On diffusivity of a tagged particle in asymmetric zero-range dynamics

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On diffusivity of a tagged particle in asymmetric zero-range dynamics

Sunder Sethuraman

^∗^,1

396 Carver Hall, Iowa State University, Ames, IA, 50011, USA

Received 13 September 2005; received in revised form 8 February 2006; accepted 3 March 2006 Available online 25 September 2006

Abstract

Consider a distinguished, or tagged particle in zero-range dynamics onZ^dwith rategwhose finite-range jump probabilitiesp possess a drift

jp(j )=0. We show, in equilibrium, that the variance of the tagged particle position at timetis at least ordert in alld1, and at most ordertind=1 andd3 for a wide class of ratesg. Also, ind=1, when the jump distributionpis totally asymmetric and nearest-neighbor, and the rateg(k)increases, andg(k)/keither decreases or increases withk, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.

Résumé

On considère « une particule marquée » pour des processus de zero-range surZ^davec un tauxgdont la probabilité de sautpest à portée finie, et possède une dérive

jp(j )=0. On montre qu’à l’équilibre la variance de la position de la particule marquée à l’instanttpour toutd1 est au plus d’ordretsid=1 etd3 pour une large classe de tauxg. Ensuite, pour le casd=1, en supposant que la probabilité de sautpest à plus proche voisin et totalement asymétrique, que le tauxg(k)est croissant, et que g(k)/kest soit croissant soit décroissant aveck, on montre que dans le cas d’approximations par diffusion la position centrée de la particule marquée converge vers un mouvement Brownien avec des coefficients de diffusion homogènes (au sens des distributions finies). Quelques caractérisations de la variance de la particule marquée sont aussi présentées.

MSC:primary 60K35; secondary 60F05

Keywords:Zero-range; Tagged particle; Diffusive; Variance; Invariance principle

1. Introduction and results

Informally, the zero-range particle system, introduced by Spitzer [35], follows the evolution of a collection of interacting random walks onZ^d. Namely, from a vertex withkparticles, one of the particles displaces byj with rate

* Tel.: +(515) 294 8140; fax: +(515) 294 5454.

E-mail address:[email protected] (S. Sethuraman).

1 Research supported in part by NSA-H982300510041 and NSF/DMS-0504193.

doi:10.1016/j.anihpb.2006.03.002

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(g(k)/k)p(j ). The function on the non-negative integersg:N→R+ is called the process “rate,” andp(·)denotes the translation-invariant single particle transition probability. The above interaction is in the “time-domain,” but not

“spatially,” hence the name “zero-range.” We note the case wheng(k)is proportional tokdescribes the situation of completely independent particles.

The problem of the asymptotics of a distinguished, or tagged particle interacting with others has a long history and was even mentioned in Spitzer’s seminal paper. Such questions are natural and important to physics and other applications (cf. Chapters 8.I, 6.II [36]). The main difficulty in the analysis is that the tagged particle motion is not in general Markovian due to the interaction with other particles. However, the basic feeling is that in some scale the tagged particle behaves as a random walk with certain “homogenized” parameters reflecting the system dynamics.

What is known in the literature, with respect to zero-range interaction, are some laws of large numbers, in equilibrium [30,32] and non-equilibrium [26], and equilibrium invariance principles when the jump probabilitypis mean-zero, jp(j )=0 [30,32] (see near (1.2) for exact statements of the equilibrium results).

The goal of this article is to further characterize the equilibrium fluctuations of the tagged particle when p is finite-range and has a drift

jp(j )=0 on which little is known. Under natural assumptions on the rate g, we give a characterization of the equilibrium tagged particle variance, and show it is at least diffusive in all dimensions (Theorem 1), and at most diffusive in dimensiond=1, and under more conditions ong, also ind3 (Theorem 2).

In addition, finite-dimensional convergence to a Brownian motion is proved in a case ind=1 with a “homogenized”

diffusion coefficient (Theorem 3).

In contrast, we remark, with respect to a tagged particle in simple exclusion, the asymptotics are more well- studied. In particular, laws of large numbers, both in equilibrium [29] and non-equilibrium [26] have been shown.

Also, equilibrium central limit theorems and invariance principles whenp is mean-zero [2,27,11,37], and whenp has a drift ind3 [33] and ind=1 whenpis in addition nearest-neighbor [9] have been proved. See also [12,15]

for fluctuations ind=1 with respect to a non-translation invariantp. Non-trivial non-equilibrium fluctuation results have even been derived ind1 whenpis symmetric (excluding thed=1 nearest-neighbor case) [25], and recently in the exceptional case ind=1 whenpis symmetric and nearest-neighbor [7]. In addition, large deviations results have been proved in some cases [24,31]. Some of these results and others are reviewed in [5], Section 4.VIII [17], Chapter 4.III [18], and Sections 4.3, 8.4 and 11.5 [10].

In this context, we note that the diffusive variance bounds for a tagged particle in zero-range given here ind=1 with respect to general rategand finite-rangepwith drift (Theorem 2) have no counterpart in the simple exclusion work at the moment. Some of the techniques used in this result may have application, after suitable modification, to simple exclusion and related models.

To state assumptions and our results, we need to define more precisely the zero-range process. LetΣ=N^Z^d be the configuration space where a configurationξ= {ξi: i∈Z^d}is given through occupation numbersξi at vertexiandN denotes the set of non-negative integers. The zero-range particle system then is a Markov processξ(t )on the space of right-continuous paths with left limitsD(R₊, Σ )with formal generator defined on certain “Lipschitz” functions,

(Lφ)(ξ )=

j

i

g(ξ_i)p(j ) φ

ξ^i,i⁺^j

−φ(ξ )

whereξ^i,i⁺^j is the configuration in which a particle fromiis moved toi+j. That is,ξ^i,i⁺^j=ξ−δi+δi+j where δk is the configuration with a single particle atk.

When a particle is distinguished, or tagged, we consider the joint Markov process(x(t ), ξ(t ))onD(R+,Z^d×Σ ) wherex(t )is the position of the tagged particle at timet. The formal generator is given by

(Lψ )(x, ξ )=

j

i=x

g(ξi)p(j ) ψ

x, ξ^i,i⁺^j

−ψ (x, ξ ) +

j

g(ξx)ξ_x−1 ξ_x p(j )

ψ

x, ξ^x,x⁺^j

−ψ (x, ξ )

+

j

g(ξ_x) ξ_x p(j )

ψ

x+j, ξ^x,x⁺^j

−ψ (x, ξ ) .

Here, the first term corresponds to particles other than at the tagged particle positionx moving, the second term corresponds to other particles moving fromx, and the last term represents motion of the tagged particle itself. One can also decompose the tagged particle motion further in terms of count processesN_j(t )which keep track of the

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number of jumps of various sizes j made up to timet and include their infinitesimal rates with respect to a more detailed generator.

To compensate for the non-Markov character of the tagged particle motion, a convenient method is to consider the “reference” process from the point-of-view of the tagged particle which has better properties. That is, letη(t )= τ_{x(t )}ξ(t )where for a configurationω∈Σ we define thek-shifted stateτ_kωby(τ_kω)_l=ω_l₊_kforl∈Z^d. We note this

“reference” processη(t )is a Markov process, obtained from the mapπ((x(·), ξ(·)))=τ_x(_·₎ξ(·), with formal generator (Lφ)(η)=

j

i=0

g(η_i)p(j ) φ

η^i,i⁺^j

−φ(η) +

j

g(η₀)η0−1 η0

p(j ) φ

η^0,j

−φ(η)

+

j

g(η0) η₀ p(j )

φ τ_j

η^0,j

−φ(η) .

In words,τj(η^0,j)is the configuration obtained by displacing the tagged particle byj and then shifting accordingly the reference frame.

The construction of these systems requires some conditions ongandp.

Basic assumptions ongandp.We will assume throughoutg(0)=0,g(k) >0 fork1,|g(k+1)−g(k)|Kfor some constantK, and lim infk→∞g(k) >0, and alsop(0)=0 andpis finite-range, that isp(i)=0 for|i|> Rfor some 1R <∞, whose symmetrizations(x)=(p(x)+p(−x))/2 is irreducible.

We remark of these assumptions, the condition lim infg(k) >0 is made to ensure invariant measures exist (cf. [1]).

When violated, the system evolution can lead to aggregation of particles in large piles, a sort of “condensation” effect (cf. [8,6]).

Under weaker assumptions, which include the above, Andjel constructs the processξ(t )semigroupT_t^Land gener- atorLon a class of “Lipschitz” functionsDdefined on a subsetΣ⊂Σ of the configuration space,

Σ=

ξ: ξ =

i∈Z^d

|ξi|βi<∞ D=

f: f (ξ)−f (ξ) c ξ−ξ for allξ, ξ∈Σ, for somec=c(f ) where one can takeβi=

n02⁻ⁿs⁽ⁿ⁾(i)for instance [1]. In a similar way, one can construct the process(x(t ), ξ(t )) semigroupT_t^Land generatorLwith respect to “Lipschitz” functionsf where

f (x, ξ)

−f

(y, ξ) c

|x−y| + ξ−ξ

for allx, y∈Z^d,ξ, ξ∈Σwithξ_x, ξ_y1 for somec=c(f ). Then, from the mapπ, processη(t )semigroupT_t^L and generatorLcan be constructed onD, the space of “Lipschitz” functions onΣ= {ξ∈Σ: ξ01}, namely those f so that|f (ξ)−f (ξ)|c ξ−ξ for allξ, ξ∈Σfor somec=c(f ).

The zero-range process ξ(t ) has a well-known explicit family product invariant measuresR_α =

i∈Z^dμ_α for 0α <lim infg(k)with marginal supported on non-negative integers,

μ_α(k)= 1 Z_α

α^k

g(k)! fork1 and μ_α(0)= 1 Z_α

whereg(k)! =g(1)· · ·g(k)andZ_α is the normalization [1]. These measures are all supported onΣ, and it can be shown that the process begun onΣnever leavesΣ. Also, it is not difficult to see thatDis a dense subset ofL²(Rα).

Let now ρ(α)=

kkμα(k), be the mean of the marginal μα, that is, the “density” of particles underRα, and let ρ^∗=limα↑lim infg(k)ρ(α). Note thatρ^∗may be finite for some type ofg’s. Asρ(α)↑ρ^∗forα↑lim infg(k), for a given 0ρ < ρ^∗, there is a unique inverseα=α(ρ).

For the reference process η(t ), the “Palm” or origin size biased measures given by dQα=(η₀/ρ(α))dRα are invariant (cf. [23,30]). Only the marginal at the origin, supported on the positive integers and denoted byμ⁰_α, differs fromμ_α, namely

μ⁰_α(k)= 1 ρZ_α

kα^k

g(k)! fork1 and μ⁰_α(k)=0 otherwise.

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Analogous to the discussion with respect toξ(t )andRα, the measureQα is supported onΣ, and it can be shown that the reference processη(t )begun onΣnever leavesΣ. Also,Dis a dense subset ofL²(Qα).

We now comment that, with respect to an invariantR_α, one can extend the zero-range process semigroupT_t^Land generatorLto a strongly continuous semigroup and closed generator with respect toL²(R_α)so that bounded functions inDform a core of the domain ofL; in fact, the same arguments show thatDitself is a core (cf. Section 2 [32]; for general reference see Chapter 1 [4]). In the same way, with respect to aQ_α, the reference semigroupT_t^Land generator Lcan be extended to a strongly continuous semigroup and closed generator onL²(Qα)with coreDfor the domain ofLdenoted Dα. We note also here constructions of these processes can be made through the martingale-problem approach [30,28]. Also, in this context, we note a Hille–Yosida type approach [16].

In addition, we note both families{R_α}and{Q_α}are in fact extremal measures in their respective convex set of invariant measures, and so process evolutions starting from these invariant states are time shift-ergodic. Extremality ofRαis proved in [32]; analogous arguments apply toQα.

Also, we note a standard computation shows that the adjointL^∗with respect toQα is itself a reference process generator but with reversed jump probabilityp^∗(·)=p(−·). Hence, in equilibriumQ_α, the time-reversed process at timet,{η(t−u): 0ut}, is also a reference process in equilibriumQ_α but with respect to jump probabilityp^∗.

In the following, to avoid degeneracies, we will work with a fixed 0< α <lim infg(k)for whichρ=ρ(α) >0, and correspondingRαandQα. For simplicity, we denote byEα[·]the expectation underQα and also under the reference process measure starting fromQαwhen there is no confusion; otherwise, the underlying measure is noted as a suffix.

Write now the tagged particle positionx(t )at timetas the sum total displacement, that is, x(t )=

j

j N_j(t )

whereNj(t )is the number of jumps of sizej it makes, or in terms of the reference process, the count of sizej-shifts made up to timet. The countNj(t )is compensated by_t

0(g(η0(s)/η0(s))p(j )ds, so that further x(t )=

j

j M_j(t )+

j

j t 0

g(η0(s))

η0(s) p(j )ds (1.1)

whereMj(t )=Nj(t )−t

0(g(η0(s)/η0(s))p(j )ds, since jumps of different sizes cannot happen simultaneously, are orthogonal martingales forj ∈Z^d. Moreover, we noteM_j²(t )−t

0(g(η₀(s))/η₀(s))p(j )dsare also martingales for j∈Z^d.

So, the tagged positionx(t )can be thought of as a function of the reference process. For most of the paper, we will think in terms of the reference frame, that is, the notationx(t )will denote the total reference shift

j N_j(t ). It will also be useful to define

M(t )=

j

j M_j(t ) and A(t )= t 0

f η(s)

ds

wheref(η)=(

jp(j ))[g(η0)/η0−α/ρ]. Note, in equilibriumQα, one obtains, asEα[g(η0)/η0] =α/ρ, that Eα

x(t )

=t Eα

g(η₀) η₀

j

jp(j )=tα ρ

j

jp(j )

and sox(t )−E_α[x(t )] =M(t )+A(t ). Also, from orthogonality of theM_j’s,E_α[|M(t )|²] =t (α/ρ)

|j|²p(j ).

Before going to the main results, we briefly review the known zero-range equilibrium tagged particle asymptotics.

In equilibriumQ_α, the law of large numbers holds [30,32]:

tlim→∞

1

tx(t )=α ρ

j

jp(j ) a.s. (1.2)

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Also, with respect to fluctuations, when the jump probabilities are mean-zero,

jp(j )=0, thenx(t )= j Mj(t ) is a martingale as the compensator terms cancel in (1.1). Then, under equilibriumQα, the quadratic variation is

E_α[|x(t )|²]

t =1

t

j

|j|²Eα

t 0

g(η₀(s)) η₀(s) p(j )ds

=α ρ

j

|j|²p(j )

and by martingale central limit theorem the invariance principle is proved [30,32]:

√1

λx(λt )⇒B(t ) asλ→ ∞

whereB(t )isd-dimensional Brownian motion with covariance matrix[(α/ρ)t

j(e_i·j )(e_k·j )p(j )]in terms of the standard basis{ei}ofZ^d.

We now arrive at our first result which gives a characterization of the tagged particle variance, and states that it is at least diffusive in all dimensions. As a comparison, we note this is not true for simple exclusion, in the cased=1 and the jump probabilitypis nearest-neighbor symmetric, for which the variance at timetis ordert^1/2[2]. See also [3,14] for variance representations with respect to exclusion processes.

Define the measureμ_αsupported on the positive integers by μ_α(k)= 1

Z_α α^k⁻¹

g(k−1)! fork2 and μ_α(1)= 1 Z_α

with normalizationZ_α. The interpretation is thatμ_α puts a particle at the origin and distributes other particles there according toμ_α. LetQ_α=

i=0μ_α×μ_α and letE_α denote expectation under the reference process begun atQ_α. Theorem 1.Under initial distributionQ_α, we have in all dimensionsd1fort0that

E_α x(t )−E_αx(t ) ²

=α ρ

j

|j|²p(j )t+E_α A(t ) ²

α

ρ

j

|j|²p(j )t. (1.3)

The lower bound is strict unlessg(k)is proportional tok. Also, the termEα[|A(t )|²]is further evaluated as

Eα A(t ) ²

=α ρ j

jp(j )

·2 t

0

E_α x(s)

−Eα

x(s)

ds. (1.4)

The first term in (1.3), (α/ρ)

|j|²p(j )t, is the mean quadratic variation of the martingale M(t ), and can be thought of as a “dynamical” contribution to the tagged particle variance. The second termE_α[|A(t )|²], on the other hand, from (1.4), as a difference in expected tagged particle positions from different initial measures, represents in a sense variation due to initial conditions. We also note wheng(k)=ck is proportional tok, the case of independent particles, thatα/ρ=c,Q_α =Q_α andf≡0, and so the tagged particle variance reduces to the variance of a single random walk with jump ratescp.

Now, to give upperbounds on the tagged particle variance, we describe a class of rate functionsg.

Assumption (SP). Let L_n be the generator of the symmetric zero-range process on a cube B_n = {i ∈ Z^d: maxl(|i_l|)n}, namely (L_nφ)(ξ )=

i,j∈Bng(ξ_i)(φ(ξ^i,j)−φ(ξ ))s(j −i). Let W (n, M) be the inverse of the spectral gap of L_n when there are M particles inB_n. Then, we assume the rateg is such that there is a constant C=C(α, g, p, d)whereE_R_α[(W (n,

i∈B_nξ_i))²]Cn⁴.

We observe ratesgwhereW (n, M)Cn²for a constantCindependent ofM, satisfy (SP) trivially, and include those rates where, for somea1 andb >0,g(k+a)−g(k)bfor allk0 [13]. Also, for the rateg(k)=1_[k1], it is knownW (n, M)C(1+M/n)²n²for some constantC [19], and so (SP) holds. It is most likely true that all ratesgsatisfy (SP), although this is open.

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Theorem 2.Under initial distributionQα, when

jp(j )=0, we have ind=1 (without further assumptions), and ind3under Assumption(SP)that there is a constantC=C(α, g, p, d)where, fort0,

Eα x(t )−Eα

x(t ) ²

Ct.

We note, unfortunately, our estimates made ind=1 and d3 do not seem to carry over straightforwardly to dimensiond=2. An open question then is to investigate the tagged particle variance ind=2 which should also be diffusive.

We now state finite-dimensional convergence to a Brownian motion in a special case ind=1.

Assumption (ID↓).The rate functiongis such thatg(k)increases andg(k)/kdecreases withk.

Assumption (ID↑).The rate functiongis such that bothg(k)andg(k)/kincrease withk.

Examples under (ID↓) include the well-studied case g(k)=1_[k1]. We note also, under (ID↑), and our basic assumptions (which preclude “super-linearity”),g(k)/kincreases to a bounded limit.

Theorem 3.Under initial distributionQ_α, ind=1when the jump probability is totally asymmetricp(1)=1andg satisfies either Assumption(ID↓)or(ID↑), we have the tagged particle varianceV (t )=E_α[(x(t )−E_α[x(t )])²]is super-additive int0, and also finite-dimensional convergence

λlim→∞

√1 λ

x(λt )−E_α x(λt )

=B(t )

to a Brownian motionBwith diffusion coefficientσ²=σ²(α, g)satisfying

Cσ²=α ρ

1+sup

t >0

2 t

t 0

E_α x(s)

−E_α x(s)

ds

α

ρ

whereCis the constant from Theorem2. Also, the lower bound is strict unlessg(k)is proportional tok.

We note an ingredient in the proof of Theorem 3 is to show the corresponding tagged particle position has weakly positively correlated increments (Lemma 4.2). In comparison, however, a tagged particle ind=1 simple exclusion with totally asymmetric nearest-neighbor transitions has negatively correlated increments, and in fact is a Poisson process (cf. Section 4.VIII [17]). But, in the above zero-range context, exactly when the system is that of independent particles (when g(k) is proportional to k) is the tagged particle a Poisson process, as otherwise σ²(ρ) > α/ρ= E_α[x(1)].

Also, we note a natural open question is to show tightness of the scaled centered positions inD[0,∞).

Remark on extensions.Presumably, one should expect diffusive variance bounds, and corresponding central limit theorems and invariance principles on the tagged position under only our initial basic assumptions ongandpin all dimensions not just the cases considered here. Also, in principle, one should expect the “finite-range” condition onp could be relaxed to a “second-moment” condition with analogous results. Finally, what possible different behaviors might arise under further weakening of the basic assumptions would be of interest and have not been considered.

We now comment on the proofs of these results, and the plan of the paper. The proof of Theorem 1, in Section 2, follows from a short argument using time-reversal. The proof of Theorem 2, which forms the bulk of the paper, follows from an analysis of certain resolvent orH₋₁norms, and is found in Section 3. On a technical level, we note the application ofH₋₁norm estimates in the asymmetric zero-range context differs somewhat from previous simple exclusion methods (cf. [33]) which use concepts of “dual” basis functions to reduce calculations. Last, the proof of Theorem 3, in Section 4, makes use of the diffusive variance bounds in Theorem 2, and follows by applying a Newman–Wright theorem. These proofs can be read independently of each other.

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2. Proof of Theorem 1

Most of the proof follows from explicit calculations. We have E_α x(t )−E_αx(t ) ²

=E_α M(t )+A(t ) ²

=E_α M(t ) ² +2Eα

M(t )·A(t )

+E_α A(t ) ²

=α ρ

|j|²p(j )t+2 t 0

E_α

M(s)·f η(s)

ds+E_α A(t ) ²

. (2.1)

Now, as noted in the introduction, under time reversal at time s, with respect to the process begun underQ_α, {η^∗(u)=η(s−u): 0us}is a reference frame zero-range process in equilibriumQ_αwith reversed ratesp(−·).

Note also that the number ofj-shifts up to timesin the forward realization equals the number of−j-shifts in the time- reversed realization up to times, that isN_j(s;η(·))=N₋_j(s;η^∗(·))where denoted in each count is the directed realization to which it refers. Also, in this notation,M^∗(s;η(·))=

j N₋_j(s;η(·))− j_s

0(g(η0(u))/η0(u))p(j )duis a martingale with respect to the reversed process with reversed ratesp(−·).

So, we have E_α

M(s)·f η(s)

=E_α M^∗

s;η^∗(·)

·f η^∗(0)

=E^∗_α M^∗

s;η(·)

·f η(0)

=

E_η^∗ M^∗

s;η(·)

·f(η)dQα(η)

after conditioning on time 0 whereE_α^∗andE^∗_ηdenote expectations with respect to the reversed process with initial dis- tributionQαand initial stateηrespectively. AsE^∗_η[M^∗(s;η(·)] =0, we haveEα[M(s)·f(η(s))] =0, and substituting into (2.1) we obtain (1.3).

To derive (1.4), we write using stationarity that Eα A(t ) ²

=2 t 0

Eα

f

η(0)

· s 0

f η(u)

du

ds=2 t 0

Eα

f η(0)

·A(s) ds

=2 t 0

E_α f

η(0)

·

(x(s)−E_α x(s)

−M(s) ds=2

t 0

E_α f

η(0)

·x(s) ds asfis mean-zero, andE_α[f(η(0))·M(s)] =E_α[f(η(0))·E_η(0)[M(s)]] =0.

Now, by a simple calculation, for a functionh∈L²(Q_α), we have E_α

g(η₀)/η₀ h(η)

=α ρE_R_α

E_η₊_δ₀ h(η)

=α ρE_Q

α

h(η) .

Hence, noting the definition off(cf. near (1.1)), E_α

f η(0)

·x(s)

=E_α f

η(0)

·E_η(0) x(s)

=α

ρ jp(j )

· E_α

x(s)

−E_α x(s)

.

We now consider whenE_α[|A(t )|²]is positive. Wheng(k)is proportional tok, as noted in the remark after the Theorem 1 statement,f≡0, and soE_α[|A(t )|²] =0 for allt0 in this case. Wheng(k)is not linear ink,fis non- trivial, and lims↓0s⁻²E_α[|A(s)|²] =E_α[|f|²]>0. Hence, in this case,E_α[|A(t )|²]> t²E_α[|f|²]/2>0 for allt >0 small. Now, supposeE_α[|A(t₀)|²] =0 for somet₀>0. Then,_t₀

0 f(η(s))ds=0 a.s. By stationarity,_t₀₊_a

a f(η(s))ds= 0 a.s. for each a >0. By simple addition and subtraction, for 0< < t₀, we have

0f(η(s))ds=_kt₀₊

kt0 f(η(s))ds a.s. for each integerk1. Then,

0

f η(s)

ds=1 n

n k=1

kt0+ kt0

f η(s)

ds a.s.

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for eachn1. Now, the limit limn→∞n⁻¹_n

k=1

kt₀+

kt₀ f(η(s))ds=Eα[

0 f(η(s))ds] =0 a.s. as the process evolution starting fromQ_α is time-shift ergodic, as noted in the introduction. Hence,

0f(η(s))ds=0 a.s. Choosing now >0 small enough so thatEα[|A()|²]>0 however yields a contradiction, finishing the proof. 2

3. Proof of Theorem 2

We first discuss some definitions and estimates involving variational formulas for some resolvent quantities involving notationE_α[f ψ] = f, ψα, andE_α[f²] = f ²₀.

The generatorL, with respect toQ_α, can be decomposed into symmetric and anti-symmetric parts,L=S+A whereS=(L+L^∗)/2 andA=(L−L^∗)/2 are well defined onD. One can check that the symmetric operatorSis in fact the generator of the reference frame zero-range process with symmetrized jump probabilitiess(·). Moreover,

−Sis a non-negative operator whose Dirichlet form can be computed, forψ∈D, as ψ, (−S)ψ

α=1 2

j

i=0

E_α g(η_i)

ψ η^i,i⁺^j

−ψ (η)2 s(j )

+1 2

j

E_α

g(η0)η₀−1 η₀

ψ η^0,j

−ψ (η)2 s(j )

+1 2

j

E_α g(η₀)

η₀ ψ

τ_j η^0,j

−ψ (η)2

s(j ). (3.1)

We now consider some useful Hilbert spaces. Forλ >0 andψ∈D, define the norm ψ _1,λby ψ ²_1,λ=

ψ, (λ−S)ψ

α.

The Hilbert spaceH1,λthen is the completion overDwith respect to this norm.

To define a dual norm, consider, forf ∈L²(Qα)andψ∈D, thatf, ψα f 0 ψ 0λ⁻^1/2 f 0 ψ 1,λ. Hence, forf∈L²(Qα), the dual norm

f ₋_1,λ=inf

κ: f, ψακ ψ _1,λfor allψ∈D

(3.2) is always finite with bound f ²₋_1,λλ⁻¹ f ²₀. In particular, forψ∈D, we have

f, ψα f ₋1,λ ψ 1,λ 1

λ f ²₀ 1/2

ψ 1,λ. (3.3)

Define, correspondingly,H₋_1,λas the Hilbert space with respect to the norm · ₋1,λ. [Although we will not need it, we remark that f ₋_1,λcan be evaluated as f ²₋_1,λ= f, (λ−S)⁻¹fα; see also [22] for other contexts.]

It will also be convenient to define (semi-)norm ψ ₁ forψ∈D by ψ ²₁= ψ, (−S)ψα, and corresponding Hilbert spaceH₁as the completion overDwith respect to this norm, after modding out by zero-norm functions. Also, forf ∈L²(Q_α), define the associatedH₋₁dual norm

f ₋₁=inf

κ: f, ψακ ψ ₁for allψ∈D

(3.4) with usual convention inf∅ = ∞. We note a large class of functions with finite norm · ₋1has been identified in [34]

(see also Proposition 3.2). LetH₋1be the corresponding Hilbert space completion over · ₋1norm finite functions, after modding out by zero-norm quantities.

There are clear relations betweenH1,λ,H₋1,λ,H1andH₋1norms, namely, forψ∈Dandf ∈L²(Qα),

ψ ₁ ψ _1,λ and f ₋_1,λ f ₋₁. (3.5)

Consider now the resolvent operator(λ−L)⁻¹:L²(Q_α)→Dαwell defined in terms of the semigroup, (λ−L)⁻¹f =

∞ 0

e⁻^λsT_s^Lfds.

(9)

Define, forf ∈L²(Qα), thatσ_t²(f )=Eα[(t

0f (η(s))ds)²], and observe from the decomposition (1.3), to get diffusive bounds on the tagged particle variance, one need only bound

σ_t²(h)=Eα

t 0

h η(s)

ds 2

< Ct

whereh(η)=g(η₀)/η₀−α/ρfor some constantC=C(α, g, p, d). The next result relatesσ_t²(f )toH₋_1,λandH₋₁ norms.

Proposition 3.1.Forf∈L²(Q_α), andt >0, there is a universal constantC₁such that σ_t²(f )C₁t

f,

t⁻¹−L₋1

f

α

C₁t f ²₋_1,t₋₁ C₁t f ²₋₁.

Proof. Letu=(λ−L)⁻¹f. The first line is a computation with the resolvent equationf =λu−Lu. First note, by multiplying through byuand integrating, thatf, (λ−L)⁻¹fα=E_α[f u] =λE_α[u²] +E_α[u(−Lu)]. Then, write

t 0

fds= t 0

λuds+M_λ(t )+u η(0)

−u η(t ) whereMλ(t )=u(η(t ))−u(η(0))−t

0Ludsis a martingale. Noteσ_t²(f )3σ_t²(λu)+3Eα[M_λ²(t )]+3Eα[(u(η(0)− u(η(t ))²]. Using stationarity, σ_t²(λu)λ²t²Eα[u²], the quadratic variation Eα[M_λ²(t )] = t Eα[u(−Lu)], and E_α[(u(η(0)−u(η(t ))²]4Eα[u²]. Hence, by collecting terms, and choosingλ=t⁻¹, we obtain the first estimate with sayC₁=15 (cf. Section 6, Appendix 1 [10]).

For the second bound, asu∈Dα, we can find a sequence of functions{ψ_n} ⊂Din the core such thatψ_n→u and Lψ_n→Lu inL²(Q_α). Then, f, ψ_nα → f, (λ−L)⁻¹fα andψ_n, (λ−S)ψ_nα = ψ_n, (λ−L)ψ_nα → f, (λ−L)⁻¹fα, and so, by substitution into (3.2),f, (λ−L)⁻¹fα f ²₋_1,λ.

The third bound is given in (3.5). 2

Proof of Theorem 2. The strategy to boundσ_t²(h)falls into two casesd=1, andd3 under (SP). We first comment on the cased=1, and then on thed3 case.

Cased=1. (1) We will find a sequence of functions (in Subsection 3.1.1){φ_λ: 0< λ <1} ⊂Dα such that sup

0<λ<1

h−Lφλ −1,λ<∞ (3.6)

and also sup

0<λ<1

φλ 2

1+λ φλ 2 0

<∞. (3.7)

(2) NoteM_t(f )=f (η(t ))−f (η(0))−t

0(Lf )(η(s))dsis a martingale forf ∈Dα with quadratic variation (by stationarity)E_α[(M_t(f ))²] =2t Eα[f (−Lf )] =2t f ²₁. Then, we can write

− t 0

Lφ_λ η(s)

ds=M_t(φ_λ)+φ_λ η(0)

−φ_λ η(t )

and so (by stationarity) σ_t²(Lφ_λ)6

t φ_λ ²₁+ φ_λ ²₀

=6t

φ_λ ²₁+1 t φ_λ ²₀

.

(3) Hence, by choosingλ=t⁻¹, we have from Proposition 3.1 that

(10)

σ_t²(h)2σ_t²(h−Lφ_t−1)+2σ_t²(Lφ_t−1) 2C1t h−Lφ_t₋1 2

−1,t⁻¹+12t

φ_t₋1 2 1+1

t φ_t₋1 2 0

. (3.8)

Then, by estimates in (1), σ_t²(h)Ct for some constant C=C(α, g, p) and t >1. For 0t1, bounds are immediate. This finishes the proof in this case.

Cased3 and(SP). By Proposition 3.1, we need only show h ₋₁<∞. One may be able to do this directly by “integration-by-parts” but as theQ_α marginal at the origin differs from the other marginals, one cannot apply immediately results in the literature. So, we “modify” the functionhand then apply these results.

Letj0=0 be a point in the support ofpwherep(j0)=0. Consider the functionφ(η)=(ηj₀−ρ)/(ρp(j0))∈D.

In Subsection 3.1.2, we show that h−Lφ ₋1<∞. Clearly φ 1<∞ and φ 0<∞. Then, by following the sequence (3.8) (noting · −1,λ · −1), we have

σ_t²(h)2C1t h−Lφ ₋₁+12t

φ ²₁+1 t φ ²₀

< Ct

for a constantC=C(α, g, p, d)andt >1. Bounds when 0t1 are clear. This finishes the proof. 2 3.1. Some estimates

We now turn to supplying the needed estimates in the two cases. We first make a calculation valid in any dimension d1. Let{ai: i∈Z^d}be numbers with

|ai|<∞, and define φ(η)=

i∈Z^d

a_i(η_i−ρ).

Note thatφ absolutely convergesQ_α a.s. and, as

a_i²<∞, that alsoφ is theL²(Q_α)limitφ=limn→∞φ⁽ⁿ⁾ of functionsφ⁽ⁿ⁾=

|i|maxna_i(η_i −ρ)∈Dwhere|i|max=max{|i₁|, . . . ,|i_d|}. Consider now,n > Rlarger than the range ofp, andi, jwhere|i|max,|i+j|maxn, that

φ⁽ⁿ⁾ η^i,i⁺^j

−φ⁽ⁿ⁾(η)=ai+j−ai

and, asτ_j(η^0,j)=τ_j(η+δ_j−δ₀)=τ_jη+δ₀−δ₋_j, thatφ⁽ⁿ⁾(τ_j(η^0,j))=

|i|maxna_i(η_i₊_j−ρ)+a₀−a₋_j and φ⁽ⁿ⁾

τ_j η^0,j

−φ⁽ⁿ⁾(η)=

|i|maxn

|i+j|maxn

(a_i−a_i₊_j)(η_i₊_j−ρ)+(a₀−a₋_j)

+

|i|maxn

|i+j|max>n

a_i(η_i₊_j−ρ)−

|i|max>n

|i+j|maxn

a_i₊_j(η_i₊_j−ρ).

These computations allow us to write Lφ⁽ⁿ⁾

(η)=

j

i=0,|i|maxn

|i+j|maxn

(a_i₊_j−a_i)g(η_i)p(j )+

j

(a_j−a₀)g(η₀)η₀−1 η₀ p(j )

−

j

|i|maxn

|i+j|maxn

(ai+j−ai)(ηi+j−ρ)g(η₀)

η₀ p(j )+

j

(a0−a₋j)g(η₀) η₀ p(j )

−

j

|i|maxn

|i+j|max>n

a_i

g(η_i)−(η_i₊_j−ρ)g(η₀) η₀

p(j )

+

j

|i|max>n

|i+j|maxn

a_i₊_j

g(η_i)−(η_i₊_j−ρ)g(η0) η0

p(j )