9.5.1 Uniqueness of statistical solutions and chaos
sec:tensor
Assuming (A2), we know from Lemma ??lem:H0that (StNL) is ac0-semigroup, that for any Φ∈U Cb(PG1,R) we may define T∞Φ∈U Cb(PG1,R) by
(T∞Φ)(f) = Φ(StNLf),
and that we build in that way a c0-semigroup (Tt∞) on U Cb(PG1,R). The Hille-Yosida theory imply that there exists an closed operator G∞ with dense domain dom(G∞) in U Cb(PG1,R) so that (Tt∞) is the semigroup associated to the generator G∞.
Now, on the one hand, for any π0 ∈ P(PG1) we may define the semigroup (St∞) on P(PG1) and the flow (¯πt) by setting ¯πt=St∞π0 and (duality formula)
eq:defbarpit
eq:defbarpit (9.5.8) ∀Φ∈U Cb(PG1;R) hSt∞π0,Φi=hπ0, Tt∞Φi. Let us explain why (eq:defbarpit
9.5.19) indeed defines uniquely a probability measure ¯πt ∈P(PG1).
First we need the following assumptions (i) E is locally compact Polish space ;
(ii) F is in duality withG1andF is dense inCb(E) in the sense of uniform convergence on any compact set, or better, F ∩C0(E) is dense inC0(E) in the sense of uniform convergence ;
For anyℓ∈N∗ we define
ϕ∈ F⊗ℓ7→ hπtℓ, ϕi:=hπ0, Tt∞Rℓϕi.
That is a positive linear form on F⊗ℓ. Thanks to assumptions (i) and (ii), the Stone-Weierstrass density theorem and the Markov-Riesz representation theorem imply that πtℓ is well defined as a element of P(Eℓ). Since now the sequence (πtℓ) is symmetric and compatible, the De Finetti, Hewitt et Savage representation theorem implies that there exists a unique probability measure ¯πt∈P(PG1) such that for anyϕ∈ F⊗ℓ
hπ¯t, Rℓϕi:=hπ0, Tt∞Rℓϕi.
On the other hand, we say that πt∈C(R+;P(PG1)) is a solution to equation (sec6:dtpi=A
10.7.4) if it is a solution of (eq:BBGKYhierarchy
10.7.5) or equivalently for any Φ∈U C1(P(E);R) there holds eq:BBGKYhierarchy2
eq:BBGKYhierarchy2 (9.5.9) d
dthπt,Φi=hπt, G∞Φi inD′([0,∞)),
where we recall that G∞ is defined for any Φ∈U C1(P(E);R) and ρ∈P(E) by (G∞Φ)(ρ) =hDΦ(ρ), Q(ρ)iG′1,G1 =hQ(ρ), DΦ(ρ)iP(E),Cb(E).
theo:BBGKYuniq Th´eor`eme 9.5.7 Assume that (A2) and (A4) hold as well as (i), (ii) above. For any initial datum π0 ∈ P(PG1) the flow π¯t is the unique solution in C([0,∞);P(PG1)) to (eq:BBGKYhierarchy
10.7.5) and (eq:BBGKYhierarchy2
9.5.20) starting from π0. Moreover, if π0 is f0-chaotic (that is if π0 =δf0 withf0∈P(E)), then πt isStNLf0-chaotic for any t≥0.
Proof of Theorem theo:BBGKYuniq
9.5.12. Step 1 : Chaos propagation. From De Finetti-Hewitt-Savage’s theorem Hewitt-Savage
[32], for any π∈P(P(E)) there exists a unique sequence (πℓ)∈P(Eℓ) such that the identities
hπℓ, ϕi = Z
P(E)hf⊗ℓ, ϕiπ(df)
= Z
P(E)
Rℓϕ(f)π(df) =hπ, Rℓϕi, hold for anyϕ∈Cb(E)⊗ℓ. As a consequence, if π0 isf0-chaotic,
hπ¯tℓ, ϕi = hπ¯t, Rℓϕi=hπ0, Tt∞Rℓϕi= (Tt∞Rϕℓ)(f0)
= Rℓϕ(StNLf0) =hStNLf0, ϕ1i...hStNLf0, ϕℓi,
which means that ¯πtℓ = ft⊗ℓ, or equivalently ¯πt = δft, and the statistical solution ¯πt is ft-chaotic.
Step 2 : Equivalence between(eq:BBGKYhierarchy
10.7.5) and (eq:BBGKYhierarchy2
9.5.20). From (eq:compatibiliteGinftyell2
10.7.3) we recognize eq:compatibiliteGinftyell1
eq:compatibiliteGinftyell1 (9.5.10) hρ⊗ℓ+1, G∞ℓ+1ϕi=hDRℓϕ(ρ), Q(ρ)i = (G∞Rϕ)(ρ), for any ϕ ∈ Cb(E)⊗ℓ and any ρ ∈ P(E), or equivalently RG∞
ℓ+1ϕ = G∞Rϕ. Since Rϕ ∈ C1,1(P(E)) for anyϕ∈ F⊗ℓ, we deduce that (eq:BBGKYhierarchy2
9.5.20) implies (eq:BBGKYhierarchy
10.7.5).
Assume conversely that πt satisfies (eq:BBGKYhierarchy
10.7.5). For a given Φ ∈ C1,1(P(E);R) and for any N ∈N∗ we define the function V 7→ϕ(V) := (πNCΦ)(V) = Φ(µNV) in Cb(EN) so that (eq:BBGKYhierarchy
10.7.5) writes bbgky1
bbgky1 (9.5.11) ∂thπN,Φ(µNV)i=hπN+1, G∞N+1Φ(µNV)i, or equivalently thanks to the Hewitt-Savage representation theorem
∂thπ, RπN
CΦi=hπ, RG∞
N+1(πNCΦ)i. On the one hand, for any ρ∈P(E)
bbgky2
bbgky2 (9.5.12) RπN
CΦ(ρ) = Z
EN
Φ(µNV)ρ⊗N(dV)→Φ(ρ), by the law of large numbers.
On the other hand, for any ρ∈P(E), we have RG∞
N+1(πNCΦ)(ρ) = hρ⊗N+1, G∞N+1(πNCΦ)i
= hDRπN
CΦ(ρ), Q(ρ)i
= XN i=1
Z
EN
Φ(µNV))Q(ρ)(dvi)Y
j6=i
ρ(dvj).
For any given i= 1, ..., N, we defineφNV−1
i =DΦ(µNV−1
i ) and we write Φ(µNV) = Φ(µNVi−1) +hφNVi−1, µNV −µNVi−1i+O
kµNVi−1−µNVk2 .
Observing that
By symmetry we may rewrite that identity as RG∞ by the law of large number again.
On the other hand, for any ρ∈P(E)
A more convenient way to set out these computations is the following XN
Again the law of large number implies bbgky3
bbgky3 (9.5.13) RG∞
N+1(πNCΦ)(ρ)∼ Z
ENhDΦ(µN),Q(µ˜ NV, ρ)iρ⊗N(dV) → hDΦ(ρ),Q(ρ, ρ)˜ i, As a conclusion, (eq:BBGKYhierarchy2
9.5.20) follows by putting together (bbgky19.5.22), (9.5.23) and (bbgky2 bbgky39.5.13). At the abstract level, we use here a variant of hypothesis (A3)and (A3′), namely
(A3′′) (G∞N+1Φ(µNV))(V, vN+1) ∼ 1 N
XN i=1
Q∗(DΦ(µNV))(vj, vN+1).
Step 3 : Uniqueness. For anyt >0 andn∈N∗owe defineε:=t/nandtk=ε k,sk=t−tk. Then for any Φ∈ Cb1(PG1;R) we define Φt:= Tt∞Φ. The very fundamental point is that thanks to Lemma??lem:H0we have Φt∈Cb1(PG1;R)⊂dom(G∞) for any t≥0. We write
hπt,Φi − hπ¯t,Φi=hπt,Φi − hπ0,Φti
=
n−1
X
k=0
hπtk+1,Φsk+1i − hπtk+1,Φski +
hπtk+1,Φski − hπtk,Φski
=T1+T2 =
n−1
X
k=0
{T1,k+T2,k}. On the one hand, we have
T1,k = hπtk+1,Φsk+1−Tε∞Φsk+1i=−hπtk+1, Z ε
0
d
ds[Ts∞Φsk+1]dsi
= −hπtk+1, Z ε
0
[G∞Φsk+1+s]dsi=− Z sk
sk+1
hπt−[s+1,ε], G∞Φsids, where [s, ε] = [s/ε]ε. Passing to the limit n→ ∞, we get
T1=− Z t
0 hπt−[s+1,ε], G∞Φsids n−→
→∞ − Z t
0 hπt−s, G∞Φsids.
On the other hand, we have T2,k =
Z ε
0
d
dτhπtk+τ,Φskidτ
= Z ε
0 hπtk+τ, G∞Φskidτ
=
Z tk+1
tk hπτ, G∞Φt−[τ,ε]idτ.
Passing to the limitn→ ∞, we get T2=
Z t
0 hπτ, G∞Φt−[τ,ε]idτ n−→
→∞
Z t
0 hπτ, G∞Φt−τidτ.
As a conclusion, for any Φ∈C1(PG1;R), we have proved hπt,Φi=hπ¯t,Φi.
From a density argument we conclude thatπt= ¯πt. ⊔⊓ Gathering Lemma lem:BBGKY10.7.4 and Theorem theo:BBGKYuniq
9.5.12 we obtain a propagation to the chaos result.
Corollaire 9.5.8 (Abstract chaos propagation) Assume(A1′),(A2),(A3′),(A3′′), (A4)as well as (i) and (ii) above. Assume furthermore thatf0N isf0-chaotic. ThenftN is StNLf0-chaotic. More generally, isf0N converges toπ0 thenfN converge toπ¯tthe associated statistical solution.
9.5.2 On statistical solutionsand the non uniqueness of its steady states
sec:?
AUTRE POSSIBILITE :
1) ENLEVER TOUT CE QUI CONCERNE l’ecriture ”directe” avec AN,ANℓ ,A∞ et la non unicit´e des etats stationnaires (qui est un peu annexe ici)
2) COMMENCER PAR UNE SECTION ”On uniqueness of statistical solutions” dans laquelle on introduit la hierarchie GNℓ et ll’hypoth`ese centrale (eq:compatibiliteGinftyell2
10.7.3), la remarque qui suit et le th´eor`eme theo:BBGKYuniq
9.5.12
3) Une section ”Chaos propagation via the BBGKY hierarchy method” dans laquelle on met le lemme10.7.4 et le corollairelem:BBGKY ??cor:BBGKY(en un seul r´esultat).
Let us consider the N-particles system associated to the Boltzmann collision process that we do not write in dual fomr as we have done before. In order to simplify the discussion we only consider the Maxwell with Grad’s cut-off model. More precisely, our model writes eq:MasterN
eq:MasterN (9.5.14) ∂tfN = 1 N
X
i<j
Z
Sd−1
b(cosθi,j)h
fN(. . . , v′i, . . . , vj′, . . .)−fi dσ,
whereθi,jstands for the angle between the vectorsv′j−vi′andvj−viwherevi′ :=v′(vi, vj, σ), v′i := v∗′(vi, vj, σ) are defined thanks to (??). We want to describe how the BBGKYeq:rel:vit (Bogoliubov, Born, Green, Kirkwood and Yvon) method introduced to derive Boltzmann’s equation from Liouville’s equation applies in our simpler space homogeneous context. Let us thus also introduce the k-th marginal fℓN = Πℓ[fN]. Integrating the master equation (10.7.1) leads toeq:MasterN
∂tfℓN = 1 N
X
i,j≤ℓ
ZijN =O(ℓ2/N) +1
N X
i≤ℓ<j
ZijN
+1 N
X
i,j>ℓ
ZijN = 0, with
ZijN :=
Z
Rd(N−ℓ−1)
Z
SN−1
bh
fN(. . . , v′i, . . . , vj′, . . .)−fNi
dσ dvℓ+1... dvN.
Only the second term does not vanish in the limitN → ∞, so that assuming thatfℓN →πℓ in the weak sense of probabilities, we find that (πℓ) is a solution to the infinite dimensional system of linear equation (the Boltzmann equation for a system of an infinite number of particles or the statistical Boltzmann equation)
eq:BBGKY
eq:BBGKY (9.5.15) ∂tfℓ=Aℓ+1(πℓ+1) withπℓ=πℓ(t, v1, ..., vℓ)≥0 and
V ∈Rdℓ7→Aℓ+1(πℓ+1)(V) = Xℓ j=1
Z
Sd−1×Rd
n
πℓ+1(Vj′)−πℓ+1(V)o
b(cosθj,ℓ+1)dvℓ+1dσ, withVj′ = (v1, ..., vj′, ..., vℓ, v′ℓ+1) andvj′,v′ℓ+1 are defined as above through (eq:rel:vit??).
sec6:SolStat Lemme 9.5.9 There exists a non chaotic stationary solution to the statistical Boltzmann equation. In other words, there existsπ ∈P(P(Rd))such that π6=δp for some p∈P(Rd) and Aℓ+1(πℓ+1) = 0 for any ℓ∈N.
Proof of Lemma sec6:SolStat
10.7.3. It is clear that any function on the form V ∈Rd(ℓ+1) 7→ πℓ+1(V) =φ(|V|2)
is a stationary solution for the equation (10.7.2), that iseq:BBGKY Aℓ+1(πℓ+1) = 0. Now we define, withd= 1 for the sake of simplicity, the sequence
V ∈Rℓ 7→πℓ(V) = cℓ (1 +|V|2)m+ℓ/2
withc1such thatπ1is a probability measure andc2=c1α2withα2 chosen in the following way :
Z
R
α2
(1 +v2+v∗2)m+1dv∗ = α2 (1 +v2)m+1
Z
R
1
(1 + 1+vv2∗2)m+1 dv∗
= α2
(1 +v2)m+1/2 Z
R
1
(1 +w∗2)2 dw∗ = 1
(1 +v2)m+1/2. By an iterative process we may chose the constants cℓ in such a way thatπℓ is a solution to Aℓ(πℓ) = 0 (because it is a function of the energy) and satisfies the compatibility condition :
πℓ(V) = Z
R
πℓ+1(V, v∗)dv∗.
We have exhibit a solution which is not chaotic. ⊔⊓
We come back to the abstract setting. We start with the N-particles system equation (10.7.1) or (eq:MasterN 10.7.2), that we write in dual formeq:BBGKY
∂thfℓN, ϕi = ∂thfN, ϕ⊗1N−ℓi
= hfN, GN(ϕ⊗1N−ℓ)i=hfℓ+1N , GNℓ+1(ϕ)i. lem:BBGKY Lemme 9.5.10 Assume that
(A1′) For any ℓ∈N∗ the sequence (fℓN) is tight in P(Eℓ);
(A3′) For any ℓ ∈ N∗ and any fixed ϕ∈ Cb(Eℓ), the sequence (GNℓ+1ϕ) of Cb(Eℓ+1) satisfies GNℓ+1ϕ→G∞ℓ+1ϕ uniformly on compact sets when N → ∞, where G∞ℓ+1ϕsatisfies the following “one typical particle compatibility condition” : for any ϕ=ϕ1⊗ ... ⊗ϕℓ ∈ Cb(E)⊗ℓ and any V = (v1, ..., vℓ+1)∈Eℓ+1
eq:compatibiliteGinftyell2
eq:compatibiliteGinftyell2 (9.5.16) (G∞ℓ+1ϕ)(V) = Xℓ
i=1
Y
j6=i
ϕj(vj)
Q∗(ϕi)(vi, vℓ+1).
Here Q∗:Cb(E)→Cb(E2)is the dual linear operator associated to Qand defined through the relation
∀ρ∈P(E), ∀ψ∈Cb(E), hQ(ρ), ψi=hρ⊗ρ, Q∗(ψ)i.
Then, up to extraction a subsequence,(fN) converges (in the sense of anyℓ-th marginals) to a solution π= (πℓ)∈P(P(E)) to the infinite hierarchy
sec6:dtpi=A
sec6:dtpi=A (9.5.17) ∂tπ=A∞π in P(P(E)), which simply means so far
eq:BBGKYhierarchy
eq:BBGKYhierarchy (9.5.18) ∂thπℓ, ϕi=hπℓ+1, G∞ℓ+1ϕi for any ℓ∈N∗. Remarque 9.5.11 The identity(eq:compatibiliteGinftyell2
10.7.3)is called “one typical particle compatibility condi-tion” because it is the natural condition in order that any solution ft to the nonlinear Boltzmann is a ”solution” to the BBGKY hierarchy (eq:BBGKYhierarchy
10.7.5). Indeed, considering such a solution ft∈C(R+;P(E)) we compute for any ϕ=ϕ1⊗ ... ⊗ϕℓ ∈Cb(E)⊗ℓ
∂thft⊗ℓ, ϕi = Xℓ
i=1
Y
j6=i
hft, ϕji
∂thft, ϕii= Xℓ
i=1
Y
j6=i
hft, ϕji
hQ(ft), ϕii
= Xℓ
i=1
Y
j6=i
hft, ϕji
hft⊗ft, Q∗(ϕi)i=hft⊗ℓ+1, G∞ℓ+1ϕi,
which precisely means that the sequence (ft⊗ℓ)ℓ≥1 is a solution to the BBGKY hierarchy of equations (eq:BBGKYhierarchy
10.7.5).
9.5.3 Uniqueness of statistical solutions and chaos
sec:tensor
Assuming(A2), we know from Lemmalem:H0??that (StNL) is a c0-semigroup, that for any Φ∈U Cb(PG1,R) we may define T∞Φ∈U Cb(PG1,R) by
(T∞Φ)(f) = Φ(StNLf),
and that we build in that way a c0-semigroup (Tt∞) on U Cb(PG1,R). The Hille-Yosida theory imply that there exists an closed operator G∞ with dense domain dom(G∞) in U Cb(PG1,R) so that (Tt∞) is the semigroup associated to the generator G∞.
Now, on the one hand, for any π0 ∈ P(PG1) we may define the semigroup (St∞) on P(PG1) and the flow (¯πt) by setting ¯πt=St∞π0 and (duality formula)
eq:defbarpit
eq:defbarpit (9.5.19) ∀Φ∈U Cb(PG1;R) hSt∞π0,Φi=hπ0, Tt∞Φi. Let us explain why (eq:defbarpit
9.5.19) indeed defines uniquely a probability measure ¯πt ∈ P(PG1).
First we need the following assumptions
(i) E is a locally compact Polish space (e.g.E =Rd) ;
(ii) F is in duality withG1andFis dense inCb(E) in the sense of uniform convergence on any compact set, or better,F ∩C0(E) is dense inC0(E) in the sense of uniform convergence ;
For anyℓ∈N∗ we define
ϕ∈ F⊗ℓ7→ hπℓt, ϕi:=hπ0, Tt∞Rℓϕi.
That is a positive linear form on F⊗ℓ. Thanks to assumptions (i) and (ii), the Stone-Weierstrass density theorem and the Markov-Riesz representation theorem imply that πtℓ
is well defined as a element of P(Eℓ). Since now the sequence (πtℓ) is symmetric and compatible, the Hewitt-Savage representation theorem implies that there exists a unique probability measure ¯πt∈P(PG1) such that for any ϕ∈ F⊗ℓ
hπ¯t, Rℓϕi:=hπ0, Tt∞Rℓϕi.
On the other hand, we say that πt∈C(R+;P(PG1)) is a solution to equation (sec6:dtpi=A
10.7.4) if it is a solution of (eq:BBGKYhierarchy
10.7.5) or equivalently for any Φ∈U C1(P(E);R) there holds eq:BBGKYhierarchy2
eq:BBGKYhierarchy2 (9.5.20) d
dthπt,Φi=hπt, G∞Φi inD′([0,∞)),
where we recall that G∞ is defined for any Φ∈C1,δ(P(E);R) and ρ∈P(E) by (G∞Φ)(ρ) =hDΦ(ρ), Q(ρ)iG1′,G1 =hQ(ρ), DΦ(ρ)iP(E),Cb(E).
theo:BBGKYuniq Th´eor`eme 9.5.12 Assume that (A2) and (A4) hold as well as (i), (ii) above. For any initial datum π0 ∈ P(PG1) the flow π¯t is the unique solution in C([0,∞);P(PG1)) to (eq:BBGKYhierarchy
10.7.5) and (eq:BBGKYhierarchy2
9.5.20) starting from π0. Moreover, if π0 is f0-chaotic (that is if π0 =δf0 withf0∈P(E)), then πt isStNLf0-chaotic for any t≥0.
Proof of Theoremtheo:BBGKYuniq
9.5.12. Step 1 : Chaos propagation. From Hewitt-Savage’s theorem Hewitt-Savage
[32], for anyπ ∈P(P(E)) there exists a unique sequence (πℓ)∈P(Eℓ) such that the identities
hπℓ, ϕi = Z
P(E)hf⊗ℓ, ϕiπ(df)
= Z
P(E)
Rℓϕ(f)π(df) =hπ, Rℓϕi, hold for anyϕ∈Cb(E)⊗ℓ. As a consequence, ifπ0 is f0-chaotic,
hπ¯tℓ, ϕi = hπ¯t, Rℓϕi=hπ0, Tt∞Rϕℓi= (Tt∞Rℓϕ)(f0)
= Rℓϕ(SNLt f0) =hStNLf0, ϕ1i...hStNLf0, ϕℓi,
which means that ¯πtℓ = ft⊗ℓ, or equivalently ¯πt = δft, and the statistical solution ¯πt is ft-chaotic.
Step 2 : Equivalence between(eq:BBGKYhierarchy
10.7.5)and (eq:BBGKYhierarchy2
9.5.20). From (eq:compatibiliteGinftyell2
10.7.3) we recognize eq:compatibiliteGinftyell1
eq:compatibiliteGinftyell1 (9.5.21) hρ⊗ℓ+1, G∞ℓ+1ϕi=hDRϕℓ(ρ), Q(ρ)i= (G∞Rϕ)(ρ),
for any ϕ ∈ Cb(E)⊗ℓ and any ρ ∈ P(E), or equivalently RG∞ℓ+1ϕ = G∞Rϕ. Since Rϕ ∈ C1,1(P(E)) for any ϕ∈ F⊗ℓ, we deduce that (eq:BBGKYhierarchy2
9.5.20) implies (eq:BBGKYhierarchy
10.7.5).
Assume conversely that πt satisfies (eq:BBGKYhierarchy
10.7.5). For a given Φ ∈ C1,1(P(E);R) and for any N ∈N∗ we define the function V 7→ ϕ(V) := (πCNΦ)(V) = Φ(µNV) inCb(EN) so that (eq:BBGKYhierarchy
10.7.5) writes bbgky1
bbgky1 (9.5.22) ∂thπN,Φ(µNV)i=hπN+1, G∞N+1Φ(µNV )i,
or equivalently thanks to the Hewitt-Savage representation theorem by the law of large numbers.
On the other hand, for any ρ∈P(E), we have by the law of large number again.
Step 3 : Uniqueness. For anyt >0 andn∈N∗owe defineε:=t/nandtk=ε k,sk=t−tk.
On the one hand, we have
T1,k = hπtk+1,Φsk+1−Tε∞Φsk+1i=−hπtk+1, Z ε
0
d
ds[Ts∞Φsk+1]dsi
= −hπtk+1, Z ε
0
[G∞Φsk+1+s]dsi=− Z sk
sk+1hπt−[s+1,ε], G∞Φsids, where [s, ε] = [s/ε]ε. Passing to the limit n→ ∞, we get
T1 =− Z t
0 hπt−[s+1,ε], G∞Φsidsn−→
→∞ − Z t
0 hπt−s, G∞Φsids.
On the other hand, we have T2,k =
Z ε
0
d
dτhπtk+τ,Φskidτ
= Z ε
0 hπtk+τ, G∞Φskidτ
=
Z tk+1
tk hπτ, G∞Φt−[τ,ε]idτ.
Passing to the limitn→ ∞, we get T2=
Z t
0 hπτ, G∞Φt−[τ,ε]idτ −→
n→∞
Z t
0 hπτ, G∞Φt−τidτ.
As a conclusion, for any Φ∈C1(PG1;R), we have proved hπt,Φi=hπ¯t,Φi.
From a density argument we conclude thatπt= ¯πt. ⊔⊓ Gathering Lemma lem:BBGKY10.7.4 and Theorem theo:BBGKYuniq
9.5.12 we obtain a propagation to the chaos result.