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3

The Simple Exclusion Process

Among the simplest and most widely studied interacting particle systems is the simple exclusion process. It represents the evolution of random walks on the lattice Zd with a hard-core interaction which prevents more than one particle per site. To describe the dynamics, fix a probability measure p(·) on Zd and distribute particles on the lattice in such a way that each site is occupied by at most one particle. Particles evolve onZdas random walks with translation–invariant transition probability p(x, y) = p(yx). Each time a particle tries to jump over a site already occupied, the jump is suppressed to respect the exclusion rule.

This informal description corresponds to a Markov process on Xd = {0,1}Zd endowed with the weak topology that makes it compact. The gener- atorLis given by

(Lf)(η) = X

x,z∈Zd

η(x)[1η(x+z)]p(z) [fx,x+zη)f(η)]. (0.1) Here,η stands for a configuration of Xd so that η(x) is equal to 1 (resp. 0) if the sitex is occupied (resp. vacant) for the configurationη. f is a cylinder function, which means that it depends onη only through a finite number of coordinates, andσx,yη is the configuration obtained fromη by interchanging the occupation variablesη(x),η(y):

x,yη)(z) =

η(z) ifz6=x, y , η(y) ifz=x , η(x) ifz=y .

(0.2) The interpretation is clear. Between 0 and dt each particle tries, inde- pendently from the others, to jump fromx to x+z at rate p(z). The jump is suppressed if it leads to an already occupied site. It is proved in Liggett (1985) that the operatorLdefined above is a Markov pre-generator and that the space C of cylinder functions, i. e. which depend on the configuration η only through a finite number of variablesη(x), forms a core forL.

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The simple exclusion process is said to be symmetric if the transition probability is symmetric (p(z) =p(z)), it is said to be mean zero if p(·) is not symmetric but has zero average: P

zzp(z) = 0. All other cases are said to be asymmetric.

Fix a cylinder functionV. The purpose of this chapter is to obtain sufficient conditions which guarantee a central limit theorem for

1 t

Z t 0

Vs)ds . (0.3)

The chapter is divided as follows. In Section 1 we discuss the main prop- erties of exclusion processes. In Section 4, we show that theL2 space can be decomposed as a direct sumn≥0Gn in a way which allows the generator of the exclusion process to be written as a sum of four pieces, two of them map- pingGn inGn, the other two mappingGn inGn+1. We recover in this manner the set-up of the previous chapter. We also prove in this section necessary and sufficient conditions for a local function to belong to H−1. In Section 2, we show that the generator of the mean zero asymmetric exclusion process satisfies a sector condition and deduce from this result a central limit theorem for (0.3). In Section 2 we prove that the pieces of the generator which changes the degree of a function satisfy a graded sector condition with respect to the symmetric part of the generator for the asymmetric exclusion process in di- mension d3. This estimate together with some bounds on the solution of the resolvent equation permits to recover a central limit theorem for (0.3) in this context. In both sections we follow the approach presented in the previous chapter.

1 Exclusion Processes

To avoid degeneracies, we assume that the transition probability p(·) is irre- ducible in the sense that the set {x : p(x) >0}generates Zd: for any pair of sites x, y in Zd, there existsM 1 and a sequence x =x0, . . . , xM =y such that p(xi, xi+1) +p(xi+1, xi) > 0 for 0 i M 1. Sometimes, we renormalize the transition probability in a slightly different way (we assume P

xp(x) =dinstead of 1). Finally, we will always suppose that the transition probability is of finite range: there existsA0 in N such thatp(z) = 0 for all sitesz outside the cube [A0, A0]d.

Denote by{S(t), t 0}the semigroup of the Markov process with gen- erator L given by (0.1). We use the same notation for semigroups acting on continuous functions or on the spaceM1(Xd) of probability measures onXd. Notice that the total number of particles is conserved by the dynamics.

This conservation is reflected in the existence of a one-parameter family of in- variant measures. For 0α1, denote byναthe Bernoulli product measure of parameter α. This means that under να the variables {η(x), xZd}are independent with marginals given by

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να{η(x) = 1} = α = 1να{η(x) = 0}.

Proposition 3.1.The Bernoulli measures{να,0α1}are invariant for simple exclusion processes.

Proof. By a simple change of variables, for any local functionsf, g and any bond{x, y},

Z

fx,yη)g(η)η(x)[1η(y)]να(dη) = Z

f(η)g(σx,yη)η(y)[1η(x)]να(dη). (1.1) This identity, the fact that 1 =P

z∈Zdp(z) = P

z∈Zdp(z) and a change in the order of summation prove that

Z

Lf dνα = 0

for all cylinder functionsf. This proves the proposition.

Notice that the family of invariant measuresνα is parameterised by the density, for

Eνα[η(0)] = να{η(0) = 1} = α . By Schwarz inequality, for any cylinder functionf,

[S(t)f(η)]2S(t)f2(η).

In particular, since να is a stationary measure for anyα[0,1], Z

[S(t)f(η)]2να(dη) Z

f(η)2να(dη).

Therefore, the semigroupS(t) extends to a Markov semigroup onL2α). Its generatorLνα is the closure ofLinL2α). Since the densityαremains fixed, to keep notation simple, we denote Lνα byL and byD(L) the domain of L in L2α).

Denote byL the generator defined by (0.1) associated to the transition probability p(x) = p(x). By (1.1), L is the adjoint of L in L2α). In particular, in the symmetric case, L is self-adjoint with respect to each να

(i.e.να is a reversible measure for the symmetric simple exclusion).

Denote bys(·) (resp. a(·)) the symmetric (resp. asymmetric) part of the transition probabilityp:

s(x) = (1/2){p(x) +p(x)}, a(x) = (1/2){p(x)p(x)}. We can decompose the operatorLin a symmetric and antisymmetric part as L=S+A where, for a cylinder functionf,

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(Sf)(η) = X

x,z∈Zd

η(x)(1η(x+z))s(z) [fx,x+zη)f(η)], (Af)(η) = X

x,z∈Zd

η(x)(1η(x+z))a(z) [fx,x+zη)f(η)].

Performing a change of variablesx0=x+z,z0 =z, we obtain (Sf)(η) = X

x,z∈Zd

η(x+z)(1η(x))s(z) [fx,x+zη)f(η)]

becauses(·) is symmetric. Adding the two previous formulae forSf and since σx,x+zη=η unlessη(x)(1η(x+z)) +η(x+z)(1η(x)) = 1, we deduce the simpler form

(Sf)(η) = (1/2) X

x,z∈Zd

s(z) [f(σx,x+zη)f(η)]

for the operatorS. In fact,S is the generator of the simple exclusion process with transition probability s(·). Since s(·) is symmetric, S is a self-adjoint operator onL2α).

For a cylinder functionf, denote byD(f) the Dirichlet form off: D(f) =hf,(Lf)iνα = hf,(Sf)iνα (1.2)

= (1/4) X

x,z∈Zd

s(z) Z

[fx,x+zη)f(η)]2να(dη).

Here,,·iναstands for the inner product inL2α). This formula holds also for functionsf in the domainD(L) of the generator, and the series defined on the right hand side converge absolutely. This can be proved by an approximation argument (cf. Lemma 4.4.3 in Liggett (1985)).

Theorem 3.2.For any α[0,1],να is ergodic forL.

Proof. Let f L2α) such that S(t)f =f for any t 0. Then f ∈ D(L) andLf= 0. Multiplying this last equation byf and integrating, by (1.2) we obtain

X

x,z

s(z) Z

[fx,x+zη)f(η)]2να(dη) = 0.

Since|a(z)| ≤s(z), the support ofp(·) is a subset of the support ofs(·). This implies that the support ofs(·) generatesZd. We deduce that for anyx, yZd

fx,yη) =f(η) ναa.e.

By De Finetti’s theorem we conclude thatf is constantνα-a.e.

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2 Central Limit Theorems for Additive Functionals

Fix 0 < α <1 and a zero-mean local function V. We prove in this section a central limit theorem for the additive functional t−1/2Rt

0Vs)ds in the context of zero-mean asymmetric simple exclusion process:

Theorem 3.3.Assume that the jump ratephas mean zero:P

x∈Zdxp(x) = 0.

Fix0< α <1and consider a zero-mean local functionV inH−1(L, να). Then,

1 t

Z t 0

Vs)ds

converges in distribution, as t ↑ ∞, to a zero-mean Gaussian variable with variance σ2(V) = 2 limλ→0kfλk21, where fλ is the solution of the resolvent equation (2.1). In the symmetric caseσ2(V) = 2kVk2−1.

Consider the resolvent equation

λfλLfλ=V . (2.1)

We have seen in Chapter 2 that a central limit theorem follows from a uniform bound on the H−1 norm of the solution of the resolvent equation:

sup

0<λ<1kLfλk−1<. (2.2) Assume that the jump rate is symmetric:p(x) =p(x). We have seen at the beginning of this chapter that in this case the generatorL is self-adjoint in L2α). In particular, by Theorem 2.2.2 and Subsection 2.6.1, (2.2) holds and so does Theorem 3.3.

To compute the variance, recall from Proposition 2.2.6 and from (2.4.2) thatσ2(V) = 2 limλ→0kfλk212kVk2−1. On the other hand, by the first part of the proof of Lemma 2.2.9, Lfλ converges weakly in H1 to V as λ0.

In particular,kVk−1 lim infλ→0kLfλk−1 = lim infλ→0kfλk1 because L is symmetric. This proves thatσ2(V) =kVk2−1as claimed in Theorem 3.3.

3 The Zero-mean asymmetric case

Assume now that the jump rate has mean zero:P

x∈Zdxp(x) = 0. In this case the generatorLsatisfies a sector condition:

Proposition 3.4.Assume that the transition probability has mean zero:P

x∈Zdx p(x) = 0. There exists a finite constantB, depending only on p(·)such that

h−Lf, gi Bh−Lf, fi1/2h−Lg, gi1/2 for all cylinder functionsf, g.

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Theorem 3.3 follows from this proposition, Theorem 2.2.2 and Subsection 2.6.2. By Proposition 2.2.6,σ2(V) = 2 limλ→0kfλk21.

We now turn to the proof of Proposition 3.4, which relies on a decomposi- tion of the generator in cycles, similar to the one presented in Section xxx for bistochastic centered random walks. The proof is divided in three steps. We first show in Lemma 3.5 that all zero-mean finite-range probability measures are convex combinations of cycle probability measures. Then, in Lemma 3.6, we prove that a sector condition holds for a finite convex combination of gen- erators if it holds individually. Finally, in Lemma 3.8 we conclude the proof of Proposition 3.4 showing that a sector condition holds for an exclusion process associated to a cycle probability.

A cycleC of length n is a sequence ofn sites of Zd starting and ending at the same point: {y0, y1, . . . , yn−1, yn =y0}. To a cycleC of lengthn, we associate a zero-mean transition probabilitypC defined by

pC(x) = 1 n

n−1X

j=0

1{x=yj+1yj}. pC has mean zero since

X

x∈Zd

x pC(x) = 1 n

X

x∈Zd n−1

X

j=0

x1{x=yj+1yj}

= 1 n

n−1

X

j=0

{yj+1yj} = yny0

n = 0.

The cycle probability measures are the finite-range zero-mean probability measures on Zd taking rational values. Indeed, fix such a probability mea- surepand denote its support byS={x1, . . . , xn}. There exists a sufficiently large positive integer M for which p(x) = m(x)/M, where m(x) are pos- itive integers. Consider the cycle C = (0, x1,2x1, . . . , m(x1)x1, m(x1)x1 + x2, . . . , m(x1)x1 +m(x2)x2, . . . , m(x1)x1 +· · ·+m(xn−1)xn−1+ (m(xn) 1)xn,0). It is easy to check that the probability measure associated to this cycle isp.

For a positive integerm, finite cyclesC ={C1, . . . , Cm}and a probability w={w1, . . . , wm}, letpC,w(·) be the probability onZd defined by

pC,w(·) = Xm k=1

wkpCk(·).

The next lemma states that all finite-range, zero-mean transition probabilities may be written as a convex combination of cycle probability measures.

Lemma 3.5.Fix a finite-range, zero-mean transition probability p on Zd. There existsm1, finite cyclesC ={C1, . . . , Cm}and a probability measure w={w1, . . . , wm}such thatp=pC,w(·).

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Proof. The proof is divided in three steps.

Claim 1:Fix a finite-range zero-mean transition probabilityp(·). Then, there exists a zero-mean probability q(·) taking only rational values and with the same support asp(·).

To prove this claim, denote by S = {x1, . . . , xn} the support of p(·) and byv1, . . . , vd then-dimensional vectorsvj= (hx1, eji, . . . ,hxn, eji). Here {e1, . . . , ed}stands for the canonical basis of Rd and,·ifor the usual inner product inRd. Sincep(·) has mean zero,p= (p(x1), . . . , p(xn)) is orthogonal tov1, . . . , vd.

Denote byVthe vector space of the real numbers over the rationals. The real numbersp(x1), . . . , p(xn) can be decomposed in terms ofQ-linearly inde- pendent reals: There exists 1 ` n, Q-linearly independent real numbers r1, . . . , r` and rational numbers{qk(x1), . . . qk(xn)}, 1k`, such that

p(x) = X` k=1

rkqk(x)

for each x in the support of p(·). Since p(·) has mean zero, by definition of qk(x),

0 = X

x∈S

x p(x) = X

x∈S

x X` k=1

rkqk(x) = X` k=1

rk

X

x∈S

x qk(x).

Sincerk areQ-linearly independent and sincexare integer-valued vectors, for

1k`, X

x∈S

x qk(x) = 0.

One could think that the claim is proved, for qk(·), 1k `, assume only rational values. However, nothing forbidsqk(·) to be negative preventing us to turnqinto a probability measure. We just know from the previous identity that the vectors qk = (qk(x1), . . . , qk(xn)), 1 k `, are orthogonal to v1, . . . , vd, as well as any linear combination of them. Sincep(·) is a probability measure and sincep=P

1≤k≤`rkqk, there is one linear combination of these vectors which is strictly positive in the sense that all its coordinates are strictly positive. Approximatingr1, . . . , r`by rationalsm1, . . . , m`we obtain a vector

˜ q = P

1≤k≤`mkqk with rational, strictly positive entries and orthogonal to v1, . . . , vd because each vector qk is orthogonal. Thus a multiple q0 of ˜q is a probability measure taking only rational values and such that

X

x∈S

x q0(x) = 0

becauseq0 is orthogonal tov1, . . . , vd. This proves the claim.

Claim 2: Every finite-range zero-mean transition probability p(·) can be de- composed asp(·)+(1w)q(·) for some 0< w1, where ˜p(·) is a probability

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measure whose support isstrictlycontained in the support ofp(·) andq(·) is a probability measure taking only rational values and whose support is equal to the support ofp(·).

Fix such probability measurep(·) and recall that we denote its support byS. By Step 1, there exists a transition probabilityq(·) taking only rational values and with the same support asp(·). Let

w = min

x∈S

p(x) q(x) ·

Since both pandqare probability measures,w1. Ifw= 1,p=qand the claim is proved. If p <1, let ˜pbe defined by

˜

p(x) = 1

1w{p(x)wq(x)}.

By definition of w, ˜pis a probability measure whose support is strictly con- tained in the support ofp(·). This proves the claim.

Claim 3: Every finite-range zero-mean transition probability p(·) assuming only rational values is a cycle measure. This has been shown right after the definition of cycle probability.

We may now conclude the proof of the lemma. By Claims 2 and 3, every finite-range zero-mean transition probability p(·) can be written as p(·) + (1w)q(·), where q(·) is a cycle probability measure and ˜p has a support strictly contained in the support ofp(·). Repeating the same argument for ˜p, after a finite number of steps, we decomposep(·) as a convex combinations of cycle probabilities and a probabilityp0(·) concentrated on at most one site.

Since any zero-mean probability concentrated on one site is the Dirac measure on the origin and since we are assuming thatp(·) puts no weight on the origin, p0(·) vanishes, which concludes the proof the lemma.

For a cycle C, denote by LC the generator associated to the transition probabilitypC:

(LCf)(η) = X

x,z∈Zd

η(x)[1η(x+z)]pC(z) [fx,x+zη)f(η)]. (3.1)

The previous lemma states that the generator L of an exclusion process associated to a zero-mean transition probability p can be written as L = P

1≤j≤mwjLCj for a positive integerm, a probability{w1, . . . , wm}and cy- cles{C1, . . . , Cm}.

LetLbe a generator which can be decomposed as a convex combination of two generators,L=θL1+ (1θ)L2 for some 0< θ <1. The next result states thatLsatisfies a sector condition if bothL1, L2do.

Lemma 3.6.Assume that there exists finite constants B1, B2 such that

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h−Ljf, gi Bjh−Ljf, fi1/2h−Ljg, gi1/2

for j= 1, 2and all functionsf, g in a common coreC for L,L1, L2. Then, h−Lf, gi Bh−Lf, fi1/2h−Lg, gi1/2

for B=θB1+ (1θ)B2 and all functionsf, g inC.

Proof. Fix two functionsf,ginC. By the triangle inequality and by assump- tion,

h−Lf, gi θh−L1f, gi + (1θ)h−L2f, gi

θB1h−L1f, fi1/2h−L1g, gi1/2 + (1θ)B2h−L2f, fi1/2h−L2g, gi1/2.

SinceL1,L2are generators,h−Ljf, fiare positive so thath−L1f, fi,h−L2f, fi are bounded above byh−Lf, fi. The previous expression is thus less than or equal to

{θB1+ (1θ)B2}h−Lf, fi1/2h−Lg, gi1/2 which concludes the proof of the lemma.

In view of the previous two lemmas, Proposition 3.4 follows from a sector condition for generators associated to cycles. This estimate is based on the following general statement.

Lemma 3.7.Fix a probability space (Ω,F, µ) and consider a familyT1, . . ., Tn of measure-preserving bijections. Assume thatT1◦ · · · ◦Tn=I, where I is the identity and consider the generator Ldefined by

(Lf)(η) = 1 n

Xn j=1

[f(Tjη)f(η)]. For all functionsf, g in a core ofL

hLf, gi 4nh−Lf, fi1/2h−Lg, gi1/2.

Proof. We first compute the Dirichlet form D(f) = h−Lf, fi associated to the generator L. Since the operators T1, . . . , Tn are measure-preserving, an elementary computation shows thatL, the adjoint ofLwith respect toµ, is given by

(Lf)(η) = 1 n

Xn j=1

[f(Tj−1η)f(η)]. In particular, an elementary computation shows that

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h−Lf, fi = 1 4n

Xn j=1

Z

{f(Tjη)f(η)}2µ(dη) (3.2)

+ 1

4n Xn j=1

Z

{f(Tj−1η)f(η)}2µ(dη).

We may turn now to the proof. Fix two functionsf,g in the core of the generator L. Since the operators are measure-preserving, after a change of variablesη=Tj+1◦ · · ·Tnξ, we obtain that

hLf, gi = 1

n Xn j=1

Z

{f(Tj◦ · · · ◦Tnη)f(Tj+1◦ · · · ◦Tnη)}g(Tj+1◦ · · · ◦Tnη)µ(dη).

SinceT1◦ · · · ◦Tn=I,Pn

j=1f(Tj◦ · · · ◦Tnη)f(Tj+1◦ · · · ◦Tnη) =f(T1

· · · ◦Tnη)f(η) = 0. The previous expression is therefore equal to 1

n Xn j=1

Z

{f(Tj◦ · · ·Tnη)f(Tj+1◦ · · · ◦Tnη)} ×

× {g(Tj+1◦ · · · ◦Tnη)g(η)}µ(dη). Applying Schwarz inequality, we obtain that this expression is less than or equal to

1 2An

Xn j=1

Z

{f(Tj◦ · · · ◦Tnη)f(Tj+1◦ · · · ◦Tnη)}2µ(dη)

+A 2

Xn j=1

Xn k=j+1

Z

{g(Tk◦ · · · ◦Tnη)g(Tk+1◦ · · · ◦Tnη)}2µ(dη)

for every A > 0. In the last step we replaced g(Tj+1◦ · · · ◦Tnη)g(η) by P

j+1≤k≤ng(Tk◦· · ·◦Tnη)g(Tk+1◦· · ·◦Tnη) and applied Schwarz inequality.

By a change of variables, we obtain that the previous sum is equal 1

2An Xn j=1

Z

{f(Tjη)f(η)}2µ(dη) + An 2

Xn k=1

Z

{g(Tkη)g(η)}2µ(dη).

In view of (3.2), this expression in bounded above by 2

Ah−Lf, fi + 2An2h−Lg, gi.

To conclude the proof of the lemma, it remains to minimize overA.

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Fix a cycleC ={0, x1, . . . , xn = 0}of lengthn and recall from (3.1) the expression of the generatorLCassociated to the cycleC. It can be written as

LC = X

z∈Zd

L0C+z,

whereL0C+zis the generator of an exclusion process in which all particles not in the setC+zare frozen while the ones inC+z may jump fromz+xj to z+xj+1. The formal generator of this process is given by

(L0C+zf)(η) = 1 n

n−1X

j=0

η(z+xj)[1η(z+xj+1)] [fz+xj,z+xj+1η)f(η)]. An elementary computation shows that the Dirichlet form associated to the generatorL0C+z is given by

h−L0C+zf, fi = 1 4n

n−1X

j=0

Eνα

{fz+xj,z+xj+1η)f(η)}2 .

Notice that the indicator function η(z+xj)[1η(z+xj+1)] gave place to a factor 1/4. Summing overzwe obtain the full Dirichlet form:

h−LCf, fi = X

z∈Zd

h−L0C+zf, fi = 1 4n

X

z∈Zd n−1X

j=0

Dz+xj,z+xj+1(f), (3.3) whereDx,y(f) stands for the piece of the Dirichlet form associated to jumps over the bond {x, y}:

Dx,y(f) = Eνα

{fx,yη)f(η)}2

. (3.4)

Lemma 3.8.For every local functions f, g,

hLCf, gi 4n2h−LCf, fi1/2h−LCg, gi1/2.

Proof. Fix two local functionsf, g. We have just seen that hLCf, gi can be written asP

z∈ZdhL0C+zf, gi. By Lemma 3.9 below, this expression is less than or equal to

4n2 X

z∈Zd

h−L0C+zf, fi1/2h−L0C+zg, gi1/2. By Schwarz inequality, this expression is bounded above by

4n2n X

z∈Zd

h−L0C+zf, fiX

z∈Zd

h−L0C+zg, gio1/2

= 4n2h−LCf, fi1/2h−LCg, gi1/2. This concludes the proof of the lemma.

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Lemma 3.9.For every local functions f, g,

hL0Cf, gi 4n2h−L0Cf, fi1/2h−L0Cg, gi1/2.

Proof. For 0j n1, denote byTjthe operatorσxj,xj+1. Fix a configura- tionξ in{0,1}Cwith a particle at the origin and observe thatTn−1◦ · · · ◦T0ξ is a configuration with a particle at the origin and where each other particle moved one step backward (two steps if the particle is originally atx1). More precisely, if {0, xj1, . . . , xjm},j1 <· · · < jm, stands for the occupied sites of the configurationξ, the occupied site ofTn−1◦ · · · ◦T0ξ are

{0, xj1−1, . . . , xjm−1} ifxj16=x1, {0, xj2−1, . . . , xjm−1, xn−1}ifxj1=x1.

In particular, for configurations ξ with a particle at the origin,Tn−1ξ =ξ if T =Tn−1◦ · · · ◦T0. This identity fails, however, if there is no particle at the origin.

By definition of the operatorsTj, we may write the scalar producthL0Cf, gi as

1 n

n−1X

j=0

Z

η(xj)[1η(xj+1)]{f(Tjη)f(η)}g(η)να(dη).

The presence of the indicators η(xj)[1η(xj+1)] in the previous formula prevents us in applying Lemma 3.7 directly. The proof, however, goes through.

Notice that we may take out the indicator [1η(xj+1)] in the previous formula since the difference f(Tjη)f(η) vanishes if η(xj+1) = 1. For 0 kn2, perform the change of variableη=Tj−1◦ · · · ◦T0Tkξ to rewrite the previous sum as

1 n(n1)

n−2X

k=0 n−1X

j=0

Z ξ(0)

f(Tj◦ · · · ◦T0Tkξ)f(Tj−1◦ · · · ◦T0Tkξ)

×g(Tj−1◦ · · · ◦T0Tkξ)να(dξ).

Observe that the indicatorη(xj) is transformed inξ(0) by the change of vari- ables. Since the configurationξhas a particle at the origin,Pn−2

k=0

Pn−1 j=0{f(Tj

· · · ◦T0Tkξ)f(Tj−1◦ · · · ◦T0Tkξ)}=f(Tn−1ξ)f(ξ) vanishes. In par- ticular, we may addg(ξ) in the previous formula and rewrite it as

1 n(n1)

n−2X

k=0 n−1X

j=0

Z ξ(0)

f(Tj◦ · · · ◦T0Tkξ)f(Tj−1◦ · · · ◦T0Tkξ)

×

g(Tj−1◦ · · · ◦T0Tkξ)g(ξ) να(dξ).

It remains to repeat the arguments presented at the end of the proof of Lemma 3.7 to estimate the previous expression by

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