1 The method of compactness for SDEs
1.1 Compactness in C [0; T ] ; Rd
Recall that a set is called relatively compact if its closure is compact. Every subset of a compact set is relatively compact.
Recall classical Ascoli-Arzelà theorem: a family of functions F C [0; T ] ; Rd is rela- tively compact (in the uniform topology) if
i) for every t 2 [0; T ], the set ff (t) ; f 2 F g is bounded ii) for every " > 0 there is > 0 such that
jf (t) f (s)j "
for every f 2 F and every s; t 2 [0; T ] with jt sj .
Recall the de…nition of the Hölder seminorm, for f : [0; T ] ! Rd, [f ]C = sup
t6=s
jf (t) f (s)j jt sj
and obviously of the supremum norm kfk1= supt2[0;T ]jf (t)j. Simple su¢ cient conditions for (i) and (ii) are
i’) there is M > 0 such that kfk1 M for all f 2 F
ii’) for some 2 (0; 1), there is R > 0 such that [f]C R for all f 2 F . Hence the sets
KM;R=n
f 2 C [0; T ] ; Rd ; kfk1 M; [f ]C Ro are relatively compact in C [0; T ] ; Rd .
The Sobolev space W ;p 0; T ; Rd , with 2 (0; 1) and p > 1, is de…ned as the set of all f 2 Lp 0; T ; Rd such that
[f ]W ;p :=
Z T 0
Z T 0
jf (t) f (s)jp
jt sj1+ p dtds < 1:
We endow W ;p 0; T ; Rd with the norm kfkW ;p = kfkLp+ [f ]W ;p. It is known that W ;p 0; T ; Rd C" [0; T ] ; Rd if ( ") p > 1
and
[f ]C" C"; ;pkfkW ;p:
Using these new spaces, simple su¢ cient conditions for (i) and (ii) are (i’) and
ii”) for some 2 (0; 1) and p > 1 with p > 1, there is R > 0 such that [f]W ;p R for all f 2 F .
Indeed, if (ii”) holds, there exists " > 0 such that ( ") p > 1, hence such that [f ]C"
C"; ;pkfkW ;p; moreover, kfkLp T1=pkfk1 T1=pM and therefore
[f ]C" C"; ;pkfkW ;p C"; ;p T1=pM + R for all f 2 F , which implies the validity of (ii’). Therefore the sets
KM;R0 =n
f 2 C [0; T ] ; Rd ; kfk1 M; [f ]W ;p Ro are relatively compact in C [0; T ] ; Rd , if p > 1.
1.2 Application to SDEs Consider the SDE
dXt= b (t; Xt) dt + (t; Xt) dBt; Xjt=0= x
with bounded continuous coe¢ cients. We may dream of a generalization of Peano theorem, namely just existence of a solution.
Let bn; n be a sequence of continuous functions, each one uniformly Lipschitz in x (with constant that may depend on n):
jbn(t; x) bn(t; y)j Lnjx yj j n(t; x) n(t; y)j Lnjx yj equibounded:
kbnk1+ k nk1 C
(here C > 0 is independent of n) and such that bn ! b, n ! , uniformly on compact sets of [0; T ] Rd. Let fXtng be the solutions of the equations
dXtn= bn(t; Xtn) dt + n(t; Xtn) dBt; Xnjt=0= x:
Let Qn be their laws on C [0; T ] ; Rd .
Lemma 1 The family fQng is tight in C [0; T ] ; Rd .
Proof. Step 1. For pedagogical reasons, we start with a partially insu¢ cient proof, to clarify the role of certain arguments.
Recalling the relatively compact sets KM;R above, it is su¢ cient to prove that, given
> 0, there are M; R > 0 such that
P Xn2 KM;Rc < (1)
for all n 2 N. Condition (1) means
P (kXnk1> M or [Xn]C > R) < : A su¢ cient condition is
P (kXnk1> M ) < =2 and P ([Xn]C > R) < =2:
Step 2. The …rst one is easy:
P (kXnk1> M ) 1 ME
"
sup
t2[0;T ]jXtnj
#
and
sup
t2[0;T ]jXtnj jxj + Z T
0 jbn(s; Xsn)j ds + sup
t2[0;T ]
Z t 0
n(s; Xsn) dBs
C + sup
t2[0;T ]
Z t 0
n(s; Xsn) dBs which implies
E
"
sup
t2[0;T ]jXtnj
#
C + E
"
sup
t2[0;T ]
Z t 0
n(s; Xsn) dBs
#
C + E
"
sup
t2[0;T ]
Z t 0
n(s; Xsn) dBs 2#1=2
C + E Z T
0 j n(s; Xsn)j2ds
1=2
C:
Hence, given > 0, we may choose M > 0 such that P (kXnk1> M ) < =2.
Step 3. The second one,
P sup
t6=s
jXtn Xsnj jt sj > R
!
< =2
however is more di¢ cult since it involves a double supremum in time and martingale in- equalities do not help. A way to prove this property is to use a quantitative Kolmogorov criterion; another is a variant of these tightness arguments based on stopping times, devel- oped by Aldous.
However we have seen above another class of compact sets based on Sobolev spaces.
The advantage of W ;p 0; T ; Rd with respect to C" [0; T ] ; Rd is that the topology is entirely de…ned by integrals, which merge with expectation better than a supremum. Let us use them in the next step.
Step 4. Recalling now the relatively compact sets KM;R0 above, using the same argu- ment as in step 1 and the result of step 2, we are left to prove that there exist 2 (0; 1) and p > 1 with p > 1, with the following property: given > 0, there is R > 0 such that
P ([Xn]W ;p > R) < =2 for every n 2 N. We have
P ([Xn]W ;p > R) 1 RE
Z T 0
Z T 0
jXtn Xsnjp jt sj1+ p dtds
= C
R Z T
0
Z T 0
E [jXtn Xsnjp] jt sj1+ p dtds:
Now, for t s,
Xtn Xsn = Z t
s
bn(r; Xrn) dr + Z t
s
n(r; Xrn) dBr
jXtn Xsnjp C Z t
s jbn(r; Xrn)j dr
p
+ C Z t
s
n(r; Xrn) dBr
p
C (t s)p+ C Z t
s
n(r; Xrn) dBr p
and, by the so called Burkholder-Davis-Gundy inequality, E
Z t s
n(r; Xrn) dBr
p
CE
" Z t
s j n(r; Xrn)j2dr
p=2#
C (t s)p=2: Therefore
P ([Xn]C" > R) C R
Z T 0
Z T 0
jt sjp=2 jt sj1+ pdtds
= C
R Z T
0
Z T 0
1
jt sj1+( 12)pdtds:
The integralRT 0
RT 0
1
jt sj1+( 12)pdtds is …nite if < 12; we need to use p > 2, because of the constraint p > 1. Thus we may …nd R > 0 such that the previous expression is smaller than =2, as required.
Remark 2 By the same computation, one can shows that B 2 W ;p 0; T ; Rd a.s., for every < 12 and p > 2, hence B 2 C" [0; T ] ; Rd a.s., for every " < 12, as we already know. It also provides a quantitative control on P ([B]C" > R), usually not stated when Kolmogorov regularity theorem is given.
Remark 3 The fractional Sobolev topology above provides one of the simplest proofs of Kolmogorov regularity theorem; obviously one has to accept the Sobolev embedding theorem, which in a sense incorporates arguments similar to those of dyadic partitions in the classical proof of Kolmogorov theorem.
Remark 4 From the previous proof we may deduce the following well-known criterium (see the books of Billingsley): if (i) holds and there are p; ; C > 0 such that
E [jXtn Xsnjp] C jt sj1+
for all n 2 N, then the sequence fQng is tight in C [0; T ] ; Rd .
We have proved that fQng is tight in C [0; T ] ; Rd . Hence there are subsequences which converge weakly. Let us take one of them and, just for simplicity of notation, let us denote it by fQng. Thus we are assuming that fQng converges weakly to some probability measure Q on Borel sets of C [0; T ] ; Rd . Our aim is to prove the existence of a solution of the SDE.
Assume for a second that the previous facts imply the existence of a continuous process X such that Xn converges to X a.s., in the uniform topology. This assertion is obviously false: convergence in law does not imply a.s. convergence. However, it is true in a more involved way, using a Skorohod representation theorem. The details are not trivial, for instance because the new processes eXtn, on a new probability space, given by such theorem, do not satisfy the SDE, a priori; one can prove that they satisfy it in a suitable weak sense.
So, let us miss these details, also because they will not enter the discussion in the case of the macroscopic limit, our …nal interest. And assume (although not true) that the original sequence Xn converges to X a.s., in the uniform topology on compact sets. It follows that
Z t s
bn(r; Xrn) dr ! Z t
s
b (r; Xr) dr Z t
s
n(r; Xrn) dBr ! Z t
s
(r; Xr) dBr
in probability, by the P -a.s. uniform convergence of bn(r; Xrn) (resp. n(r; Xrn)) to b (r; Xr) (resp. (r; Xr)) and the equiboundedness of bn(r; Xrn) (resp. n(r; Xrn)). It follows that Xt satis…es the SDE.
1.3 The zero-noise example
Concerning this last issue, the convergence, there is an interesting …nite dimensional ex- ample which anticipates what happens in the case of macroscopic limits. It is the case of the sequence of equations
dXt"= b (t; Xt") dt + "dBt; Xjt=0= x
when b is bounded continuous. Under this assumption there is existence and uniqueness in law, but also in the strong sense, for every " > 0. We claim that the family fQ"g of the laws of fXt"g is tight in C [0; T ] ; Rd and each limit measure Q of the family has the following property:
Q (Cx) = 1
where Cx C [0; T ] ; Rd is the set of all solutions of the deterministic equation dXt
dt = b (t; Xt) ; Xjt=0= x:
The proof of tightness of fQ"g is identical to the one given above, we leave it as an exercise.
Let fQ"ng be a weakly converging subsequence and Q be its limit. Consider the functional : C [0; T ] ; Rd ! R de…ned as
(f ) = sup
t2[0;T ]
ft x Z t
0
b (s; fs) ds : It has the property that (f ) = 0 if and only if f 2 Cx.
The functional is continuous on C := C [0; T ] ; Rd , with the uniform topology (here we use that b is continuous). Recall that by Portmanteau theorem one has Q (A) lim inf Q"n(A) for every open set A C. Hence, for every > 0,
Q (f 2 C : (f ) > ) lim inf Q"n(f 2 C : (f ) > ) :
If we prove that this lim inf is zero, then Q (f 2 C : (f ) > ) = 0 for every > 0, hence Q (f 2 C : (f ) = 0) = 1, which proves Q (Cx) = 1 (we leave as an exercise to prove that Cx is a closed set, hence Borel).
We have
Q"n(f 2 C : (f ) > ) = P ( (X"n) > )
= P sup
t2[0;T ]
Xt"n x Z t
0
b (s; Xs"n) ds >
!
= P sup
t2[0;T ]j"nBtj >
!
= P sup
t2[0;T ]jBtj > ="n
! "n E
"
sup
t2[0;T ]jBtj
#
! 0:
2 Compactness criteria in in…nite dimensions
2.1 Classical Aubin-Lions lemma
If (E; d) is a metric space, there is a natural extension to the space C ([0; T ] ; E): a family of functions F C ([0; T ] ; E) is relatively compact if
i) for every t 2 [0; T ], the set ff (t) ; f 2 F g is relatively compact ii) for every " > 0 there is > 0 such that
d (f (t) ; f (s)) "
for every f 2 F and every s; t 2 [0; T ] with jt sj .
There are generalizations to Lp-spaces. One is named Kolmogorov-Riesz, not discussed here. We mention the following one.
Lemma 5 (Aubin-Lions) Let E0 E E1 be three Banach spaces, with compact em- bedding E0 E (bounded sets of E0are relatively compact in E) and continuous embedding E E1. Let p; q 1. Then
Lp(0; T ; E0) \ W1;q(0; T ; E1) is compactly embedded into
Lp(0; T ; E) :
This theorem is very powerful for application to PDEs.
2.2 An example
Let D be a bounded regular domain in Rd; consider the PDE
@u
@t = u + b (u) ru + c (u) ujt=0 = u0; uj@D = 0:
where b; c : L2(D) ! R are continuous, b bounded and c with linear growth. We want to prove the existence of a solution of class
u 2 C [0; T ] ; L2(D) \ L2 0; T ; W1;2(D) .
Let us develop the so called a priori estimates: if a solution u exists and is su¢ ciently regular, then
d dt
Z
u2(t; x) dx = 2 Z
u ( u + b (u) ru + c (u)) dx:
Using also the boundary condition, one has Z
u udx = Z
jru (t; x)j2dx Z
u (b (u) ru + c (u)) dx Cb
Z
juj jruj dx + Cc Z
juj (1 + juj) dx 1
2 Z
jruj2dx +Cb2 2
Z
juj2dx + (Cc+ 1) Z
juj2dx + CcjDj and thus we have
Z
u2(t; x) dx + 2 Z t
0
Z
jru (s; x)j2dxds = Z
u20(x) dx + C Z t
0
Z
ju (s; x)j2dxds:
It follows, by Gronwall lemma and then again by the same inequality, sup
t2[0;T ]
Z
u2n(t; x) dx + Z T
0
Z
jrun(s; x)j2dxds C
with a constant C > 0 independent of n. This proves in particular that the sequence fung is bounded in L2 0; T ; W1;2(D) , which is one half of the assumptions of Aubin-Lions lemma. The second half is somewhat easier: from the di¤erential equation and the fact that the linear operator is bounded from W1;2(D) to the negative order Sobolev space W 1;2, it follows that @u@tn is bounded in L2 0; T ; W 1;2 . The we apply Aubin-Lions lemma with p; q = 2,
E0 = W1;2(D) ; E = L2(D) ; E1 = W 1;2
and use in essential way the compactness of the embedding W1;2(D) L2(D) (Rellich theorem). We deduce that fung is relatively compact in L2 0; T ; L2(D) in the strong topology (this is the important information in order to pass to the limit in nonlinear terms).
The sequence is also weakly relatively compact in L2 0; T ; W1;2(D) and weak star rel- atively compact in C [0; T ] ; L2(D) . Then there exists a subsequence which converges in all these three topologies to a function u which is then of class L2 0; T ; W1;2(D) and C [0; T ] ; L2(D) . Thanks to the strong convergence in L2 0; T ; L2(D) one can pass to the limit and prove that u is a solution (this step depends on the sequence of approximating problems and thus we have to miss the details).
2.3 Fractional version of Aubin-Lions lemma and a zero-noise example Aubin-Lions lemma is not suitable for stochastic problems since processes usually are not di¤erentiable. As we have seen in the case of SDEs, a criterion based on fractional Sobolev spaces is more suitable. The de…nition of
Lemma 6 Let E0 E E1 be three Banach spaces, E0; E1 re‡exive, with compact embedding E0 E and continuous embedding E E1. Let p; q > 1, 2 (0; 1). Then
Lp(0; T ; E0) \ W ;q(0; T ; E1) is compactly embedded into
Lp(0; T ; E) :
Let us see the (expected) scheme of its applicability to the following zero-noise problem:
in a bounded regular domain D Rdwe consider the SPDE
du" = ( u"+ b (u") ru"+ c (u")) dt + "dBt u"jt=0 = u0; u"j@D = 0:
with b and c as in the example 2.2 above. Here, for sake of simplicity, assume that B is a one-dimensional Brownian motion, but this issue can be generalized a lot.
The rigorous analysis of such a problem requires several details which are not important for our main aim of particle systems and would deteriorate the understanding of the main points we want to emphasize. Thus the following discussion is intentionally approximate from the viewpoint of rigor, including what we call "proof" below.
First, let us accept the following claim, for shortness of expositions: for every ", there ex- ists at least one solution u"with paths of class Y = C [0; T ] ; L2(D) \L2 0; T ; W1;2(D) . Consider the law Q"of u", as a probability measure on Borel subsets of Y . Denote by Cu0 the set of all solutions of class Y of the deterministic PDE of example 2.2. We want to prove that the family fQ"g is tight in L2 0; T ; L2(D) and all its limit points Q are supported on Cu0.
Lemma 7 fQ"g is tight in L2 0; T ; L2(D) .
Proof. We want to apply the fractional Aubin-Lions lemma with p = q = 2, E0 = W1;2(D), E = L2(D), E1 = W 1;2 as in the deterministic case. Given > 0 we have to
…nd two constants M; R > 0 such that P
Z T 0
Z
ju"(s; x)j2+ jru"(s; x)j2 dxds > M < (2)
P Z T
0 ku"(s)k2W 1;2ds + Z T
0
Z T 0
ku"(t) u"(s)k2W 1;2
jt sj1+2 > R
!
< (3)
for some 2 (0; 1).
By Itô formula
d ju"(t; x)j2 = 2u"( u"+ b (u") ru"+ c (u")) dt +2u""dBt+ "2dt:
Hence
Z
ju"(t; x)j2dx + 2 Z t
0
Z
jru"(s; x)j2dxds
= Z
u20(x) dx + 2 Z t
0
Z
u"(b (u") ru"+ c (u")) dxdt +2
Z t 0
Z
u"dx"dBt+ "2jDj t
(as an example, let us mention that concerning rigor, the result of this formula is correct but the proof requires additional arguments, for instance because we have treated u"(t; x) as an Itô process, a fact which is true only when u" is a real valued process, while here it is only a distribution).
As above in the deterministic case Z
u20(x) dx + Z t
0
Z
jruj2dxdt + C Z t
0
Z
juj2dxdt + C + 2 Z t
0
Z
u"dx"dBt which implies
Z
ju"(t; x)j2dx + 2 Z t
0
Z
jru"(s; x)j2dxds
Z
u20(x) dx + C Z t
0
Z
juj2dxdt + C +2
Z t 0
Z
u"dx"dBt: If we know thatR
u"dx is of class M2 (this depends on the properties of u" that have been proved about its existence, an issue that we have missed), we take expectation and prove E
Z
ju"(t; x)j2dx +2E Z t
0
Z
jru"(s; x)j2dxds Z
u20(x) dx+C Z t
0
E Z
juj2dx dt+C
which provides …rst a uniform-in-" bound on EhR
ju"(t; x)j2dxi
by Gronwall lemma,
sup
t2[0;T ]
E Z
ju"(t; x)j2dx C (4)
then the bound
E Z t
0
Z
jru"(s; x)j2dxds C (5)
by the inequality itself. If know only thatR
u"dx is of class 2, we have to apply a stopping time of the form
N = inf t > 0 : Z
ju"(t; x)j dx > N
so that the process R
u"(t ^ N; x) dx is now of class M2; then we apply the same com-
putations an remove N with Fatou lemma at the end, in the …nal inequalities (4)-(5), by taking the limit as N ! 1.
This is the …rst half of our e¤ort, because, splitting (2) in two parts for simplicity of exposition, we have
P Z T
0
Z
jru"(s; x)j2dxds > M 1 ME
Z T 0
Z
jru"(s; x)j2dxds C M and thus for M large this probability is arbitrarily small. The proof for P RT
0
R ju"(s; x)j2dxds > M is similar, using (4).
Concerning the second half, namely (3), the part concerningRT
0 ku"(s)k2W 1;2ds follows from the previous computations because ku"(s)k2W 1;2 C ku"(s)k2L2. For the other part we have
P Z T
0
Z T 0
ku"(t) u"(s)k2W 1;2
jt sj1+2 > R
! 1
RE Z T
0
Z T 0
ku"(t) u"(s)k2W 1;2
jt sj1+2
= 1
R Z T
0
Z T 0
Eh
ku"(t) u"(s)k2W 1;2
i
jt sj1+2 so we have to estimate Eh
ku"(t) u"(s)k2W 1;2
i
. From the equation satis…ed by u" we have
u"(t) u"(s) = Z t
s
( u"+ b (u") ru"+ c (u")) dr + " (Bt Bs)
ku"(t) u"(s)kW 1;2
Z t
s k u"+ b (u") ru"+ c (u")kW 1;2dr + " jBt Bsj (t s)1=2
Z t
s k u"+ b (u") ru"+ c (u")k2W 1;2dr
1=2
+ C" jBt Bsj
ku"(t) u"(s)k2W 1;2 2 (t s) Z t
s k u"+ b (u") ru"+ c (u")k2W 1;2dr + C" jBt Bsj2 Eh
ku"(t) u"(s)k2W 1;2
i
2 (t s) Z T
0
Eh
k u"+ b (u") ru"+ c (u")k2W 1;2
i
dr+C" (t s) : It is not di¢ cult to prove that
Eh
k u"+ b (u") ru"+ c (u")k2W 1;2
i
CEh
ku"k2W1;2
i + C
hence, using (5), we get Eh
ku"(t) u"(s)k2W 1;2
i
C (t s) and therefore
1 R
Z T 0
Z T 0
E h
ku"(t) u"(s)k2W 1;2
i
jt sj1+2
C R
if we choose < 1=2. Thus we may choose R large to make the probability above small.
