply the same result: there is no reason why f_n(x_n) should converge to f (x) if we have only pointwise convergence, or L^p convergence, and so on.

Can we say that V_N⁰ S_t^N converges uniformly to some limit? This is a very di¢ cult question. The only easy thing we can say is that it weakly converges:

V_N⁰ S_t^N; ! ²1

Z

r (x) t(dx) = ²₁ Z

(x) rut(x) dx in the case when _t has a di¤erentiable density u_t(x). Indeed

V_N⁰ S_t^N; = S_t^N; V_N( ) r ! ²1

Z

r (x) t(dx) = ²₁ Z

(x) rut(x) dx:

Implicitly we have understood that, in this problem, sooner or later we have to prove that _t has a density. One way is to investigate the convergence of the molli…ed empirical measure h^N_t (x) = WN S_t^N (x).

After these preliminary remarks on the di¢ culty of taking the limit in the non-linear term, let us describe the brilliant trick devised by Karl Oelschläger, under an additional assumption. Assume that (take ²₁ = 1 here for simplicity of exposition)

V_N = W_N W_N

where W_N are classical molli…ers and, from now on, we write W_N (x) = W_N ( x) : Then we have

S_t^N; r t V_N⁰ S_t^N = r tS_t^N; W_N W_N⁰ S_t^N

= WN r tS_t^N ; r W^N S_t^N where we have used the property

W_N f; g = hf; W^N gi easy to prove. Now, assume we can prove that the quantity

W_N r tS_t^N ; r WN S_t^N r t W_N S_t^N ; r WN S_t^N

is small for large N (namely that commuting W_N with the pointwise product r t is irrelevant, in the limit). If so, we have to deal with

r t WN S_t^N ; r W^N S_t^N

and here we just need weak L²-convergence of r WN S_t^N and strong L²-convergence of W_N S_t^N. Recall indeed that if f_n! f in L² R^d and g_n* g in L² R^d , then

jhfn; g_ni hf; gij jhfn; g_ni hf; gnij + jhf; gni hf; gij kfn f k kgnk + jhf; gni hf; gij

and now the …rst term goes to zero by strong convergence of f_n to f and boundedness of gn; the second term goes to zero by weak convergence of gn to g.

In the next section we describe a modi…ed model where it is possible to prove such properties and then to pass to the limit rigorously.

2.3 Bounds on particle position

Our ultimate aim is to prove tightness of the laws Q^N of the empirical measures S^N. [In fact, at the end, we follow a di¤erent approach, namely we prove tightness of molli…ed empirical measure, but it is convenient for pedagogical reasons to argue about tightness of

the laws Q^N of S^N] To reach this result, the …rst and more important estimate, following the computations made for the Mean Field case, would be to prove that

E

"

sup

t2[0;T ]

X_t^i;N

# C:

But now

X_t^i;N X₀^i;N + 1 N

XN j=1

Z t 0

V_N⁰ X_s^i;N X_s^j;N ds + B_tⁱ

and V_N⁰ is unbounded with respect to N , so we cannot repeat the estimates of the mean

…eld case. Notice that 1 N

X

j

V_N⁰ X_tⁱ X_t^j dt = ruN t; X_tⁱ

where

u_N(t; x) = V_N S_t^N (x) = 1 N

X

j

V_N x X_t^j : Hence the interacting particle system can be written in the form

dX_t^i;N = ru^Nt X_tⁱ dt + dB_tⁱ: Thus

X_t^i;N X₀^i;N + Z t

0 ru^Ns X_sⁱ ds + B_tⁱ : If we prove suitable estimates on ru^Nt , we have a tool to prove Eh

sup_{t2[0;T ]} X_t^i;N i C.

The next section shows that there is hope to prove an estimates on ru^Nt . To compare the empirical function u^N_t used here with the function u_t of the next section, notice that u^N_t (x) = VN S_t^N (x), if it converges, it should converge to the solution utof the Porous Media equation, by the general conjecture underlying this section.

2.4 Energy estimates on the Porous Media equation

We …rst describe an a priori estimate for the Porous Media equation since, we think, it is conceptually the idea behind the next computation on S_t^N. For simplicity of notations, consider the equation

@u

@t =

2

2 u + div (uru) :

For this equation, the standard energy estimate reads d

dt 1 2

Z

u²dx = Z

u@u

@tdx = Z

u

2

2 u + div (uru) dx

=

2

2 Z

jruj²dx Z

u jruj²dx hence

1 2

Z

u²_tdx + Z t

0 2

2 Z

jruj²dx + Z

u jruj²dx ds = 1 2

Z u²₀dx where we stress the remarkable fact that

Z

u div (uru) dx = Z

ru (uru) dx = Z

u jruj²dx:

This provides estimates on ru. Recall that at the end of the previous subsection we identi…ed as a possible tool exactly the control of ruN. Thus we shall try now to repeat these energy computations on uN.

3 Rigorous results on intermediate interaction

In this section we prove the following result. We consider the SDE dX_t^i;N = 1

N XN j=1

V_N⁰ X_t^i;N X_t^j;N dt + dB_tⁱ with

VN(x) = N V N ^=dx

with 2 (0; 0), for a suitable ₀ < 1 stated by the next theorem. Recall that we write W (x) = W ( x).

Theorem 9 Assume that

V = ²₁W W

where W is a smooth compact support probability density function and ²₁ is a positive constant. Set V_N(x) = N V N ^=dx , W_N(x) = N W N ^=dx (so that V_N = ²₁W_N WN) and set w^N_t (x) := WN S_t^N (x). Assume

d d + 2: Finally, assume that the initial conditions X₀^i;N satisfy

E Z

R^d

w^N₀ (x)²dx C; E X₀^{i;N 2} C

and S₀^N; ! hu0; i in probability, for every 2 Cc¹, with u₀2 L² R^d , 0 u₀ 1.

Then the family of laws eQ^N of the functions w^N_t (x) are tight on the space Y described below, which includes L²_loc [0; T ] R^d .

They have the unique limit eQ = _u, and u_t is the unique weak solution of the Porous Media equation

@u

@t =

2

2 u +

21

2 u² with initial condition u0.

The proof of this theorem requires several steps that we divide in the next subsections.

3.1 Energy estimates on the molli…ed empirical measure Recall Lemma 7:

d S_t^N; _t = S_t^N;@ _t

@t dt S_t^N; r t V_N⁰ S_t^N dt + dM_t^;N+

2

2 S_t^N; _t dt:

and the de…nitions

w^N_t (x) = WN S_t^N (x) = 1 N

X

j

WN x X_t^j u^N_t (x) = V_N S_t^N (x) :

In this section we prove the following two results.

Lemma 10 For the molli…ed empirical measure w_t^N(x), under the assumption V = ²₁W W , we have

1 2

Z

R^d

w^N_t (x)²dx +

2

2 Z t

0

Z

R^d rw^Ns (x)²dx ds + Z t

0

Z

R^d

V_N⁰ S_s^N (x)²S_s^N(dx) ds

= 1 2

Z

R^d

w^N₀ (x)²dx + Z

R^d

Z t 0

w_s^N(x) dM_s^N dx + Z

R^d 2

N² XN

i=1

Z t

0 rWN x X_s^{i;N 2}ds

! dx:

Corollary 11 Under the further assumption _d+2^d , there is a constant C > 0 such that 1

2E Z

R^d

w^N_t (x) ²dx +

2

2 E Z t

0

Z

R^d rw^Ns (x)²dx ds + E Z t

0

Z

R^d ru^Ns (x) ²S_s^N(dx) ds 1

2 Z

R^d

w^N₀ (x)²dx + C:

Let us prove the lemma. Replace _t(y) by W_N(x y), with x given as a parameter, and think S_t^N; _t as an integration in the y variable, so that

S_t^N; _t = S_t^N; W_N(x ) = w^N_t (x) and so on for the other terms:

S_t^N; W_N(x ) = W_N S_t^N (x) = w_t^N(x)

S_t^N; rWN(x ) V_N⁰ S_t^N = Z

ryW_N(x y) V_N⁰ S_t^N (y) S^N_t (dy)

= div

Z

WN(x y) V_N⁰ S_t^N (y) S_t^N(dy)

= div WN V_N⁰ S_t^N S_t^N : We get

dw_t^N(x) = div W_N V_N⁰ S_t^N S^N_t (x) dt + dM_t^N(x) +

2

2 w^N_t (x) dt where

M_t^N(x) = 1 N

XN i=1

Z t

0 rWN x X_s^i;N dBⁱ_s:

Let us apply Itô formula to w_t^N(x) ², with x treated again as a parameter. We have d w^N_t (x) ² = 2w_t^N(x) dwN(t; x) + d w^N(x) _t

= 2w_t^N(x) div W_N V_N⁰ S_t^N S_t^N (x) dt + ²w_t^N(x) w^N_t (x) (t; x) dt +2w_t^N(x) dM_t^N + d M^N(x) _t

and

d M^N(x) _t=

2

N² XN i=1

rWN x X_t^{i;N 2}dt:

Think to the previous identity as integrated in time, as it is rigorously. Then we integrate in dx, namely we compute the di¤erential of the "energy" ¹₂R

R^d w^N_t (x)²dx:

1 2

Z

R^d

w^N_t (x)²dx = 1 2

Z

R^d

w₀^N(x) ²dx +

Z

R^d

Z t 0

w_s^N(x) div W_N V_N⁰ S_s^N S_s^N (x) ds dx +

2

2 Z

R^d

Z t 0

w^N_s (x) w_s^N(x) ds dx +

Z

R^d

Z t 0

w_s^N(x) dM_s^N dx

+ Z

R^d 2

N² XN

i=1

Z t

0 rWN x X_s^{i;N 2}ds

! dx:

Let us exchange integration (the functions are integrable under minor assumptions on W_N).

We have

Z

R^d

Z t 0

w_s^N(x) w^N_s (x) ds dx

= Z t

0

Z

R^d

w_s^N(x) w^N_s (x) dx ds

=

Z t 0

Z

R^d rw^Ns (x)²dx ds

that we shall write on the left-hand-side of the identity above. Moreover, and this is the main point of the computation,

Z

R^d

Z t 0

w_s^N(x) div W_N V_N⁰ S_s^N S_s^N (x) ds dx

= Z t

0

Z

R^d

w_s^N(x) div W_N V_N⁰ S_s^N S_s^N (x) dx ds

=

Z t 0

Z

R^drw^Ns (x) WN V_N⁰ S_s^N S_s^N (x) dx ds and, by a rule already used above,

= Z t

0

Z

R^d

W_N rwN(s) (x) V_N⁰ S_s^N (x) S_s^N(dx) ds:

From the assumption V = ²₁W W it follows V_N = ²₁W_N W_N (we check it below) and thus

W_N rws^N(x) = W_N rW^N S_s^N = rWN WN S_s^N

= r V^N S_s^N = V_N⁰ S_s^N:

(we have used the properties (f g) h = f (g h) and f rg = rf g, easy to prove).

We deduce

Z

R^d

Z t 0

w_s^N(x) div W_N V_N⁰ S_s^N S_s^N (x) ds dx

=

Z t 0

Z

R^d

V_N⁰ S_s^N (x)²S_s^N(dx) ds:

Let us prove that VN = ²₁W_N WN:

2

1 W_N WN (x) = ²₁ Z

WN( x + y) WN(y) dy = ²₁N Z

N W N ^=d(y x) W N ^=dy dy

= ²₁N Z

W y N ^=dx W (y) dy = ²₁N W W N ^=dx

= N V N ^=dx = V_N(x) : The lemma is proves.

Let us now prove the corollary. We have to prove that the last term in the identity of the lemma is bounded by a constant, in expected value (the martingale termRt

0 w_N(s; x) dM_s^N has zero average). The required bound will follow from the following result. Notice that we shall perform just identities and even without expected value, hence the result, in terms of range of , is optimal.

Lemma 12 If _d+2^d , there is a constant C > 0 such that, for all s t in [0; T ], Z

R^d 2

N² XN i=1

Z t

s rW^N x X_r^{i;N 2}dr

!

dx C (t s) : Proof. We have

Z

R^d 2

N² XN i=1

Z t

s rWN x X_r^{i;N 2}dr

! dx

=

2

N² XN i=1

Z t s

Z

R^d rW^N x X_r^{i;N 2}dx dr

=

2

N² XN i=1

Z t s

Z

R^djrWN(x)j²dx dr

because, by the change of variable x ! x Xr^i;N in the integral in dx, we have Z

R^d rWN x X_r^{i;N 2}dx = Z

R^djrWN(x)j²dx: