Regularity of the diffusion coefficient
1 Additive functionals
Consider an exclusion process onZdas introduced in Chapter 3. Fix 0≤α≤1 and a mean-zero local function V in H−1(L, να). In Theorems 3.3.3, 3.3.13 we proved that the additive functionalt−1/2Rt
0V(ηs)dsconverges to a mean- zero Gaussian variable with variance σ2(V) = 2 limλ→0kfλk21, where fλ is the solution of the resolvent equation 3.2.1. In this section we examine the dependence of the varianceσ2(V) on the densityα.
Consider a functionV(α, η) which may depend both on the densityαand on the configurationη. Recall the dual representation ofV defined in Section 3.4:
V(α, η) = X
A∈E
V(α, A)ΨA.
For each 0≤α≤1,V(α,·) :E →Ris a finite supported function. Denote by V(j)(α,·) thej-th derivative of V(α,·) inα:
V(j)(α, A) = dj
dαjV(α, A)·
Recall from Section 3.5 the definition of the Hilbert spaces Hk,0(S, µ), Hk,1(S, µ), Hk,−1(S, µ), k ≥ 0, denoted simply by Hk,0, Hk,1, Hk,−1 and recall the definition of the norms||| · |||k,0,||| · |||k,1,||| · |||k,−1. Assume that (H1) There exists a finite setA such that the support of V(α,·) is contained
inA for all 0≤α≤1. In other words, the functions V(α,·), 0≤α≤1, have a common finite support;
(H2) For eachAin E,V(·, A) is an infinitely differentiable function;
(H3) For eachj≥0,V(j)(α,·) belongs toHk,−1for each 0≤α≤1 and sup
0≤α≤1
|||V(j)(α,·)|||k,−1 < ∞ for allk≥0.
150 6 Regularity of the diffusion coefficient
Since V(α,·) has finite support condition (H3) is satisfied as soon as it holds fork= 0.
We showed in Lemma 3.7 that a finite supported function f : E → R belongs toH−1=H−1(S, µ) if and only if
X
A∈En
f(A) = 0
for allnd≤2. In this formula,En stands for all subsets ofZdwithnpoints. In particular, D(α, φ) = 0. Moreover, any local function V(α, η) whose Fourier representationV(α, A) does not depend onαand such that
X
A∈En
V(A) = 0
for allnd≤2 satisfies assumptions (H1)-(H3).
V(α, η) = Ψ{x}(α, η) − Ψ{y}(α, η) for anyx6=yis such an example in any dimension.
To stress its dependence on the densityα, denote byσ2(α, V) the variance of central limit theorem for the additive functional Rt
0dsV(α, ηs) proved in Chapter 3. We prove in this section that σ2(α, V) depends smoothly on the density:
Theorem 6.1.Consider an exclusion process as defined in Chapter 3. In the asymmetric case assume thatd≥3. Fix a functionV(α, η)satisfying hypothe- ses (H1)-(H3). The diffusion coefficientσ2(α, V), as function ofα, is of class C∞ in the open interval (0,1).
In the symmetric case, the smoothness can be extended up to the bound- ary. This is also the case for special local functionsV in the asymmetric case.
This question, however, will not be elucidated because we prove in Section 4 that the bulk diffusion coefficient is smooth up to the boundary.
Consider the resolvent equation associated toV: forλ >0, denote by fλ
the solution of the resolvent equation:
λfλ(α,·) − Lfλ(α,·) = V(α,·).
We use the dual representation to carry out the estimates. Recall the defini- tion of the operatorTintroduced at the beginning of Section 3.4. Letfλ=Tfλ
through the unitary isomorphism. Of coursefλ=fλ(α) depends onα. Apply- ingTon both sides of the resolvent equation, we obtain
λfλ(α,·) − Lαfλ(α,·) = V(α,·). (1.1) Theorem 3.3.3 and Theorem 3.3.13 give an explicit formula for the variance in terms of the solution of the resolvent equation:
(1/2)σ2(V, α) = lim
λ→0kfλk21 = lim
λ→0hV, fλiα = lim
λ→0hV,fλi
= lim
λ→0
X
n≥1
X
A∈En
V(α, A)fλ(α, A).
The sum starts from n = 1 because V(α, φ) = 0. Moreover, all but a finite number of terms vanish because the local functionsV(α,·) have a finite com- mon support. In view of this identity, to prove Theorem 6.1 we just need to show that there exists a subsequenceλk↓0 for whichP
A∈EnV(α, A)fλk(α, A) converges uniformly inαto a smooth function for eachn≥1.
To prove the existence of such subsequence, it is enough to show that the functionsfλ(·, A) are smooth for eachλ >0 (1.2) and that
sup
0<λ≤1
sup
0≤α≤1
dj dαj
X
A∈En
V(α, A)fλ(α, A) < ∞ for eachj≥0,n≥1. To prove this bound, it is enough to show that
sup
0<λ≤1
sup
0≤α≤1
X
A∈En
V(j)(α, A)f(k)λ (α, A) < ∞
for each j ≥0, k ≥ 0,n ≥ 1. Here, g(j) stands for the j-th derivative of a function g: [0,1]× E →Rwith respect to the first coordinate.
We claim that
X
A∈En
V(j)(α, A)f(k)λ (α, A)
≤ C(j, V, p)kπnf(k)λ (α,·)k1. (1.3) for some finite constantC(j, V, p) depending only onj, the transition proba- bilitypandV.
Fix j, k ≥ 0 and assume first that nd ≤ 2. Since V(α,·) belongs to H−1(S, να), by Lemma 3.7, P
A∈EnV(α, A) = 0. In particular,
X
A∈En
V(j)(α, A) = 0.
Denote byS(V) the support ofV:S(V) ={A∈ E:V(α, A)6= 0 for someα}.
By the previous identity, X
A∈En
V(j)(α, A)f(k)λ (α, A)
= 1
M
X
A,B∈En∩S(V)
f(k)λ (α, A)
V(j)(α, A)−V(j)(α, B)
= 1
M
X
A,B∈En∩S(V)
V(j)(α, A)
f(k)λ (α, A)−f(k)λ (α, B) ,
152 6 Regularity of the diffusion coefficient
whereM stands for the cardinality ofEn∩S(V). By assumptions (H1), (H2) and by Schwarz inequality, the square of this expression is bounded by
Cj(V) X
A,B∈En∩S(V)
f(k)λ (α, A)−f(k)λ (α, B) 2
for some finite constantCj(V) depending only onj and V. SinceS(V) is a finite set, constructing paths from A to B and applying Schwarz inequality, we obtain that the previous expression is less than or equal to
Cj(V) X
A∈En
X
B∼A
f(k)λ (α, B)−f(k)λ (α, A) 2.
Here and below the constantsCj(V) may change from line to line. The second sum is carried over all sets B which may be obtained from A by moving a single particle: A ∼ B if B = Ax,y for some pair {x, y} such that x ∈ A, y6∈A, s(y−x)>0. It is easy to see from the explicit formula (3.4.8) for the H1norm, that the previous expression is bounded by Cj(V, p)kπnf(k)λ (α,·)k21. This proves the claim in the casend≤2.
Assume now that nd > 2. In this case, the evolution of n symmetric exclusion particles inZdis transient. In particular, by Proposition 3.2.24 there exists a finite constantC(p), depending only on the transition probabilityp(·), such that for any functiong:En→Rand any setA inEn,
g(A)2 ≤ C(p) X
x,y∈Zd
X
A∈En
s(y−x)
g(Ax,y)−g(A) 2 = C(p)kπng(α,·)k21.
Therefore, since V(·,·) is a finitely supported function, smooth in the first coordinate,
X
A∈En
V(j)(α, A)f(k)λ (α, A)
≤C(j, V) X
A∈En∩S(V)
f(k)λ (α, A)
≤C(j, V, p)kπnf(k)λ (α,·)k1
becauseS(V) is a finite set. This proves Claim (1.3) in the casend >2.
In view of (1.3), to prove Theorem 6.1 we need to show that {fλ(·, A) : λ >0, A∈ E}are smooth functions and that
sup
0≤α≤1
sup
0<λ≤1
|||f(j)λ (α,·)|||0,1 < ∞ (1.4) for allj≥0. We prove this bound in Section 2 for symmetric simple exclusion processes and in Section 3 for asymmetric processes in dimensiond≥3. The asymmetric mean-zero case can be treated as the asymmetric case and does not require any restriction on the dimension (cf. Remark 6.4).
2 The symmetric case
Recall the defintion of the operatorLαintroduced in (3.4.2). In the symmetric case, the operatorLα=Sdoes not depend on the parameterαand the proof of (1.2), (1.4) is elementary.
We say that a functionh(α,·) with values inL2(E) is differentiable atαif γ−1[h(α+γ)−h(α)] converges, asγ↓0, strongly inL2(E) to some function that we denote byh0(α).
By assumptionV(α,·) is infinitely differentiable in the first coordinate and uniformly finitely supported in the second. In particular,V(j)(α,·) belongs to L2(E) for eachj ≥0 and we may examine the resolvent equation forV(j)(α,·).
Forλ > 0,j ≥1, 0≤α≤1, denote bygλ,j(α) the solution of the resolvent equation
λgλ,j(α) − Sgλ,j(α) = V(j)(α). (2.1) Recall thatfλ(α) stands for the solution of (1.1). We claim that fλ(α) is infinitely differentiable and thatf(j)λ (α) =gλ,j(α) for allj≥1.
Indeed, let us first prove the claim forj= 1. forh >0, letvh=h−1{fλ(α+
h)−fλ(α)}, Vh=h−1{V(α+h)−V(α)}. By Assumption,Vh converges to V(1)(α) inL2(E) ash↓0. Since
λ
vh−gλ,1(α) − S
vh−gλ,1(α) =
Vh−V(1)(α) ,
taking scalar product on both sides with respect tovh−gλ,1(α), we obtain that vhconverges togλ,1(α) inL2(E) ash↓0. Therefore,fλ(α) is differentiable and its derivative is equal to gλ,1(α). An induction argument permits to extend the statement toj ≥2.
We just showed thatfλ(α) is a smooth function inα. Sincef(k)λ (α) is the solution of the resolvent equation (2.1), taking the scalar product on both sides with respect toπnf(k)λ , applying Schwarz inequality and summing over n, we obtain that
|||f(k)λ (α)|||0,1 ≤ |||V(k)(α)|||0,−1.
By Assumption (H3), sup0≤α≤1|||V(k)(α)|||0,−1 < ∞. Hence (1.4) holds and the proof is complete in the symmetric case.
3 The asymmetric case in d≥ 3
Recall the definition of the operatorLαintroduced in (3.4.2). Since the coef- ficients of Lαare not smooth at the boundary of [0,1], we reparametrize the family of equations byα= sin2t,t∈[0, π/2], to get
L(t) = S + cos(2t)A + sintcostJ, whereJ=J++J−. Consider the resolvent equation
154 6 Regularity of the diffusion coefficient
λgλ(t)−L(t)gλ(t) =W(t),
whereW(t) =V(α(t)). Of course, sinceL(t) =Lα(t),gλ(t) =fλ(α(t)).
To prove that the functions {fλ(·, A) : λ > 0, A ∈ E} are smooth in any compact interval of (0,1), it is enough to show a similar statement for the functions {gλ(·, A) : λ >0, A∈ E}. Moreover, to prove 1.4 on compact interval of (0,1), it suffices to show that
sup
0≤t≤π/2
sup
0<λ≤1|||g(j)λ (t,·)|||0,1 < ∞
for allj≥0. From these results and the relation betweentandα, we deduce boundness in||| · |||1,0norm of{f(j)λ (α, A), λ >0}in the interior of the domain.
An extra argument, presented at the end of the proof, extends the smoothness up to the boundary for functionsV satisfying the extra condition xxx.
We now start our way through the proof thatgλ is a sequence of smooth functions with bounded derivatives inH0,1. By Lemma 2.2.11,
sup
0<λ≤1
sup
0≤t≤π/2
|||gλ(t)|||k,1 ≤ C(k) sup
0≤α≤1
|||V(α,·)|||k,−1
for all k ≥ 1. By assumption (H3), the right hand side is bounded for all k≥1.
We now turn to the proof of the differentiability of gλ(·). Notice that differentiating formallyL(t) in twe get the operator
L0(t) = −2 sin(2t)A + cos(2t)J.
Lemma 6.2.Suppose that U(t, A) satisfies the assumption (H1) and that
|||U(t,·)|||k+1,−1 < ∞ for some k ≥ 0. For λ > 0, let uλ be the solution of the resolvent equation
λuλ(t)−L(t)uλ(t) = U(t). (3.1) Then, uλ belongs toHk+1,0 andL0(t)uλ toHk,0.
Proof. By Lemma 2.2.11 and Lemma 3.3.14,
λ|||uλ(t)|||k+1,0 + |||uλ(t)|||k+1,1 ≤ C0|||U(t,·)|||k+1,−1
for some constantC0 which depends only onp. Hence, by Lemma 6.5 below, L0(t)uλ(t) belongs toHk,0.
Besides the assumptions of Lemma 6.2, suppose furthermore thatU(t) is a differentiable function oft. In view of the previous lemma, we may consider the resolvent equation
λvλ(t) − L(t)vλ(t) = U0(t) + L0(t)uλ(t). (3.2)
Proposition 6.3.Suppose that U(t)is differentiable and that sup
0≤t≤π/2
|||U(t)|||k,−1 < ∞, sup
0≤t≤π/2
|||U(t)|||k,0 < ∞
for all k ≥ 0. For λ > 0, let uλ(t) be the solution of the resolvent equation (3.1). Then, uλ(t) is differentiable and its derivative u0λ(t) is the solution of (3.2). Moreover, L0(t)uλ(t)is differentiable and
sup
0≤t≤π/2
|||L0(t)uλ(t)|||k,−1 < ∞, sup
0≤t≤π/2
|||L0(t)uλ(t)|||k,0 < ∞ for allk≥0.
Proof. Forh >0, letah=h−1{cos 2(t+h)−cos 2t},bh=h−1{sin(t+h) cos(t+
h)−sintcost},vh=h−1{uλ(t+h)−uλ(t)},Uh=h−1{U(t+h)−U(t)}. With this notation, it follows from the explicit formula for L(t) that
λvh − L(t)vh = Uh + ahAuλ(t) + bhJuλ(t) + hahAvh + hbhJvh. (3.3) Claim 1:vhbelongs toHk,0for anyk≥0. Moreover, lim suph→0|||vh|||k,0<∞.
For 0< n1< n2, to be fixed later, letT :L2(E)→L2(E) be the diagonal operator defined by:
Tf = X
n≥0
tnπnf,
where,
tn=
nk1 forn≤n1, nk forn1< n≤n2, nk2 forn > n2.
Notice that|||Tg|||0,0≤nk1|||g|||k,0for all functionsginHk,0.
ApplyT on both sides of (3.3) and take the scalar product withTvh. By Schwarz inequality and Lemma 6.5, the first three terms on the right hand side of (3.3) are bounded by
(3/2)δn2k1 |||vh|||2k,0 + (1/2)δ−1n2k1 |||Uh|||2k,0 + 12n2k1 δ−1|||uλ(t)|||2k+1,0 , because sup0<h<1a2h≤4, sup0<h<1b2h≤1. SinceT commutes with the anti- symmetric diagonal operatorA,hTvh, TAvhivanishes.
It remains to estimate on the right hand sidehbhhTvh, TJvhiand on the left hand side hTvh, TL(t)vhi. The latter term is equal to hTvh,STvhi+ sintcosthTvh, TJvhi. Repeating the proof of Lemma 2.2.11, we otbtain that
hTvh, TJvhi ≤ (1/4)|||vh|||2k,1
provide we choose n1 = n1(p, k) large enough. On the other hand, letting n2↑ ∞, hTvh,STvhiconverges to an expression which is bounded below by
|||vh|||2k,1.
156 6 Regularity of the diffusion coefficient
Since sup0≤t≤π/2h|bh(t)| ≤ 1 for h ≤ 1, putting all previous estimates together, we obtain that
λ|||vh|||2k,0 + (1/2)|||vh|||2k,1 ≤
(3/2)δn2k1 |||vh|||2k,0 + (1/2)δ−1n2k1 |||Uh|||2k,0 + 12n2k1 δ−1|||uλ(t)|||2k+1,0 forh≤1. Choosingδ=δ(λ, p, k) so that (3/2)δn2k1 =λ/2, we deduce that
|||vh|||2k,0 ≤ C0(p, λ, k)n
|||Uh|||2k,0 + |||uλ(t)|||2k+1,0o .
SinceUhconverges toU0(t),|||Uh|||2k,0≤ |||U0(t)|||2k,0+ 1 forhsmall enough. This concludes the proof of Claim 1.
Claim 2: uλ(t) is differentiable and its derivativeu0λ(t) is the solution of the resolvent equation (3.2).
Subtracting (3.2) from (3.3), we obtain that λ{vh−vλ(t)} − L(t){vh−vλ(t)} = {Uh−U0(t)}
+{ah−a0(t)}Auλ(t) + {bh−b0(t)}Juλ(t) + hahAvh + hbhJvh, wherea0(t) =−2 sin(2t),b0(t) = cos(2t). We claim that the right hand side of this identity vanishes inHk,0, ash↓0, for anyk≥0. Indeed, by assumption, Uh converges toU0(t). To handle the second and third terms, it is enough to show that Auλ(t), Juλ(t) belong to Hk,0. This follows by Lemma 6.5 since uλ(t) belongs to Hk+1,0 by Lemma 6.2. On the other hand, by Claim 1 and by Lemma 6.5, Avh,Jvh belong toHk,0 and have uniformly bounded norms in h. Sinceah,bhare bounded, the last two terms vanish in Hk,0 ash↓0.
Let Tk : Hk,0 → H0,0 be the diagonal operator defined by Tkg = P
n≥0nkπng. The same arguments show that Tk applied to the right hand side of the previous identity vanishes inH0,0, ash↓0, for anyk≥0.
Hence, applying Tk on both sides of the equation taking scalar product with respect to Tk{vh−u0λ(t)}in the previous formula we show thatvh con- verges to u0λ(t) in L2(E) as h ↓ 0. This concludes the proof of the Claim 2.
Claim 3: For any k ≥ 0, supλ≤1sup0≤t≤π/2|||L0(t)uλ(t)|||k,−1 < ∞ and L0(t)uλ(t) is differentiable.
Sinceuλ(t) is the solution of (3.1), by Lemma 3.3.17, Corollary 3.3.15 and the explicit form of the operatorL(t),
|||L0(t)uλ|||k,−1 ≤ C0|||U(t)|||k+1,−1 + C0|||uλ|||k+2,1
for some finite constant C0 depending only on p(·). By Lemma 2.2.11, the previous expression is less than or equal to C0|||U(t)|||k+2,−1 uniformly in λ andt.
Since the coefficients ofL(t) are smooth, to prove thatL0(t)uλ(t) is differ- entiable, we just need to check thatAuλ(t),Juλ(t) belong toHk,0for allk≥0 and are differentiable. This follows from the differentiability of uλ(t) proved in Claim 1 and from Lemma 6.5.
The previous lemma applied toU=Wshows that the family of functions gλ is differentiable for each fixed λ and that the derivative g0λ satisfies a resolvent-type equation.
Proof of Theorem 6.1: We first show that {gλ(t), λ > 0} is a family of smooth functions whose derivatives satisfy the bound
sup
0<λ≤1
sup
0≤t≤π
2
|||g0λ(t)|||1,k <∞ (3.4) for each k ≥ 0, By assumption (H3) |||W(t)|||−1,k is bounded uniformly in t. Hence, by Lemma 2.2.11, |||gλ|||1,k is bounded, uniformly in λ and t. By Proposition 6.3,gλ is differentiable and its derivativeg0λ satisfies
λg0λ − L(t)g0λ = W0(t) + L0(t)gλ.
By Lemma 3.3.17, Corollary 3.3.15 and the explicit form of the operatorL(t),
|||L0(t)gλ|||−1,k ≤ C0|||W(t)|||−1,k+1 + C0|||gλ|||1,k+2 + 2|||gλ|||1,k+1
for some finite constant C0 depending only on p(·). By Lemma 2.2.11, the previous expression is less than or equal to C0|||W(t)|||−1,k+2 uniformly in λ and t. We may therefore apply Lemma 2.2.11 again to show that |||g0λ(t)|||1,k
is uniformly bounded in (t, λ) for all k≥1.
To iterate the argument, we just need to prove by induction the existence of constants{an,i, n≥1,0≤i < n}such that
λg(j)λ (t) − L(t)g(j)λ (t) = W(j)(t) +
j−1
X
i=0
aj,iL(j−i)(t)g(i)λ (t), (3.5)
whereg(i)λ ,L(i)(t) stands for thei-th derivative ofgλ(t),L(t). This is elemen- tary and left to the reader.
The previous argument shows thatuλ(t) is a sequence of smooth functions on [0,1] with their derivatives having the uniform bounds
sup
0<λ≤1
sup
0≤t≤π/2
|||g(j)λ (t)|||1,k<∞
for each j ≥ 0. We have seen just after (1.4) that these uniform estimates guarantee the smoothness ofσ2(V, α(t)) as a function oft defined in [0, π/2].
Since α = sin2t, this translates immediately into smoothness in α for α ∈ (0,1).
Remark 6.4 Mean-zero asymmetric exclusion processes.
We conclude this section with an estimate on the operatorsA,J.
158 6 Regularity of the diffusion coefficient
Lemma 6.5.There exists a finite constant C0 depending only on the transi- tion probability psuch that
kAfk ≤ 2nkfk and kJ±fk ≤ 2(n+ 1)kfk (3.6) for allf:En→RinL2(En).
Proof. Fix a function inf:En →Rin L2(En). By Schwarz inequality, kAfk2 ≤ n X
A∈En
X
x∈Ay6∈A
s(y−x){f(Ax,y)−f(A)}2
because|a(z)| ≤s(z) andP
x∈A,y6∈As(y−x)≤ |A|=n. It remains to bound the expression inside the square by 2{f(Ax,y)2+f(A)2}and to perform a change of variablesB =Ax,yto conclude the proof of the first claim.
The proof of the second claim is similar. One just need to rewrite the operatorJ− as (J−f)(A) =−2P
y6∈A,x∈Aa(y−x)f(A∪ {y}), which is allowed becauseP
za(z) = 0. Details are left to the reader.
4 Regularity of the self-diffusion coefficient in the symmetric case
5 Comments and References