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 V^{0} for rV , we have

V^{0}(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_{t}^{i}

in the form

dX_{t}^{i;N} = N^{1=d}
XN
j=1

V^{0} N^{1=d} X_{t}^{i;N} X_{t}^{j;N} dt + dB_{t}^{i}:
Setting further

V_{N}(x) = N V N^{1=d}x

we have

V_{N}^{0} (x) = N N^{1=d}V^{0} N^{1=d}x
hence we may write the previous SDE as

dX_{t}^{i;N} = 1
N

XN j=1

V_{N}^{0} X_{t}^{i;N} X_{t}^{j;N} dt + dB_{t}^{i}:

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_{N}^{0} X_{t}^{i;N} X_{t}^{j;N} dt + dB_{t}^{i}: (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_{M}^{0} X_{t}^{i;N;M} X_{t}^{j;N;M} dt + dB_{t}^{i} (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=d}x :

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} = _{N}^{1} PN

i=1 X_{t}^{i;N;M} weakly converges to the
measure-valued solution _{t}of the Mean Field equation

@ _{t}

@t =

2

2 ^{t}+ div _{t} V_{M}^{0} _{t} :

Assume (we need it below) that _{t} has a density ut(x). More precisely, both _{t}and 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_{M}^{0} 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 C0^{1},

u^{M}_{t} ; = u^{M}_{0} ; +

2

2 Z t

0

u^{M}; ds
Z t

0

V_{M}^{0} u^{M}; u^{M}r 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_{M}^{0} u^{M}; u^{M}r ds:

In addition, if we prove that V_{M}^{0} u^{M} ! v in a suitable sense, we get
hu^{t}; i = hu^{0}; 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 C0^{1},
V_{M}^{0} u^{M}; = u^{M}; V_{M}^{0} ( ) :
Moreover,

V_{M}^{0} ( ) (x) =
Z

V_{M}^{0} (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=d}x if V is proportional to a smooth probability density

V (x) = ^{2}_{1}f (x)

where ^{2}_{1} > 0 and f is a pdf. In this case, since is smooth and compact support,
RV_{M}(y x) (y) dy ! ^{2}1 (x), hence (V_{M}^{0} ( ) ) (x) ! ^{2}1r (x) and …nally

V_{M}^{0} u^{M}; ! ^{2}1hu; r i = ^{2}1hru; i :
We have found

v = ^{2}_{1}ru
and therefore the limit PDE

@u

@t =

2

2 u +

2 1

2 u^{2}: (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_{M}^{0} (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,

V^{0}(x) = 1_{jxj 1}(1 jxj) x
jxj
and we rescale it as above:

V_{M}^{0} (x) = M M^{1=d}V^{0} M^{1=d}x

= M^{2}V^{0}(M x) :

The following functions de…ne part of V^{0} with a generic rescaling, where in the sequel we
shall take R = M ^{1}, = M^{2}.

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

V^{0}(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 ^{=d}x

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_{N}^{0} 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_{s}^{i}:

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}^{t}E

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_{N}^{0} 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_{n}^{2} 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} = _{x}_{n}, = _{x}, x_{n} ! x, then n * . If f_{n} converges uniformly to f ,
we see that hf^{n}; ni = f^{n}(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_{N}^{0} 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_{N}^{0} S_{t}^{N}; ! ^{2}1

Z

r (x) t(dx) = ^{2}_{1}
Z

(x) rut(x) dx
in the case when _{t} has a di¤erentiable density u_{t}(x). Indeed

V_{N}^{0} S_{t}^{N}; = S_{t}^{N}; V_{N}( ) r ! ^{2}1

Z

r (x) t(dx) = ^{2}_{1}
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 ^{2}_{1} = 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_{N}^{0} S_{t}^{N} = r tS_{t}^{N}; W_{N} W_{N}^{0} 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 L^{2}-convergence of r WN S_{t}^{N} and strong L^{2}-convergence of
W_{N} S_{t}^{N}. Recall indeed that if f_{n}! f in L^{2} R^{d} and g_{n}* g in L^{2} R^{d} , then

jhfn; g_{n}i hf; gij jhfn; g_{n}i 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} X_{0}^{i;N} + 1
N

XN j=1

Z t 0

V_{N}^{0} X_{s}^{i;N} X_{s}^{j;N} ds + B_{t}^{i}

and V_{N}^{0} is unbounded with respect to N , so we cannot repeat the estimates of the mean

…eld case. Notice that 1 N

X

j

V_{N}^{0} X_{t}^{i} X_{t}^{j} dt = ruN t; X_{t}^{i}

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^{N}t X_{t}^{i} dt + dB_{t}^{i}:
Thus

X_{t}^{i;N} X_{0}^{i;N} +
Z t

0 ru^{N}s X_{s}^{i} ds + B_{t}^{i} :
If we prove suitable estimates on ru^{N}t , 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^{N}t . 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

u^{2}dx =
Z

u@u

@tdx = Z

u

2

2 u + div (uru) dx

=

2

2 Z

jruj^{2}dx
Z

u jruj^{2}dx
hence

1 2

Z

u^{2}_{t}dx +
Z t

0 2

2 Z

jruj^{2}dx +
Z

u jruj^{2}dx ds = 1
2

Z
u^{2}_{0}dx
where we stress the remarkable fact that

Z

u div (uru) dx = Z

ru (uru) dx = Z

u jruj^{2}dx:

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_{N}^{0} X_{t}^{i;N} X_{t}^{j;N} dt + dB_{t}^{i}
with

VN(x) = N V N ^{=d}x

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 = ^{2}_{1}W W

where W is a smooth compact support probability density function and ^{2}_{1} is a positive
constant. Set V_{N}(x) = N V N ^{=d}x , W_{N}(x) = N W N ^{=d}x (so that V_{N} = ^{2}_{1}W_{N}
WN) and set w^{N}_{t} (x) := WN S_{t}^{N} (x). Assume

d
d + 2:
Finally, assume that the initial conditions X_{0}^{i;N} satisfy

E Z

R^{d}

w^{N}_{0} (x)^{2}dx C; E X_{0}^{i;N} ^{2} C

and S_{0}^{N}; ! hu0; i in probability, for every 2 Cc^{1}, with u_{0}2 L^{2} R^{d} , 0 u_{0} 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 L^{2}_{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 u^{2}
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_{N}^{0} 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 = ^{2}_{1}W
W , we have

1 2

Z

R^{d}

w^{N}_{t} (x)^{2}dx +

2

2 Z t

0

Z

R^{d} rw^{N}s (x)^{2}dx ds +
Z t

0

Z

R^{d}

V_{N}^{0} S_{s}^{N} (x)^{2}S_{s}^{N}(dx) ds

= 1 2

Z

R^{d}

w^{N}_{0} (x)^{2}dx +
Z

R^{d}

Z t 0

w_{s}^{N}(x) dM_{s}^{N} dx +
Z

R^{d}
2

N^{2}
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) ^{2}dx +

2

2 E Z t

0

Z

R^{d} rw^{N}s (x)^{2}dx ds + E
Z t

0

Z

R^{d} ru^{N}s (x) ^{2}S_{s}^{N}(dx) ds
1

2 Z

R^{d}

w^{N}_{0} (x)^{2}dx + 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_{N}^{0} S_{t}^{N} =
Z

ryW_{N}(x y) V_{N}^{0} S_{t}^{N} (y) S^{N}_{t} (dy)

= div

Z

WN(x y) V_{N}^{0} S_{t}^{N} (y) S_{t}^{N}(dy)

= div WN V_{N}^{0} S_{t}^{N} S_{t}^{N} :
We get

dw_{t}^{N}(x) = div W_{N} V_{N}^{0} 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} dB^{i}_{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_{N}^{0} S_{t}^{N} S_{t}^{N} (x) dt + ^{2}w_{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

N^{2}
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" ^{1}_{2}R

R^{d} w^{N}_{t} (x)^{2}dx:

1 2

Z

R^{d}

w^{N}_{t} (x)^{2}dx = 1
2

Z

R^{d}

w_{0}^{N}(x) ^{2}dx
+

Z

R^{d}

Z t 0

w_{s}^{N}(x) div W_{N} V_{N}^{0} 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

N^{2}
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^{N}s (x)^{2}dx 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_{N}^{0} S_{s}^{N} S_{s}^{N} (x) ds dx

= Z t

0

Z

R^{d}

w_{s}^{N}(x) div W_{N} V_{N}^{0} S_{s}^{N} S_{s}^{N} (x) dx ds

=

Z t 0

Z

R^{d}rw^{N}s (x) WN V_{N}^{0} 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_{N}^{0} S_{s}^{N} (x) S_{s}^{N}(dx) ds:

From the assumption V = ^{2}_{1}W W it follows V_{N} = ^{2}_{1}W_{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_{N}^{0} 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_{N}^{0} S_{s}^{N} S_{s}^{N} (x) ds dx

=

Z t 0

Z

R^{d}

V_{N}^{0} S_{s}^{N} (x)^{2}S_{s}^{N}(dx) ds:

Let us prove that VN = ^{2}_{1}W_{N} WN:

2

1 W_{N} WN (x) = ^{2}_{1}
Z

WN( x + y) WN(y) dy = ^{2}_{1}N
Z

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

= ^{2}_{1}N
Z

W y N ^{=d}x W (y) dy = ^{2}_{1}N W W N ^{=d}x

= N V N ^{=d}x = 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

N^{2}
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

N^{2}
XN
i=1

Z t

s rWN x X_{r}^{i;N} ^{2}dr

! dx

=

2

N^{2}
XN
i=1

Z t s

Z

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

=

2

N^{2}
XN
i=1

Z t s

Z

R^{d}jrWN(x)j^{2}dx 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^{d}jrWN(x)j^{2}dx: