Equilibrium fluctuations for the bulk density

5

Equilibrium fluctuations for the bulk density

July 17, 2007

1 The density field

In chapter 4 we studied the asymptotic evolution of a tagged particle in an infinite exclusion process in equilibrium. We will study now the asymptotic evolution of a fluctuation of the density of particles in equilibrium. Because of the conservation of the total number of particles, a density fluctuation will behave diffusively, and it is possible to tag its position and its time evolution with a special object that is called second class particle. We will see that the problem is very close to the one of the tagged particle, and it will be resolved with similar techniques.

Let us consider the simple exclusion process on Z

as defined in chapter 3. We consider this dynamics in a stationary state with density α. For any ε > 0, let us define the density field Y

as a random process with values in the distributions on R

:

Y

(G, t) = ε

X

x

G(εx)[η

(x) − α] (1.1) where G is a continuous function with compact support on R

. Since its

variance is

Y

(G, t)

= χ(α)ε

X

x

G(εx)

where as usual χ(α) = α(1 − α), the definition (1.1) can be extended to any G ∈ L

( R

). By the central limit theorem follows that, for G and t fixed, Y

(G, t) converges in law to a centered gaussian variable with variance χ(α)kGk

.

For F and G continuous compact support functions with compact support

on R

we have that

hY

(G, t)Y

(F, t)i −→

ε→0

χ(α) Z

R^d

F(q)G(q) dq

So we say that the sequence of random fields Y

(·, t) converge in law the centered gaussian random field with covariance χ(α)δ(q − q

). This is the standard white noise on R

multiplied by p

χ(α).

look for some stan- dard reference on white noise

Since we are on the stationary state, by translational invariance and sta- tionarity, all this is also valid if we modify (1.1) in

Y

(G, t) = ε

X

x

G(εx − vε

t)[η

(x) − α] (1.2) for any constant v ∈ R

.

In order to study the time evolution of Y

, let us compute the time corre- lations

hY

(G, t + s)Y

(F, s)i = ε

X

x,y

G(εx)F (εy) h[η

(x) − α][η

(y) − α]i

= ε

X

x,y

G(εx)F (εy)R(x − y, ε

t) (1.3) where R(x, t) = hη

(x)η

(0)i − α

.

1.1 The second class particle

The second class particle is a special particle that has the same jump rates of the other particles (with the exclusion rule), furthermore when a normal (first class) particle attempts to jump in the site occupied by the second class particle, the particles exchange site. Therefore the evolution of the first class particles is unaffected by the presence of the second class particles. We will show here that the asymptotic evolution of the second class particle is closely related to the equilibrium fluctuations of the density.

Let ν

the Bernoulli measure on Z

with parameter α conditioned to have the site 0 empty. Let X

be the position at time t of the second class particle, starting at time 0 from 0, when the other particles are distributed initially with ν

. There is an exact relation between the transition probability for the second class particle and the correlation function R(x, t).

Proposition 5.1.

E

(1

) = χ(α)

R(x, t) (1.4) Proof: Let σ

η(x) = η(x) if x 6= 0 and σ

η(0) = 0.

E

(η

(x)η

(0)) = Z

ν

(dη)η(0)E

(η

(x))

= Z

ν

(dη)η(0) [E

(η

(x)) − E

(η

(x))] + Z

ν

(dη)η(0)E

(η

(x))

= Z

ν

(dη)η(0)E

(1

) + Z

ν

(dξ) α

1 − α (1 − ξ(0))E

(η

(x))

1 The density field 133

(where in the last term we have performed the change of variable ξ = σ

η)

= αE

(1

) + α

1 − α α − α

1 − α E

(η

(x)η

(0)) Reordering the term one obtains (1.4)

1.2 The symmetric case

In the symmetric case one can see the dynamics of the exclusion process just as an exchange of occupation sites: site x and site y exchange occupation with rate p(x − y) = p(y − x). It is clear that in this case the second class particle moves like a simple symmetric random walk with rates p, unaffected from the presence of the other particles. The standard invariance principle for simples random walks the says that the process εX

converges in law to a brownian motion on R

with covariance matrix σ

= P

x

x

x

p(x). In particular, by (1.4) we have

R(ε

x, ε

t) −→

ε→0

χ(α)

(2πdet(σ

)t)

exp

− x · (σ

)

x 2t

i.e. the kernel of the operator χ(α)e

. By (1.3), for any continuous com- pact support F and G

ε→0

lim hY

(G, t + s)Y

(F, s)i = χ(α) Z

R^d

G(u)(e

F )(u) du. (1.5) A generalized (stationary) Ornstein-Uhlenbeck process corresponding to the operators

∇ · σ∇ and p

χ(α)∇ is a gaussian random field on R

× R with covariance given by the right hand side of (1.5). Equivalently this is the (sta- tionary) solution of the linear stochastic partial differential equation

∂

ξ = 1

2 ∇ · σ

∇ξ + p

χ(α) ∇ · σ W ˙ (1.6)

where ˙ W = ( ˙ W

, . . . , W ˙

) are independent standard white noises on R

× R . Since ξ is only a distribution, equation (1.6) should be intended in a weak sense.

In order to conclude that Y

converges to this generalized Ornstein- Uhlenbeck process (cf ref) we need some tightness of the sequence of the laws (that we make precise later) and the remark that the limit points satisfies (1.6), which in the symmetric case follows by straightforward calculations.

1.3 The asymmetric case

If p is not symmetric, the motion of the second class particle is affected by the presence of the other particles. If γ = P

x

xp(x) = 0, the asymptotic behavior

is still centered and diffusive, with a diffusion matrix that depend on α, the average density of the first class particles. We will call this matrix the viscosity matrix and denote it with D(α). Then the argument above applies with σ

replaced by D(α) and the limit equation is

∂

ξ = 1

2 ∇ · D(α)∇ξ + p

χ(α) ∇ · D(α)

W ˙ (1.7) where D(α)

denotes the square root of the symmetric matrix D(α).

If γ = P

x

xp(x) 6= 0, then the second class particle picks an asymptotic velocity v = γ(1 − 2α). This can be explained by the following simple argu- ment. The velocity of the second class particle will be γ if there were not any other particle. The exclusion rule reduce this velocity by a factor (1 −α) as for the tagged particle, and then we have still to subtract γα because with this rate the first class particles take its position and the second class particle move backward. Consequently in order to see a diffusive behavior we have to look at the re-centered position ˜ X

= X

−vt. We will see that, if γ 6= 0, only in di- mension 3 or higher the second class particle behave diffusively (D(α) < +∞), while in dimension 1 and 2 it actually superdiffuse. What is the nature of the motion in a proper superdiffusive scale is one of the most challenging open problem in this field (see bibliografical notes ).

Consequently we have to recenter the density field and consider (1.2) with v = γ(1 − 2α). We will prove that it converges to (1.7) if d ≥ 3 . This will be done by a direct analysis of the time evolution of Y

.

We state now more precisely the convergence theorem. For any k ≥ 0 and F, G ∈ C

( R

) consider the scalar product

(G, F )

= Z

R^d

G(q) |q|

− ∆

F (q) dq (1.8)

and denote by i

the corresponding closure. For any positive k we denote by i

its dual space with respect to the L

( R

) ≡ i

scalar product.

We will show in section 6 that the probability distribution Q

of Y

under P

, is supported and is tight in D([0, T ], i

) for any k > d + 1.

We are now ready to state the main theorem. If γ 6= 0 assume that d ≥ 3.

Theorem 5.2. There exists a strictly positive symmetric matrix D

(α) such that the sequence Q

converges weakly to the probability measure Q of the corresponding Ornstein–Uhlenbeck process with operators

∇ · D(α)∇ and p χ(α) · D(α)

∇

By (1.4) and the arguments above, the viscosity matrix D(α) is identified as

D

(α) = 1

α(1 − α) lim

t→∞

1 t

X

x∈Z^d

x

x

E

α

((η

(x − vt) − α)(η

(0) − α)) , (1.9)

Other expressions for this matrix can be given (cf. [ ? ]), while in chapter 6 we

prove that D

(α) are smooth functions of the density α ∈ [0, 1].

2 Time evolution of density fluctuations 135

2 Time evolution of density fluctuations

Fix a smooth function G : R

→ R with compact support and define the martingale M

by the time evolution equation

Y

(G) − Y

(G) = Z

0

∂

+ ε

L

Y

(G) ds + M

. An elementary computation shows that

(∂

+ ε

L)Y

(G) = −ε

X

x

(v · ∇G)[ε(x − vtε

)](η(x) − α)

− ε

X

x

G[ε(x − vtε

)] X

y

j

, (2.1) where j

, the instantaneous current between sites x and y, is given by j

= −s(x − y)[η(y) − η(x)] − a(x − y) n

η(y)(1 − η(x)) + η(x)(1 − η(y)) o . Let G

(x) = G[ε(x − vtε

)]. Since the current is anti-symmetric (j

=

−j

) and since its expectation with respect to ν

is equal to 2a(y − x)χ(α), an elementary computation gives that

− X

x

G

(x) X

y

j

= X

x,y

s(x − y)[G

(y) − G

(x)][η(x) − α]

− X

x,y

a(x − y)[G

(x) − G

(y)][η(y) − α][η(x) − α]

+ (1 − 2α) X

x,y

a(x − y)[G

(x) − G

(y)][η(x) − α]

because

η(y)(1 − η(x)) + η(x)(1 − η(y)) − 2α(1 − α)

= −2(η(y) − α)(η(x) − α) + (1 − 2α) n

(η(y) − α) + (η(x) − α) o . Finally, since a(·) is anti-symmetric, a Taylor expansion and a change of vari- ables in the summation permit to conclude that

(∂

+ ε

L)Y

(G) (2.2)

= ε

X

x

X

i,j

σ

(∂

∂

G)[ε(x − vtε

)](η(x) − α)

− ε

X

x,z

a(z)(z · ∇G)[ε(x − vtε

)]Φ

+ R

(η) .

In this formula, Φ

is the zero-mean local function defined by

Φ

= (η(x) − α)(η(y) − α)

and R

(η) is a remainder which vanishes in L

(ν

) as ε ↓ 0. In fact

< R

(η)

>= O(ε

).

The first term on the right hand side of (2.2) is already a function of the fluctuation field. Our problems come from the second term, where Φ is not a gradient and it does not permit to perform directly another integration by part. The idea is to decompose Φ, in some approximate sense, in a gradient term (dissipation) and a fluctuation term that is in the range of the generator L. More precisely we will prove in Theorem 5.4 in the next section, that there exists a sequence of local functions {v

, m ≥ 1} and constants D

such that

m→∞

lim lim

ε→0

E

h sup

0≤t≤T

ε

Z

X

x,z

a(z)(z · ∇G)[ε(x − vtε

)]

n

Φ

− Lv

− X

z⁰

a(z

)D

[η(z

) − η(0)] o

(τ

η

) ds

i

= 0 . (2.3) This result shows that we may replace Φ

in the second term on the right hand side of (2.2) by Lv

− P

z⁰

a(z

)D

[η(z

) − η(0)]. The difference η(z

) − η(0) enables a second summation by parts which cancels a factor ε

, while the term Lv

produces an extra martingale.

Let F (x) = P

z

a(z)(z · ∇G)(x). For each m ≥ 1, consider the martingale M

defined by

M

= ε

X

x

Z

F(ε(x − vsε

))ε

Lv

(τ

η

) ds + ε

X

x

Z

∂

F (ε(x − vsε

))v

(τ

η

) ds

− ε

X

x

n F (ε(x − vtε

))v

(τ

η

) − F(εx)v

(τ

η

) o . Since v

are local functions, it is easy to see that the L

norm of the third term of the right hand side vanishes as ε → 0. The second term is equal to

ε

X

x

Z

(v · ∇F)(ε(x − vtε

))v

(τ

η

) dt . (2.4) We will show in the next chapters that the functions v

involved have the property that the expectation of the square of (2.4) goes to 0 as ε → 0.

We have shown so far that Z

0

∂

+ ε

L

Y

(G) ds = Z

0

Y

(AG) ds + M

+ R

(t) ,

2 Time evolution of density fluctuations 137 where

m→∞

lim lim

ε→0

E

sup

0≤t≤T

R

(t)

2

= 0 and the second order differential operator A is given by

A = X

i,j

D

∂

∂

,

with the matrix D

given by

D

= σ

− X

z,z⁰

a(z)a(z

)z

z

D

.

We turn now to the calculation of the quadratic variation of the martingale M

+ M

. This is given by

Z

X

y

X

z

p(z)η(y)[1 − η(y + z)]ε

H

(η

) − H

(η

)

ds , (2.5)

where

H

(η) = Y

(G) − ε

X

x

Z

F (ε(x − vtε

))v

(τ

η

) . Elementary computations show that (2.5) is equal to

Z

ε

X

y

X

z

p(z)η(y)[1 − η(y + z)]

ε

[G

(y + z) − G

(y)] − X

x

F

(x) v

(τ

σ

η

) − v

(τ

η

) !

ds , (2.6) where F

(x) = F (ε(x − vtε

)).

Since v

is a local function, the sum inside the square in the above ex- pression extends over a finite number of x depending only on the support of v

. We can therefore substitute F

(x) by F

(y), with an error that is small in view of Theorem 5.3. In the same way, we replace the discrete derivative of G by the actual derivative, obtaining that (2.5) is equal to

Z

ε

X

y

X

z

p(z)η(y)[1 − η(y + z)]

"

(z · ∇G)(ε(y − vtε

)) − F

(y) X

x

v

(τ

σ

η

) − v

(τ

η

) #

ds

plus a remainder R

(t) which vanishes in L

as ε ↓ 0. Recall the definition of F and take the limit as ε → 0. By the law of large numbers, we obtain that the previous expression converges to

t Z

dy X

z

p(z)(z · ∇G)(y)

η(0)(1 − η(z))

1 − a(z) h

Γ

(η

) − Γ

(η) i

, where Γ

(η) denotes the formal sum P

x

v

(τ

η).

Since we performed this calculations in equilibrium and since for the in- variant product measure the static fluctuations converges to the Gaussian field with covariance operator α(1 − α)(−∆)

, defining

b

= lim

m→∞

X

z

p(z)

η(0)(1 − η(z))

1 − a(z) h

Γ

(η

) − Γ

(η) i

z

z

, the fluctuation-dissipation relation for the limit Ornstein-Uhlenbeck process gives

b

= α(1 − α)D

.

3 The Fluctuation-Dissipation Theorem

The fluctuation-dissipation theorem is the main tool we need in order to prove (2.3). We will state it here and prove it later.

The first step is to characterize the class of local functions v with null average such that the space time variance

E

α



 ε

Z

0

X

x

G(εx)v(τ

η

) ds

!



 ≤ C(v) (3.1)

is bounded by a constant C(v, G) independent of ε.

It is clear that this is not true for all local function with null average. In fact take v(η) = η(0) − α, then

ε

Z

0

X

x

G(εx)v(τ

η

) ds = ε

Z

0

ε

X

x

G(εx)(η

(x) − α) dτ (3.2) and we expect that the variance of the right hand side of (3.2) diverges with ε → 0. This is in fact a necessary condition for the validity of Theorem 5.2, i.e. density fluctuations move smoothly at the time scale ε

t.

An important class of functions that verify (3.1) is given by gradients of the type v(η) = φ(τ

η)−φ(η), with φ a given local function with null average.

In fact in this case

3 The Fluctuation-Dissipation Theorem 139

ε

Z

0

X

x

G(εx)v(τ

η

) ds = − Z

0

ε

X

x

∇

G(εx)φ(η

) dτ + R

(3.3) where the remainder R

has bounded variance vanishing in ε → 0. The by using Schwartz inequality we can bound

E

α



 ε

Z

0

X

x

∇

G(εx)φ(τ

η

) ds

!





≤ t

2 ε

X

x,y

∇

G(εx)∇

G(εy)| < φ(τ

η)φ(τ

η) > |

= ε

X

x

(∇

G(εx))

X

y

| < φ(τ

η)φ(η) > | ≤ k∇

Gk

C(φ)

(3.4)

Another class of functions that satisfies (3.1) is given by the functions in the range of the generator L, i.e. v = LF . In this case

ε

Z

0

X

x

G(εx)v(τ

η

) ds

=ε

X

x

G(εx)(F(τ

η

) − F (τ

η

)) + M

(3.5)

The first term on the RHS of (3.5) has vanishing variance as ε → 0. The second term is a martingale with bounded quadratic variation, by the same compu- tation done in the previous section shows (cf. equation (2.5) and following ones).

Recalling that our goal is to prove (2.3), we need then to prove that (i) Φ

= (η(0) − α)(η(z) − α) have finite space-time variance in the sense of

(3.1),

(ii) Φ

can be approximated by functions of the type LF plus linear combi- nation of gradients η(x) − η(0), in the sense of (2.3).

These are now the difficult points of the proof, since even point 1 can be proven only in dimension d ≥ 3 as we will see below. Let us clarify now why we expect a decomposition as in point 2. Recall the orthogonal decomposition of L

(α) = ⊕

G

as defined in section 4. An explicit calculation shows that, if F ∈ ⊕

G

, then the projection on G

of LF is a linear combination of gradients like P

x

b(x)(η(x) − η(0)) (in fact it is enough to consider function F ∈ G

). Ideally we would like to solve the equation in C

= ⊕

G

Φ

= LF − π

(LF ) (3.6)

where π

denotes the projection in G

. Similarly as what happens in the

previous chapters, we cannot solve (3.6) exactly, but we want to solve ap-

proximately in such a way the remainder has vanishing space-time variance.

We cannot compute directly these space-time variances, but we can estimate them by those relative to the symmetric simple exclusion, in the sense made precise by Theorem 5.3 below.

We need to introduce some new formalism. For local functions u, v in C

, define the scalar product · , · by

u, v = X

x∈Z^d

{ < τ

u, v > − < u >< v > } , (3.7) where {τ

, x ∈ Z

} is the group of translations and < · > stands for the expectation with respect to the measure ν

. That this is in fact an inner product can be seen by the relation

u, v = lim

V↑Z^d

1

|V | h X

x∈V

τ

(u− < u >), X

x∈V

τ

(v− < v >) i . Since u − τ

u , v = 0 for all x in Z

, this scalar product is only positive semidefinite. We will show later that these gradients are the only elements of the kernel of ·, · . Denote by L

(ν

) the Hilbert space generated by the local functions in C

and the inner product ·, · .

Let L = S + A the generator defined by 0.1. For two local functions u, v ∈ C

, let

u, v

= u, (−S)v

and let H

be the Hilbert space generated by zero-mean local functions and the inner product ·, ·

. To introduce the dual Hilbert spaces of H

, for a local function u, consider the semi-norm k · k

given by

kuk

= sup

v

n 2 u, v − v, v

o ,

where the supremum is carried over all local functions v ∈ C

. Denote by H

the Hilbert space generated by C

and the semi-norm k · k

.

Indeed we will show that if f ∈ C

then kf k

< ∞. The next theorem states that a local function f in C

has a finite space-time variance in the diffusive scaling.

Theorem 5.3. Let d ≥ 3 and fix T > 0, a vector v

in R

, a smooth function G : R

→ R with compact support and a local function f in C

. There exists a finite constant C

such that

lim sup

ε→0

E

α

h sup

0≤t≤T

ε

Z

0

X

x∈Z^d

G(ε[x − sv

])f (τ

η

) ds

i

≤ C

T kGk

kf k

.

We will prove in section xxx that for every local function f in C

and every

ε > 0, there exists a local function u

, which may be taken in C

, such that

4 Duality revised 141 k(Lu

− π

Lu

) − f k

≤ ε . (3.8) The fluctuation-dissipation theorem stated below follows from this result, the estimate stated in Theorem 5.3 and some elementary computations.

Theorem 5.4. Fix T > 0, a vector v

in R

, a local function w in C

and a smooth function G : R

→ R with compact support. There exist a sequence of local functions u

and D

(α) such that

lim sup

m→∞

lim sup

ε→0

E

α

h sup

0≤t≤T

ε

Z

0

X

x∈Z^d

G(ε[x − rv

])τ

W

(η

) ds

i

= 0 , where

W

(η) = w − Lu

+ X

z∈Z^d

a(z)D

(α){η(z) − η(0)} .

4 Duality revised

In order to prove the fluctuation-dissipation theorem 5.4, we need to analyze, in the proper space, the resolvent equation u

− Lu

= w with w ∈ C

. As in the previous chapters we will use duality, and we will see that the problem is very close to the one of the tagged particle.

From now on we will consider functions f ∈ C

= ⊕

G

. For each n ≥ 2, let be E

, E , Ψ

, G

as defined in section 4. Then a local function f can be written as f = P

n≥0

P

A∈En

f(A)Ψ

. Since f is a local function, f : E → R is a function on E with finite support.

Fix two local functions u, v and write them in the basis {Ψ

, A ∈ E}:

u = X

A∈E

u(A)Ψ

, v = X

A∈E

v(A)Ψ

.

An elementary computation shows that u, v = X

x∈Z^d

X

n≥2

X

A∈En

u(A) v(A + x) .

where u, v is defined by (3.7). In this formula, B +z is the set {x +z; x ∈ B}.

We say that two finite subsets A, B of Z

are equivalent if one is the

translation of the other. This equivalence relation is denoted by ∼ so that

A ∼ B if A = B + x for some x in Z

. Let ˜ E

be the quotient of E

with

respect to this equivalence relation: ˜ E

= E

/

, ˜ E = E/

. If, for some n,

f : E

→ R is a summable function,

X

A∈En

f(A) = X

A∈˜ E˜n

˜ f( ˜ A)

where, for any equivalence class ˜ A and a summable function f : E → R ,

˜ f( ˜ A) = X

z∈Z^d

f(A + z) , (4.1)

A being any representative from ˜ A. In particular, for two local functions u, v, u, v = X

x,z∈Z^d

X

n≥2

X

A∈˜ E˜n

u(A + z) v(A + x + z) = X

n≥1

X

A∈˜ E˜n

˜ u( ˜ A) ˜ v( ˜ A) .

We say that a function f : E → R is translation invariant if f(A + x) = f(A) for all sets A in E and all sites x of Z

. Of course, functions ˜ f on ˜ E are the same as translation invariant functions on E. Fix a subset A of Z

with n points.

There are n sets in the class of equivalence of A that contain the origin.

Therefore, summing a translation invariant function f over all equivalence classes ˜ A in ˜ E

is the same as summing f over all sets B in E

which contain the origin and dividing by n:

X

A∈˜ E˜n

f( ˜ A) = 1 n

X

A∈En

A30

f(A)

provided that f(A) = f(A + x) for all A, x. Let E

be the class of all finite subsets of Z

= Z

\{0} and let E

be the class of all subsets of Z

with n points. These are the same we used for the tagged particle in chapter ??.

Then, we may write

u, v = X

n≥2

1 n

X

A∈En

A30

˜

u(A) ˜ v(A)

= X

n≥0

1 n + 1

X

A∈E∗,n

˜

u(A ∪ {0}) ˜ v(A ∪ {0}) .

In conclusion, if for a finitely supported function f : E → R , we define Tf : E

→ R by

(Tf)(A) = ˜ f(A ∪ {0}) = X

z∈Z^d

f([A ∪ {0}] + z) ,

we have that

u, v = X

n≥1

1 n + 1

X

A∈E∗,n

Tu(A) Tv(A) . (4.2)

Observe that T transform a function on E

into a function on E

.

4 Duality revised 143 For n ≥ 2, denote by π

the projection that corresponds to E

i.e.

(π

f)(A) = f(A) 1 {A ∈ E

} and denote by < · , · > the usual scalar product on each set E

: for f, g: E

→ R ,

< f, g > = X

A∈E∗,n

f(A) g(A) .

In view of formula (4.2), it is natural to introduce, for an integer k ≥ −1, the Hilbert spaces L

(E

, k) generated by finite supported functions f : E

→ R and the scalar product · , ·

defined by

f, g

= X

n≥0

(n + 1)

< π

f, π

g > . With this notation, for local functions f , g in ⊕

G

,

f, g = Tf, Tg

.

To summarize some observations on the transformation T, we need some notation. For a subset A of Z

∗

Recall the definitions of A

given by (4.1) and of θ

A given by (??).

Remark 5.5.

(a) The restriction of f to E

is irrelevant for the definition of Tf(A) if A is nonempty.to be erased

(b) Not every function f

: E

→ R is the image by T of some function f : E → R since

(Tf)(A) = (Tf)(S

A) (4.3)

for all z in A.

(c) Let f

: E

→ R be a finitely supported function satisfying (4.3): f

(A) = f

(θ

A) for all z in A. Define f : E → R by

f(B) =

|B|

f

(B \ {0}) if B 3 0 ,

0 otherwise . (4.4)

An elementary computations shows that Tf = f

. With this choice, which is natural but not unique, f(0) = f

(φ).to be erased

(d) T maps E

into E

lowering the degree of a function by one. Thus the translations in the inner product ·, · effectively reduce the degree by one while replacing the space Z

by Z

∗

. to be erased

(e) Formula (4.2) shows also that a local function f is in the kernel of the inner product ·, · if and only if Tf vanishes, i.e., if and only if

X

x∈Z^d

f(A + x) = 0

for all finite subsets A such that |A| ≥ 1. Examples of such functions are the constants or the difference of the translations of a local function:

τ

f − τ

f .

Most of the time, for a function f : E → R , we denote Tf by ¯ f. Real functions on E or on E

are indistinctively denoted by the symbols f, g.

Fix a function u of degree n ≥ 2 and denote by u its Fourier coefficients.

Recall the definition of S given by (??) and (4.3) (Su)(A) = (1/2) X

x,y∈Z^d

s(y − x)[u(A

) − u(A)] (4.5) An elementary computation, based on the fact that

X

z∈Z^d

f([B ∪ {y}] + z) = Tf(θ

B)

for all subsets B of Z

, sites y not in B and finitely supported functions f : E → R , shows that for every set B in E

TSu (B) = S

Tu (B) + S

Tu (B) , (4.6) where

(S

u) (B) = (1/2) X

x,y∈Z^d∗

s(y − x)[u(B

) − u(B)]

and

(S

u) (B) = X

y6∈B

s(y)[u(θ

B) − u(B)] .

This computation should be understood as follows. We introduced an equivalence relation in E when we decided not to distinguish between a set and its translations. This is the same as assuming that all sets contain the origin.

If n particles evolve as exclusion random walks on Z

, one of them fixed to be at the origin, two things may happen. Either one of the particles which is not at the origin jumps or the particle we assumed to be at the origin jumps. In the first case, this is just a jump on Z

and is taken care by the generator S

. In the second case, however, since we are imposing the origin to be always occupied, we need to translate back the configuration to the origin. This part corresponds to the generator S

.

In section 4 we have defined, on local function u = P

A∈E

u(A)Ψ

, the operator L

= S + (1 − 2α)A + p

χ(α)(J

+ J

) such that Lu = X

A∈E

(L

u)(A)Ψ

,

We can compute the operator L

on function on E

such that, for any function u : E → R , we have

TL

u = L

Tu .

The calculation gives

4 Duality revised 145 L

= S

+ S

+ (1 − 2α)(A

+ A

) + p

χ(α){J

+ J

} (4.7) where S

, S

are defined above, and, for A ∈ E

, v : E

→ R a finitely supported function,

(A

v)(A) = X

x∈A,y6∈A x,y6=0

a(y − x){v(A

) − v(A))

(A

v)(A) = X

y6∈A y6=0

a(y){v(θ

A) − v(A)} ,

(L

v)(A) = 2 X

x∈A,y∈A

a(y − x) v(A\{y}) + 2 X

x∈A

a(x){v(A\{x}) − v(θ

[A\{x}])} , (L

v)(A) = 2 X

x6∈A,y6∈A x,y6=0

a(y − x) v(A ∪ {y}) .

Observe that the form of L

is very similar as the one obtained for the tagged particle ??.??.

For two local functions u, v ∈ C

, define

u, v

= u, (−S

)v

and let H

be the Hilbert space generated by local functions f ∈ C

and the inner product ·, ·

. Since S

keeps the degree of the functions mapping E

in E

, by (??), (4.2) and (4.6) the previous scalar product is equal to

X

n≥1

1

n + 1 < π

u , (−S

)π

v > .

This formula leads to the following definitions. For each n ≥ 0, denote by

< ·, · >

the scalar product on E

defined by

< f, g >

= < f, (−S

)g >

and denote by H

(E

) the Hilbert space on E

induced by the finitely supported functions and the scalar product < ·, · >

. The associated norm is denoted by kfk

=< f, (−S

)f >. Furthermore, for an integer k ≥ −1, denote by H

= H

(E

, S

, k) the Hilbert space induced by the finite supported functions f, g: E

→ R and scalar product

f, g

= f, (−L

)g

= X

n≥1

(n + 1)

< π

f, (−S

)π

g > .

The associated norm is denoted by k · k

so that kfk

= f, f

.

Observe that for every n ≥ 1 and every finitely supported functions f, g: E

→ R ,

< f, g >

= 1 4

X

x,y6=0

s(y − x) X

A∈E∗,n

[g(A

) − g(A)] [f(A

) − f(A)]

+ 1 2

X

y∈Z^d

s(y) X

A∈E∗,n

A63y

[g(θ

A) − g(A)] [f(θ

A) − f(A)] .

To introduce the dual Hilbert spaces of H

, H

, for a local function u ∈ C

, consider the semi-norm k · k

given by

kuk

= sup

v∈C1

n 2 u, v − v, v

o ,

where the supremum is taken over all local functions v. Denote by H

the Hilbert space generated by the local functions in C

and the semi-norm k·k

. Recall the definition of the spaces G

introduced at the beginning of sub- section 3.1. Since S

keeps the degree of a function and since the spaces G

are orthogonal, for local functions of fixed degree, we may restrict the supremum to local functions of the same degree so that

kf k

= X

n≥2

kπ

f k

, where π

f stands for the projection of f on G

.

In the same way, for an integer n ≥ 1 and a finitely supported function u : E

→ R , let

kuk

= sup

v

n 2 < u, v > − < v, v >

o ,

where the supremum is carried over all finitely supported functions v : E

→ R . Denote by H

= H

(E

) the Hilbert space induced by the finitely supported functions u : E

→ R and the semi-norm k · k

.

For a integer k ≥ −1, define the H

= H

(E

, S

, k) norm of a finite supported function u : E

→ R by

kuk

= sup

v

n 2 u, v

− v, (−S

)v

o ,

where the supremum is carried over all finitely supported functions v : E

→ R . Denote by H

= H

(E

, S

, k) the Hilbert space induced by this semi- norm and the space of finite supported functions. Here again, since S

does not change the degrees of a function, for every finitely supported u : E

→ R ,

kuk

= X

n≥1

(n + 1)

kπ

uk

and for any local function u ∈ C

,

kuk

= kTuk

.

6 Tightness 147

5 Resolvent approximation

To conclude our proof we have to consider the equation in E

λu

− L

u

= w (5.1)

with w a finitely supported function on E

. In our case w = δ

. The remainder of the proof continues exatly as done in section 4.7.

6 Tightness

Recall that we have defined, for any k ∈ R , i

as the closure of C

( R

) with respect to the scalar product

(g, f)

= Z

R^d

g(q) |q|

− ∆

f (q) dq

It is convenient to represent the scalar product (·, ·)

in the orthonormal basis of the Hermite polynomials, which are the eigenfunctions of |q|

− ∆.

Let n be a multi-index of ( Z

)

and |n| = P

i=1

n(i). We denote by λ

= 2n(i) + 1 for n(i) ∈ Z

and λ

= P

i=1

λ

. Define h

(q) = Q

i=1

h

(q

) where h

is the m

normalized Hermite polynomial of order m in R . We have then for every k ≥ 0 and f ∈ L

kf k

= Z

R^d

f (q)(|q|

− ∆)

f (q) dq = X

n∈(Z⁺)^d

λ

Z

R^d

f (q)h

(q) dq

This is valid also for negative k. So the i

-norm of a distribution ξ on R

can be written as

kξk

= X

n∈(Z⁺)^d

λ

ξ(h

)

(6.1)

Observe that, for k

> k, the injection J of i

in i

is compact. In fact it can be approximated by the finite range operators J

ξ = P

|n|≤m

ξ (h

)h

, and it is easy to see that the operator norm of the difference is bounded by

kJ − J

k ≤ (2m + d)

By the compactness of the injections i

, → i

for k < k

, and stan- dard compactness arguments, the tightness of the distribution of Y

is a con- sequence of the following proposition.

Proposition 5.6. For any k > d + 1 and every T > 0, we have that (i)

sup

ε∈(0,1)

E

sup

t∈[0,T]

||Y

||

!

< +∞

(ii) For any R > 0,

δ→0

lim lim sup

ε→0

P



 sup

t,s∈[0,T]

|t−s|≤δ

kY

− Y

k

> R



 = 0

It is easy to see, by using (6.1) and that k > d + 1, that Proposition 5.6 is a consequence of the following

Proposition 5.7. For any smooth function G on R

with compact support

sup

ε

E

sup

t∈[0,T]

Y

(G)

!

≤ T C

(6.2)

δ→0

lim lim sup

ε→0

P



 sup

t,s∈[0,T]

|t−s|≤δ

|Y

(G) − Y

(G)| > R



 = 0 (6.3)

Let just sketch the proof of (6.3), the proof of (6.2) will follow a similar argument.

By the same calculation made in the previous section we have Y

(G) − Y

(G) =

Z

∂

+ ε

L

Y

(G) dτ + M

= Z

s

Y

(AG) dτ + M

+ M

+ R

(s, t) ,

(6.4)

where M

and M

are the differences of the corresponding martingales defined in the previous section, and

m→∞

lim lim

ε→0

E

sup

0≤s≤t≤T

R

(s, t)

2

= 0 The first term is easy to deal since

E



 sup

t,s∈[0,T]

|t−s|≤δ

| Z

s

Y

(AG) dτ |



 ≤ δT

Y

(AG)

≤ δT CkAGk

. (6.5)

About the difference martingale M f

= M

+ M

, for any finite m has a bounded quadratic variation given by (2.5) or (2.6), and it is not difficult to show exponential bounds for it. So it follows that for any m

δ→0

lim lim sup

ε→0

P



 sup

t,s∈[0,T]

|t−s|≤δ

| M f

| > R



 = 0.