On Pr R d , the set of all probability measures on Borel sets of R d , we may introduce several metrics. There is a general kind of metric given by

1 Compact sets in the space of measure valued processes

On Pr R ^d , the set of all probability measures on Borel sets of R ^d , we may introduce several metrics. There is a general kind of metric given by

( ; ) = X 1

i=1

2 ⁱ jh ; i i h ; i ij 1 + jh ; i i h ; i ij (or ( ; ) = P ₁

i=1 2 ⁱ (jh ; i i h ; i ij ^ 1)) where f i g is a suitable sequences of bounded continuous functions. Another one, but on Pr ₁ R ^d , the set of probability measures with

…nite …rst moment R

R

jxj (dx) < 1, is the Wasserstein metric W 1 ( ; ) = sup fjh ; i h ; ij : Lip ( ) 1g

(in fact this is not the usual de…nition but a theorem, named Kantorovich-Rubinstein char- acterization). In both cases we have a complete separable metric space and the convergence is equivalent to the weak convergence of probability measures. As a consequence of the general version of Ascoli-Arzelà theorem in metric spaces, we have the following fact.

Proposition 1 Let E = Pr R ^d with the metrics d = above or E = Pr 1 R ^d with the Wasserstein metric d = W 1 . A family of functions F = _t ; t 2 [0; T ] ₂ C ([0; T ] ; E) is relatively compact in C ([0; T ] ; E) if

i) for every t 2 [0; T ], for every > 0, there exists r ;t > 0 such that

t (B (0; r ;t )) > 1 for every 2

ii) for every > 0, there exists > 0 such that d _t ; _s < for every 2 and t; s 2 [0; T ] such that jt sj < .

To prove the second condition in examples, in the case of the Wasserstein metric, we may use the following fact.

Lemma 2 If, for some 2 (0; 1) and p 1 with p > 1, one has Z T

0

Z T 0

W ¹ t ; _s ^p

jt sj ^{1+ p} dtds C for every 2 , then (ii) holds.

Proof. Clearly, (ii) holds if there exists ; C > 0 such that sup _s6=t ^W

(

;

)

jt sj C, for every 2 . Choose > 0 such that ( ) p > 1. Then, for some constant C > 0,

D

t ; E D

s ; E

C jt sj hD

t ; Ei

W

and therefore

W ¹ t ; _s C jt sj sup nhD

t ; Ei

W

: Lip ( ) 1 o

: It follows that

sup

s6=t

W ¹ S _t ^N ; S _s ^N

jt sj C sup

Lip( ) 1

Z T 0

Z T 0

t ; _s ; ^p

jt sj ^{1+ p} dtds C

Z T 0

Z T 0

sup _{Lip( ) 1 t} ; _s ; ^p

jt sj ^{1+ p} dtds

= C Z T

0

Z T 0

W ¹ t ; _s ^p jt sj ^{1+ p} dtds:

To prove the second condition in examples, for the metric , it is useful to have the following criterion.

Lemma 3 Assume that, for every 2 C b R ^d , one has the following property: for every

> 0, there exists > 0 such that D

t ; E D

s ; E

<

for every 2 and t; s 2 [0; T ] such that jt sj < . Then condition (ii) above holds. In particular, a su¢ cient condition is that there exist 2 (0; 1) with the following property:

for every 2 C b R ^d , one has

hD

; Ei

C C

for every 2 . Another su¢ cient condition is that there exist 2 (0; 1) and p 1 with p > 1 with the following property: for every 2 C b R ^d , one has

hD

; Ei

W

C

for every 2 .

Proof. Given > 0, let k 2 N be such that 2 ^k =2. Notice that X 1

i=k+1

2 ^{i t} ; _{i s} ; _i 1 + _t ; _i h s ; _i i

X 1 i=k+1

2 ⁱ 2 ^k =2

hence

d _t ; _s

X k i=1

2 ^{i t} ; _{i s} ; _i

1 + _t ; _i h s ; _i i + =2:

Since k is …nite, there exists > 0 such that D

t ; _i E D

s ; _i E

< =2

for every 2 and t; s 2 [0; T ] such that jt sj < and i = 1; :::; k. Then d _t ; _s

2 X k

i=1

2 ⁱ + 2 <

for every 2 and t; s 2 [0; T ] such that jt sj < . This is property (ii).

Remark 4 The previous result is somewhat in the same spirit as Aubin-Lions lemma: the regularity in time necessary for tightness is su¢ cient when the space regularity is measured in a very weak sense. Compactness in time and in space are dealt with separately.

Lemma 5 Let t 2 [0; T ] be given. If Z

R

jxj t (dx) C _t

for every 2 , then condition (i) is ful…lled, for that value of t.

Proof.

t (B (0; r) ^c ) 1 r

Z

R

jxj t (dx) C _t r hence, given > 0, we may choose r > ^C

.

Remark 6 We may replace R

R

jxj t (dx) in the hypothesis of the lemma by R

R

g (jxj) t (dx) with any increasing function g which diverges to in…nity as jxj ! 1.

Corollary 7 Let 2 (0; 1) and p 1 with p > 1, be given. For any M; R > 0 the set K ^M;R C [0; T ] ; Pr 1 R ^d de…ned as

K M;R = (

: sup

t2[0;T ]

Z

R

jxj t (dx) M;

Z T 0

Z T 0

W 1 t ; _s ^p

jt sj ^{1+ p} dtds R )

for some 2 (0; 1) and p 1 with p > 1, is relatively compact in C [0; T ] ; Pr ₁ R ^d . Corollary 8 Let 2 (0; 1) and p 1 with p > 1, and g be an increasing function which diverges to in…nity as jxj ! 1, be given. For every M > 0 and function 7! C from C _b R ^d to (0; 1), the set K M;C C [0; T ] ; Pr R ^d de…ned as

K ^M;C = (

: sup

t2[0;T ]

Z

R

g (jxj) t (dx) M; [h ; i] W

C for every 2 C b R ^d )

for some 2 (0; 1) and p 1 with p > 1, is relatively compact in C [0; T ] ; Pr R ^d .

2 Mean …eld

Consider the interacting particle system

dX _t ^i;N = 1 N

X N j=1

K X _t ^i;N X _t ^j;N dt + dB _t ⁱ

with i = 1; :::; N , K : R ^d ! R ^d bounded Lipschitz, > 0. We assume to have a …ltered probability space ( ; F; F t ; P ) and independent Brownian motions B _t ⁱ in R ^d . Assume that the initial conditions X ₀ ⁱ are i.i.d. F ⁰ -measurable r.v.’s, with law ₀ 2 Pr ¹ R ^d .

Denote by S _t ^N the empirical measure and by Q ^N the law of S ^N on C [0; T ] ; Pr ₁ R ^d . Theorem 9 Q ^N

N 2N is relatively compact.

Proof. Step1. Given > 0, we have to …nd a compact set K E such that Q ^N (K ) > 1

for every N 2 N. We look for a set of the form K M;R as described in Corollary 7. We have Q ^N K M;R ^c = P S ^N 2 K ^c M;R

P sup

t2[0;T ]

Z

R

jxj S t ^N (dx) > M

! + P

Z T 0

Z T 0

W ¹ S _t ^N ; S _s ^{N p}

jt sj ^{1+ p} dtds > R

! :

We separately prove that both probabilities are smaller than =2, for every N 2 N. In both cases we apply Chebishev inequality. We have

Z

R

jxj S t ^N (dx) = 1 N

X N i=1

X _t ^i;N

hence

P sup

t2[0;T ]

Z

R

jxj S t ^N (dx) > M

! 1

M E

"

sup

t2[0;T ]

Z

R

jxj S t ^N (dx)

#

= 1

M E

"

sup

t2[0;T ]

1 N

X N i=1

X _t ^i;N

#

= 1

M 1 N

X N i=1

E

"

sup

t2[0;T ]

X _t ^i;N

# C

M

where the property E h

sup _{t2[0;T ]} X _t ^i;N i

C will be checked below in Step 2. Similarly, we have

P Z T

0

Z T 0

W 1 S _t ^N ; S _s ^{N p}

jt sj ^{1+ p} dtds > R

! 1

R E Z T

0

Z T 0

W 1 S _t ^N ; S _s ^{N p} jt sj ^{1+ p} dtds

= 1

R Z T

0

Z T 0

E W ¹ S ^N _t ; S _s ^{N p} jt sj ^{1+ p} dtds:

Thus we have to prove that E

h

W ¹ S _t ^N ; S _s ^{N p} i

C jt sj ¹⁺

for some p 1 and > 0. We shall prove this below in Step 3. If so, choose > 0 so small that p < and get

P Z T

0

Z T 0

W 1 S _t ^N ; S _s ^{N p}

jt sj ^{1+ p} dtds > R

! C

R :

Now, given > 0 above, we may …nd M; R > 0 such that Q ^N K ^c M;R < . Step2. We simply have

X _t ^i;N X ₀ ⁱ + 1 N

X N j=1

Z t 0

K X _s ^i;N X _s ^j;N ds + B _t ⁱ X ₀ ⁱ + kKk ₁ T + B _t ⁱ

hence, recalling that E X ₀ ⁱ = R

jxj 0 (dx) < 1,

E

"

sup

t2[0;T ]

X _t ^i;N

#

C + E

"

sup

t2[0;T ]

B _t ¹

# C:

Step3 . In order to estimate W 1 S _t ^N ; S _s ^N we …rst estimate S _t ^N ; S _s ^N ; with Lip( ) 1. We have

S _t ^N ; S ^N _s ; = 1 N

X N i=1

X _t ^i;N X _s ^i;N 1

N X N i=1

X _t ^i;N X _s ^i;N 1

N X N i=1

X _t ^i;N X _s ^i;N :

Hence

W 1 S _t ^N ; S _s ^N 1 N

X N i=1

X _t ^i;N X _s ^i;N : Therefore (treating _N ¹ P N

i=1 as an integration w.r.t. a probability measure on the natural numbers 1; :::; N , hence using Hölder inequality)

W ¹ S _t ^N ; S _s ^{N p} 1 N

X N i=1

X _t ^i;N X _s ^{i;N p}

E h

W ¹ S _t ^N ; S _s ^{N p}

i 1

N X N i=1

E h

X _t ^i;N X _s ^{i;N p} i

: Thus we have to estimate E h

X _t ^i;N X _s ^{i;N p} i

. The system is reversible, hence we could reduce to E h

X _t ^1;N X s ^1;N p i

, but it is un-essential, just conceptually interesting (a par- adigm is that the quantitative analysis of particle i = 1 is su¢ cient).

From the SDE we have X _t ^i;N X _s ^i;N = 1

N X N j=1

Z t s

K X _r ^i;N X _r ^j;N dr + B _t ⁱ B _s ⁱ

hence

X _t ^i;N X _s ^i;N 1 N

X N j=1

Z t s

K X _r ^i;N X _r ^j;N dr + B _t ⁱ B _s ⁱ kKk ₁ jt sj + B _t ⁱ B _s ⁱ

which implies

E h

X _t ^i;N X _s ^{i;N p} i

C jt sj ^p + CE h

B ⁱ _t B _s ^{i p} i C jt sj ^p + C jt sj ^p=2 C jt sj ^p=2

renaming each time the constants. Choosing p > 2 and setting = ^p ₂ 1 > 0, the proof is now complete because we have proved

E h

W ¹ S _t ^N ; S _s ^{N p} i

C jt sj ¹⁺ .

It may be noticed that, from the proof of this theorem and the results of the previous

section, we may extract a general criterion of compactness.

Proposition 10 Assume that Q

2 is a family of probability measures on C [0; T ] ; Pr ₁ R ^d . If, for some p; ; C > 0

Q

"

sup

t2[0;T ]

Z

R

jxj t (dx)

# C Q [W ¹ ( _t ; _s ) ^p ] C jt sj ¹⁺

for all 2 , then Q ₂ is relatively compact on C [0; T ] ; Pr 1 R ^d . Here the notation _t stands for the generic element of C [0; T ] ; Pr 1 R ^d .

Let us now examine the limit points of Q ^N _{N 2N} , the family of laws of the empirical process. The ain is to prove that, in the limit, we get solutions of the nonlinear Fokker- Planck equation

@ _t

@t =

2

2 ^t div ( _t K _t )

with initial condition ₀ . However, these solutions are, at least a priori, only probability measures, depending on time (and a priori also on randomness). We need the de…nition of measure-valued solutions similar to the one given in the linear case.

Theorem 11 If Q is a weak limit point of a subsequence of Q ^N _{N 2N} , then Q is supported on the measure-valued solutions of the nonlinear Fokker-Planck equation

@ _t

@t =

2

2 ^t div ( _t K _t ) with initial condition ₀ .

Proof. For every 2 C c ¹ R ^d , consider the functional ( ) := sup

t2[0;T ] h t ; i h 0 ; i Z t

0 s ;

2

2 ds +

Z t

0 h s K _s ; r i ds : It is continuous on C [0; T ] ; Pr 1 R ^d . Hence, if Q ^N

is a subsequence which weakly converges to Q, by Portmanteau theorem we have

Q ( ( ) > ) lim inf

k!1 Q ^N

( ( ) > ) : But

Q ^N

( ( ) > ) = P S ^N

>

= P sup

t2[0;T ]

D

S ^N _t

; E

h 0 ; i Z t

0

S _s ^N

;

2

2 ds +

Z t 0

S _s ^N

K S _s ^N

; r ds

!

:

We need now a key ingredient: the equation satis…ed by S _t ^N . It is ... There- ofore

Q ^N

( ( ) > )

= P sup

t2[0;T ]

D

S ₀ ^N

; E

h 0 ; i M _t ^;N

!

: