1
Compact sets in the space of measure valued functions
On Pr Rd , the set of all probability measures on Borel sets of Rd , we may introduce
several metrics. There is a general kind of metric given by
1
X
( ; )=
2
i=1
P1
i
jh ; i i h ; i ij
1 + jh ; i i h ; i ij
(or ( ; ) = i=1 2 i (jh ; i i h ; i ij ^ 1)) where f i g is a suitable sequences of bounded
d
continuous functions.
R Another one, but on Pr1 R , the set of probability measures with
…nite …rst moment Rd jxj (dx) < 1, is the Wasserstein metric
n
o
W1 ( ; ) = sup jh ; i h ; ij : [ ]Lip 1
where [ ]Lip is the Lipschitz seminorm [ ]Lip = sups6=t j (t)jt sj(s)j (in fact this is not the
usual de…nition but a theorem, named Kantorovich-Rubinstein characterization). 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 Rd with the metrics d = above or E = Pr1 Rd with the
Wasserstein metric d = W1 . A family of measure-valued functions F
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 ))
for every 2 F
ii) for every > 0, there exists
t; s 2 [0; T ] such that jt sj < .
>1
> 0 such that d ( t ;
s)
<
for every
2 F and
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
Z
0
for every
T
Z
0
T
1 with
p > 1, and a constant C > 0 one has
W1 ( t ; s )p
dtds
jt sj1+ p
2 F , then (ii) holds, with E = Pr1 Rd .
1
C
Proof. Clearly, (ii) holds if there exists ; C > 0 such that sups6=t
2 F . Choose
> 0 such that (
jh t ; i
s)
h
s;
sup
jt
s6=t
s)
sj
ij
C jt
Z
C sup
T
[ ]L ip 1 0
C
Z
T
0
= C
sj [h t ; i]W
n
sj sup [h t ; i]W
C jt
It follows that
W1 ( t ;
s)
C, for every
) p > 1. Then, for some constant C > 0,
and therefore
W1 ( t ;
W1 ( t ;
jt sj
Z
0
Z
T
Z
T
0
sup[
Z
T
0
o
1 :
: [ ]Lip
jh t ; i h s ; ijp
dtds
jt sj1+ p
]L ip 1 jh t ;
jt
0
T
;p
;p
i
sj1+
p
h
s;
ij
p
dtds
W1 ( t ; s )p
dtds:
jt sj1+ p
To prove the second condition in examples, for the metric , it is useful to have the
following criterion.
Lemma 3 Assume that, for every
> 0, there exists > 0 such that
2 Cb Rd , one has the following property: for every
jh t ; i
h
s;
ij <
for every
2 F and t; s 2 [0; T ] such that jt sj < . Then condition (ii) above holds,
with E = Rd . In particular, a su¢ cient condition is that there exist 2 (0; 1) with the
following property: for every 2 Cb Rd there is a constant C > 0 such that
[h ; i]C
C
for every
2 F . Another su¢ cient condition is that there exist 2 (0; 1) and p 1 with
p > 1 with the following property: for every 2 Cb Rd there is a constant C > 0 such
that
[h ; i]W ;p C
for every
2 F.
Proof. Given
> 0, let k 2 N be such that 2
1
X
i=k+1
2
i
k
jh t ; i i h s ; i ij
1 + jh t ; i i h s ; i ij
2
=2. Notice that
1
X
i=k+1
2
i
2
k
=2
hence
d ( t;
s)
k
X
i
2
i=1
Since k is …nite, there exists
> 0 such that
jh t ;
for every
ii
h
s;
2 F and t; s 2 [0; T ] such that jt
d ( t;
for every
jh t ; i i h s ; i ij
+ =2:
1 + jh t ; i i h s ; i ij
s)
2
i ij
< =2
sj <
k
X
2
i=1
2 F and t; s 2 [0; T ] such that jt
i
and i = 1; :::; k. Then
+
2
<
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 there exists a constant Ct > 0 such that
Z
jxj t (dx) Ct
Rd
for every
2 F , then condition (i) is ful…lled, for that value of t.
Proof.
t (B (0; r)
hence, given
c
1
r
)
> 0, we may choose r >
Ct
Z
Rd
jxj
t (dx)
Ct
r
.
R
R
Remark 6 We may replace Rd jxj t (dx) in the hypothesis of the lemma by Rd g (jxj)
with any increasing function g such that limr!+1 g (r) = +1.
t (dx)
Corollary 7 Let 2 (0; 1) and p 1 with p > 1, be given. For any M; R > 0 the set
KM;R C [0; T ] ; Pr1 Rd de…ned as
)
(
Z
Z TZ T
W1 ( t ; s )p
dtds R
KM;R =
: sup
jxj t (dx) M;
jt sj1+ p
0
0
t2[0;T ] Rd
is relatively compact in C [0; T ] ; Pr1 Rd .
3
Corollary 8 Let 2 (0; 1) and p 1 with p > 1, and g be an increasing function such
that limr!+1 g (r) = +1. For every M > 0 and every function 7! C from Cb Rd to
(0; 1), the set KM;C
C [0; T ] ; Pr Rd de…ned as
(
)
Z
KM;C =
: sup
t2[0;T ]
g (jxj)
Rd
t (dx)
M; [h ; i]W
;p
C for every
2 Cb R d
is relatively compact in C [0; T ] ; Pr Rd .
2
Mean …eld model. Compactness of the empirical measure
Consider the interacting particle system
dXti;N
N
1 X
K Xti;N
=
N
Xtj;N dt + dBti
j=1
with i = 1; :::; N , K : Rd ! Rd bounded Lipschitz, > 0. We assume to have a …ltered
probability space ( ; F; Ft ; P ) and independent Brownian motions Bti in Rd .
Remark 9 Each particle Xti;N interacts with every other particle Xtj;N such that the distance Xti;N Xtj;N is in the support of K. Even if K is compact support, in the limit when
N ! 1 we must expect that the number of particles Xtj;N interacting with Xti;N goes to
in…nity since, as we shall see, the empirical measure will converge to a probability measure,
so particles remain relatively concentrated. For this reason, this model is also called "long
range". It is not strictly correct if we want to describe local interactions, like membrane
interactions between cells.
On the initial
X0i;N , we assume they are F0 -measurable and:
i
h conditions
a) supi;N E X0i;N < 1
! h 0 ; i in probability, for every 2
b) there is 0 2 Pr1 Rd such that S0N ;
d
Cb R .
This is true in particular when X0i;N = X0i , where X0i i2N is a sequence of i.i.d. F0 measurable r.v.’s with law 0 2 Pr1 Rd .
Denote by StN the empirical measure and by QN the law of S N on C [0; T ] ; Pr1 Rd .
Theorem 10 Under assumption (a), QN
Proof. Step1. Given
N 2N
is relatively compact on C [0; T ] ; Pr1 Rd .
> 0, we have to …nd a compact set K
QN (K ) > 1
4
E such that
for every N 2 N. We look for a set of the form KM;R as described in Corollary 7. We have
c
QN KM;R
c
= P S N 2 KM;R
!
Z
P
sup
jxj StN (dx) > M + P
Z
t2[0;T ] Rd
T
0
Z
T
0
W1 StN ; SsN
jt
sj1+
p
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
Rd
jxj StN (dx) =
N
1 X i;N
Xt
N
i=1
hence
P
sup
Z
t2[0;T ] Rd
jxj StN
(dx) > M
!
1
E
M
"
sup
Z
t2[0;T ] Rd
"
=
i=1
i
#
(dx)
#
N
1 X i;N
sup
Xt
t2[0;T ] N i=1
#
"
N
1 1 X
i;N
E sup Xt
MN
t2[0;T ]
1
E
M
=
h
jxj StN
C
M
C will be checked below in Step 2. Similarly,
where the property E supt2[0;T ] Xti;N
we have
!
Z TZ T
Z TZ T
p
p
W1 StN ; SsN
W1 StN ; SsN
1
P
dtds > R
E
dtds
R
jt sj1+ p
jt sj1+ p
0
0
0
0
Z Z
p
1 T T E W1 StN ; SsN
=
dtds:
R 0 0
jt sj1+ p
Thus we have to prove that
h
E W1 StN ; SsN
p
i
C jt
sj1+
for some p
1 and > 0. We shall prove this below in Step 3. If so, choose
small that p < and get
!
Z TZ T
p
W1 StN ; SsN
C
:
P
1+ p dtds > R
R
jt sj
0
0
5
> 0 so
c
< .
Now, given > 0 above, we may …nd M; R > 0 such that QN KM;R
Step2. We simply have
Xti;N
X0i
N Z
1 X t
K Xsi;N
+
N
0
Xsj;N
ds +
Bti
j=1
+ kKk1 T + Bti
R
= jxj 0 (dx) < 1,
#
"
X0i
hence, recalling that E X0i
"
E
sup
t2[0;T ]
Xti;N
C+ E
Bt1
sup
t2[0;T ]
#
C:
StN ;
Step3. In order to estimate W1 StN ; SsN we …rst estimate
Lip( ) 1. We have
StN ;
SsN ;
=
N
1 X
N
1
N
1
N
Hence
i=1
N
X
i=1
N
X
Xti;N
Xsi;N
Xsi;N :
i=1
N
1 X i;N
Xt
N
W1 StN ; SsN
with
Xsi;N
Xti;N
Xti;N
SsN ;
Xsi;N :
i=1
PN
Therefore (treating N1 i=1 as an integration w.r.t. a probability measure on the natural
numbers 1; :::; N , hence using Hölder inequality)
p
W1 StN ; SsN
h
E W1 StN ; SsN
N
1 X i;N
Xt
N
Xsi;N
p
i=1
i
N
pi
1 X h i;N
E Xt
Xsi;N
:
N
i=1
pi
Xsi;N . The system is reversible, hence we could re-
p
h
Thus we have to estimate E Xti;N
h
pi
duce to E Xt1;N Xs1;N , but it is unessential, just conceptually interesting (a paradigm
is that the quantitative analysis of particle i = 1 is su¢ cient).
6
From the SDE we have
Xti;N
Xsi;N =
N Z
1 X t
K Xri;N
N
s
Xrj;N dr +
Bti
Bsi
j=1
hence
Xti;N
N Z
1 X t
K Xri;N
N
s
Xsi;N
Xrj;N
Bti
dr +
Bsi
j=1
kKk1 jt
sj +
Bti
Bsi
C jt
sjp + CE
C jt
sjp + C jt
which implies
E
h
Xti;N
Xsi;N
pi
C jt
h
Bti
Bsi
sjp=2
p
i
sjp=2
renaming each time the constants. Choosing p > 2 and setting
now complete because we have proved
h
i
p
E W1 StN ; SsN
C jt sj1+ .
=
p
2
1 > 0, the proof is
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 of a family of empirical measures
StN , independenly of the precise equations satis…ed by the particles:
Proposition 11 If, for some p; ; C > 0
#
"
Z
N
jxj St (dx)
E sup
C
t2[0;T ] Rd
h
Q W1 StN ; SsN
p
i
C jt
sj1+
for every N 2 N, then the famyly of laws QN of S N is relatively compact in Pr C [0; T ] ; Pr1 Rd
The fact that QN are laws of S N is not essential. A general criterion is:
Proposition 12 Set C = C [0; T ] ; Pr1 Rd . Assume that G
probability measures on C. If, for some p; ; C > 0
!
Z
Z
sup
jxj t (dx) dQ ( ) C
C
t2[0;T ] Rd
7
Pr (C) is a family of
.
Z
C
W1 ( t ;
s)
p
dQ ( )
C jt
sj1+
for every Q 2 G, then G is relatively compact.
Here the notation
3
stands for the generic element of C [0; T ] ; Pr1 Rd .
Passage to the limit
Let us now examine the limit points of QN N 2N , the family of laws of the empirical
process. The aim is to prove that, in the limit, we get solutions of the nonlinear FokkerPlanck equation
2
@ t
=
div ( t K
(1)
t
t)
@t
2
with initial condition 0 . However, these solutions are, at least a priori, only probability
measures, depending on time (and a priori also on randomness). Thus we use the concept
of measure-valued solutions, completely similar to the one given in a previous section in
the linear case.
Let us …rst give the intuition about the result. Preliminarily, notice that
N
1 X
K x
N
Xtj;N
=
j=1
Z
y) StN (dy) =: K
K (x
StN (x)
hence the SDE of the interacting particles can be rewritten as
dXti;N = K
Let
t (x)
Xti;N dt + dBti :
StN
Rd . By Itô formula we have
be a test function of calls Cb1:2 [0; T ]
d
t
Xti;N
=
@ t
Xti;N dt + r
@t
+r
t
Xti;N
t
dBti +
8
Xti;N
K
2
2
t
StN
Xti;N dt:
Xti;N dt
Therefore, being StN ;
=
t
d StN ;
1
N
=
t
PN
Xti;N ,
t
i=1
N
1 X@ t
Xti;N dt
N
@t
+
+
i=1
N
X
1
N
i=1
N
X
1
N
i=1
r
t
Xti;N
K
r
t
Xti;N
dBti
N
1 X 2
+
N
2
Xti;N dt
StN
Xti;N dt
t
i=1
StN ;
=
@ t
@t
dt + StN ; r
where
Mt ;N =
t
K
dt + dMt ;N +
StN
N Z
1 X t
r
N
0
s
Xsi;N
2
2
StN ;
t
dt
dBsi :
i=1
Therefore we have proved:
Lemma 13 The empirical measure StN satis…es the identity
d StN ;
t
=
StN ;
@ t
@t
dt + StN ; r
t
K
StN
dt + dMt ;N +
2
2
StN ;
t
dt:
Remark 14 We cannot write Mt ;N in terms of StN , at least directly. However, this is
true for the quadratic variations:
h
M
;N
;M
;N
i
t
2
=
N2
N Z
X
i=1
0
t
r
i;N
s Xs
2
2
ds =
N
Z tD
0
SsN ; jr
E
2
j
ds:
s
By a representation theorem for stochastic integrals, there exists an auxiliary Brownian
motion t such that
Z t rD
E
;N
SsN ; jr s j2 d s :
Mt
=p
N 0
In this somewhat arti…cial way we may consider the identity satis…ed by StN as a closed
equation.
9
From the identity of the lemma, if we assume true for a second that StN weakly converges
to a limit measure t , we see (at least intuitively) that t satis…es the nonlinear FokkerPlanck equation (1). To be rigorous, however, we have to deal with the weak convergence
of QN , the laws of S N .
Theorem 15 If Q is a weak limit point of a subsequence of QN N 2N , then Q is supported
on the measure-valued solutions of the nonlinear Fokker-Planck equation
2
@ t
=
@t
2
with initial condition
d StN ;
t
@ t
@t
StN ;
=
t)
0.
Rd , consider the functional
2 Cb1:2 [0; T ]
Proof. For every
div ( t K
t
( ) := sup h t ;
dt + StN ; r
ti
t2[0;T ]
h
0;
Z
0i
t
s;
0
2
dt + dMt ;N +
StN
K
t
2
@ s
+
@t
2
s
+r
2
s
StN ;
(K
t
s)
dt:
ds :
It is continuous on C [0; T ] ; Pr1 Rd . Hence, if QNk is a subsequence which weakly
converges to Q, by Portmanteau theorem we have
Q(
lim inf QNk (
( )> )
k!1
( ) > ):
But
QNk (
S Nk >
D
StNk ;
sup
( )> ) = P
= P
t2[0;T ]
t
E
h
0;
0i
Z
t
SsNk ;
0
2
@ s
+
@t
2
s
From the identity of Lemma 13 we get
QNk (
( )> )=P
sup
t2[0;T ]
P
P
D
D
S0Nk
S0Nk
0;
0;
0
0
E
E
D
S0Nk
+ sup Mt ;Nk >
t2[0;T ]
> =2 + P
0
!
sup Mt
t2[0;T ]
10
0;
E
;Nk
Mt ;Nk >
!
> =2 :
!
+r
s
K
SsNk
ds >
The …rst term goes to zero, as k ! 1, by the assumption on S0N . For the second term we
have
!
"
#
2
4
P
sup Mt ;Nk > =2
sup Mt ;Nk
2E
t2[0;T ]
t2[0;T ]
MT ;Nk
CE
2
because Mt ;Nk is a martingale. Now, it is well known that
Z
E
t
0
r
Z
dBsi
i;Nk
s Xs
t
0
r
dBsj
Xsi;Nk
s
=0
if i 6= j because B i and B j are independent. Hence
E
MT
;Nk
2
=
Nk
2 X
Nk2
E
" Z
0
i=1
2
t
r
Xsi;Nk
s
dBsi
#
:
By the isometry formula for Itô integrals, this is equal to
=
Nk Z t
2 X
Nk2 i=1
0
r
s
2
Xsi;Nk
ds
which is bounded by
2
C
as k ! 1.
Therefore Q (
Nk
( ) > ) = 0. Since this holds true for every
Rd , Q is supported on the set of measure-valued func-
Hence, for every 2 Cb1:2 [0; T ]
tions
such that
ti = h
0;
0i +
> 0, we deduce
( ) = 0) = 1:
Q(
h t;
kr k21 T ! 0
Z
0
t
s;
2
@ s
+
@t
2
s
+r
s
(K
s)
ds:
This implies that Q is supported on the set of measure-valued solutions of the nonlinear
Fokker-Planck equation, by an argument of countable density of functions.
11
4
Uniqueness for the PDE and global limit of the empirical
measure
Theorem 16 The nonlinear (1), with initial condition
one measure-valued solution.
0
2 Pr Rd , has one and only
Proof. Existence has been proved above: the support of any limit measure Q of the
family QN N 2N is non empty and it is made of such solutions. The proof of uniqueness
is postponed.
Corollary 17 The family QN N 2N of laws of the empirical processes StN weakly converge
to
where 2 C [0; T ] ; Pr1 Rd is the unique solution of equation (1).
Proof. From the compactness of QN N 2N we know that from every subsequence we
may extract a further subsequence which converges to some limit measure Q, supported
on measure-valued solutions of (1). By the previous uniqueness theorem, Q =
where
d
2 C [0; T ] ; Pr1 R
is the unique solution of equation (1). Since this holds true
for all limit points Q, and the weak convergence in Pr C [0; T ] ; Pr1 Rd
is a metric
N
convergence, we deduce that the full sequence Q N 2N converges, to .
12