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El e c t ro n ic

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b a b i_{l i t y}

Vol. 14 (2009), Paper no. 34, pages 960–977. Journal URL

http://www.math.washington.edu/~ejpecp/

Central Limit Theorem for a Class of Linear Systems

Yukio Nagahata

Department of Mathematics, Graduate School of Engineering Science

Osaka University, Toyonaka 560-8531, Japan.

email:nagahata@sigmath.es.osaka-u.ac.jp

http://www.sigmath.es.osaka-u.ac.jp/~nagahata/

Nobuo Yoshida

∗

Division of Mathematics Graduate School of Science

Kyoto University, Kyoto 606-8502, Japan. email:nobuo@math.kyoto-u.ac.jp

http://www.math.kyoto-u.ac.jp/~nobuo/

Abstract

We consider a class of interacting particle systems with values in[0,_∞)Zd

, of which the binary contact path process is an example. Ford≥3 and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.

Key words:central limit theorem, linear systems, binary contact path process, diffusive behavior, delocalization.

AMS 2000 Subject Classification:Primary 60K35; Secondary: 60F05, 60J25. Submitted to EJP on November 11, 2008, final version accepted April 3, 2009.

1

Introduction

We writeN_{={0, 1, 2, ...},}N∗_{={1, 2, ...}and}Z_{={±x ; x ∈}N_{}. Forx = (x1, ..,xd)∈}Rd_,|x|stands for the ℓ1-norm: _|x| = Pd_i=1|xi|. For η = (ηx)x∈Zd ∈R

Zd

, _|η_| = P_x∈Zd|η_x|. Let (Ω,F,P) be a probability space. We write P[X] =R X d P and P[X :A] =R

AX d P for a r.v.(random variable) X and an eventA.

1.1

The binary contact path process (BCPP)

We start with a motivating simple example. Letη_t= (η_t,x)_x∈Zd∈NZ d

,t _≥0 be binary contact path process (BCPP for short) with parameterλ >0. Roughly speaking, the BCPP is an extended version of the basic contact process, in which not only the presence/absence of the particles at each site, but also their number is considered. The BCPP was originally introduced by D. Griffeath[4]. Here, we explain the process following the formulation in the book of T. Liggett[5, Chapter IX]. Letτz,i, (z ∈Zd_{, i∈}N∗_{) be i.i.d. mean-one exponential random variables and T}z,i = τz,1_+...+τz,i_{. We} suppose that the process(η_t)starts from a deterministic configurationη₀= (η_0,x)_x∈Zd ∈NZ

d with

|η_0|<_∞. At timet=Tz,i,η_t−is replaced byη_t randomly as follows: for eache_∈Zd _with|e|=1,

η_t,x = ¨

η_t−,x+η_t−,z if x=z+e,

η_t−,x if otherwise with probability λ

2dλ+1,

(all the particles at sitezare duplicated and added to those on the sitez= x+e), and

η_t,x = ¨

0 if x =z,

η_t−,x if x 6=z with probability

1 2dλ+1

(all the particles at sitezdisappear). The replacement occurs independently for different(z,i)and independently from_{τz,i_}z,i. A motivation to study the BCPP comes from the fact that the projected

process _€

η_t,x∧1Š

x∈Zd, t≥0 is the basic contact process[4].

Let

κ₁= 2dλ−1

2dλ+1 and ηt= (exp(−κ1t)ηt,x)x∈Zd.

Then,(_|η_t|)_t≥0is a nonnegative martingale and therefore, the following limit exists almost surely:

|η_∞|def=lim t |ηt|. Moreover,P[_|η_∞|] =1 if

d_≥3 andλ > _2d(11

−2πd), (1.1) whereπ_d is the return probability for the simple random walk onZd _{[4, Theorem 1]. It is known} thatπ_d≤π₃=0.3405... ford_≥3[7, page 103].

We denote the density of the particles by:

ρ_t,x = ηt,x

|η_t|1{|ηt|>0}, t>0,x ∈Z

Interesting objects related to the density would be

ρ∗_t =max x∈Zd

ρ_t,x, and Rt=

X

x∈Zd

ρ2_t,x. (1.3)

ρ∗_t is the density at the most populated site, whileRt is the probability that a given pair of particles at timet are at the same site. We call_Rt thereplica overlap, in analogy with the spin glass theory. Clearly, (ρ∗_t)2 _{≤ Rt ≤} ρ∗_t. These quantities convey information on localization/delocalization of the particles. Roughly speaking, large values ofρ∗_t orRt indicate that the most of the particles are concentrated on small number of “favorite sites" (localization), whereas small values of them imply that the particles are spread out over a large number of sites (delocalization).

As a special case of Corollary 1.2.2 below, we have the following result, which shows the diffusive behavior and the delocalization of the BCPP under the condition (1.1):

Theorem 1.1.1. Suppose (1.1). Then, for any f ∈Cb(Rd),

lim t→∞

X

x∈Zd

f€x/ptŠρ_t,x = Z

Rd

f dν in P(_{· ||}η_∞|>0)-probability,

where C_b(Rd_{)stands for the set of bounded continuous functions on}Rd_{, andνis the Gaussian measure}

with _Z

Rd

x_idν(x) =0,

Z

Rd

x_ix_jdν(x) = λ

2dλ+1δi j, i,j=1, ..,d. Furthermore,

Rt =O(t−d/2) as tր ∞in P(· ||η_∞|>0)-probability.

1.2

The results

We generalize Theorem 1.1.1 to a certain class of linear interacting particle systems with values in [0,∞)Zd_[

5, Chapter IX]. Recall that the particles in BCPP either die, or make binary branching. To describe more general “branching mechanism", we introduce a random vectorK= (K_x)_x∈Zd which is bounded and of finite range in the sense that

0_≤K_{x ≤}b_K1_{|x|≤r

K} a.s. for some non-random bK,rK∈[0,∞). (1.4)

Letτz,i, (z_∈Zd_,i∈N∗_{) be i.i.d. mean-one exponential random variables andT}z,i_=τz,1_+...+τz,i_. Let alsoKz,i = (K_xz,i)_x∈Zd (z∈Z

d_{, i}

∈N∗_{) be i.i.d. random vectors with the same distributions as} K, independent of _{τz,i_}z∈Zd,i∈N∗. We suppose that the process(ηt)t≥0 starts from a deterministic

configurationη₀ = (η_0,x)_x∈Zd ∈[0,∞) Zd

with|η_0|< _∞. At time t = Tz,i, η_t− is replaced byη_t, where

η_t,x = ¨

K₀z,iη_t−,z if x=z,

η_t−,x+Kz_x,₋i_zη_t−,z if x₆=z. (1.5) The BCPP is a special case of this set-up, in which

K= ¨

0 with probability _2dλ1₊₁

€

δ_x,0+δ_x,eŠ

x∈Zd with probability λ

2dλ+1, for each 2d neighboreof 0.

A formal construction of the process(η_t)_t≥0can be given as a special case of[5, page 427, Theorem 1.14] via Hille-Yosida theory. In section 1.3, we will also give an alternative construction of the process in terms of a stochastic differential equation.

We set

κ_p = X x∈Zd

P[(Kx −δx,0)p], p=1, 2, (1.7)

η_t = (exp(₋κ₁t)η_t,x)_x∈Zd. (1.8)

Then,

(_|η_t|)_t≥0 is a nonnegative martingale. (1.9) The above martingale property can be seen by the same argument as in[5, page 433, Theorem 2.2 (b)]. For the reader’s convenience, we will also present a simpler proof in section 1.3 below. By (1.9), following limit exists almost surely:

|η_∞|def=lim

t |ηt|. (1.10)

To state Theorem 1.2.1, we define

G(x) = Z ∞

0

P_S0(St=x)d t, (1.11)

where ((S_t)_t≥0,P_Sx) is the continuous-time random walk on Zd _{starting from x ∈} Zd_{, with the} generator

LSf(x) = 1₂

X

y∈Zd

€

P[Kx−y] +P[Ky−x]

Š

f(y)₋f(x). (1.12)

As before,C_b(Rd_{)stands for the set of bounded continuous functions on}Rd_.

Theorem 1.2.1. Suppose (1.4) and that

the set_{x _∈Zd_{; P[Kx]6=0}contains a linear basis of}Rd_{, (1.13)}

X

y∈Zd

P[(K_y−δ_y,0)(K_x+y−δx+y,0)] =0 for all x∈Zd\{0}. (1.14)

Then, referring to (1.7)–(1.12), the following are equivalent:

(a) κ2

2 G(0)<1,

(b) sup t≥0

P[_|η_t|2]<_∞,

(c) lim t→∞

X

x∈Zd

f€(x₋mt)/ptŠη_t,x =_|η_∞|

Z

Rd

f dν inL2(P)for all f _∈C_b(Rd),

where m=P_x∈Zdx P[K_x]∈Rd andν is the Gaussian measure with

Z

Rd

x_idν(x) =0,

Z

Rd

x_ix_jdν(x) = X x∈Zd

Moreover, if κ2

2 G(0)<1, then, there exists C ∈(0,∞)such that

X

x,ex∈Zd

f(x_−ex)P[η_t,xη_t,ex]_≤C t−d/2_|η_0|2 X

x∈Zd

f(x) (1.16)

for all t>0and f :Zd _{→[0,∞)with}P

x∈Zd f(x)<∞.

The main point of Theorem 1.2.1 is that the condition (a), or equivalently (b), implies the central limit theorem (c) (See also Corollary 1.2.2 below). This seems to be the first result in which the central limit theorem for the spatial distribution of the particle is shown in the context of linear systems. Some other part of our results ((a) ⇒ (b), and Theorem 1.2.3 below) generalizes [4, Theorem 1]. However, this is merely a by-product and not a central issue in the present paper.

The proof of Theorem 1.2.1, which will be presented in section 3.1, is roughly divided into two steps:

(i) to represent the two-point function P[η_t,xη_t,ex]in terms of a continuous-time Markov chain on Zd_×Zd _{via the Feynman-Kac formula (Lemma 2.1.1 and Lemma 2.1.4 below),}

(ii) to show the central limit theorem for the “weighted" Markov chain, where the weight comes from the additive functional due to the Feynman-Kac formula (Lemma 2.2.2 below).

The above strategy was adopted earlier by one of the authors for branching random walk in random environment[10]. There, the Markov chain alluded to above is simply the product of simple random walks onZd_{, so that the central limit theorem with the Feynman-Kac weight is relatively easy. Since} the Markov chain in the present paper is no longer a random walk, it requires more work. However, the good news here is that the Markov chain we have to work on is “close" to a random walk. In fact, we get the central limit theorem by perturbation from that for a random walk case.

Some other remarks on Theorem 1.2.1 are in order:

1) The condition (1.13) guarantees a reasonable non-degeneracy for the transition mechanism (1.5). On the other hand, (1.14) follows from a stronger condition:

P[(K_x−δ_x,0)(K_y−δ_y,0)] =0 forx,y _∈Zd _{withx 6= y, (1.17)}

which amounts to saying that the transition mechanism (1.5) updates the configuration by “at most one coordinate at a time". A typical examples of suchK’s are given by ones which satisfy:

P(K=0) + X a∈Zd_\{0}

P€K= (δ_x,0+K_aδ_x,a)_x∈Zd

Š =1.

These include not only BCPP but also models with asymmetry and/or long (but finite) range.

Here is an explanation for how we use the condition (1.14). To prove Theorem 1.2.1, we use a certain Markov chain onZd_×Zd_{, which is introduced in Lemma 2.1.1 below. Thanks to (1.14), the} Markov chain is stationary with respect to the counting measure onZd_×Zd_{. The stationarity plays} an important role in the proof of Theorem 1.2.1– see Lemma 2.1.4 below.

Therefore, κ2

2 G(0)< 1 is possible only if d ≥3. As will be explained in the proof,

κ2

2 G(0) <1 is

equivalent to

P_S0

–

exp

‚

κ₂

2

Z ∞

0

δ₀(St)d t

Ϊ

<_∞.

3)If, in particular,

P[K_x] = ¨

c>0 for|x|=1,

0 for_|x_{| ≥}2, (1.18)

then, (S_t)_t≥0 law= (bS_{2d c t})_t≥0, where (Sb_·) is the simple random walk. Therefore, the condition (a) becomes

κ₂

4d c(1−π_d)<1. (1.19)

By (1.6), the BCPP satisfies (1.13)–(1.14). Furthermore,κ₂=1 and we have (1.18) withc= λ

2dλ+1.

Therefore, (1.19) is equivalent to (1.1).

4)The dual process of (η_t) above (in the sense of[5, page 432]) is given by replacing the linear transform in (1.5) by its transpose:

η_t,x = ¨ P

y∈ZdKz

,i

y−xηt−,y if x =z,

η_t−,x if x ₆=z. (1.20)

As can be seen from the proofs, all the results in this paper remain true for the dual process. 5)The central limit theorem for discrete time linear systems is discussed in[6].

We define the density and the replica overlap in the same way as (1.2)–(1.3). Then, as an immediate consequence of Theorem 1.2.1, we have the following

Corollary 1.2.2. Suppose (1.4), (1.13)–(1.14) and that κ2

2G(0)<1. Then, P[|η∞|] =1and for all

f _∈C_b(Rd_),

lim t→∞

X

x∈Zd

f €(x−mt)/ptŠρ_t,x = Z

Rd

f dν in P(_{· ||}η_∞|>0)-probability,

where m=P_x∈Zdx P[Kx]∈Rd andν is the same Gaussian measure defined by (1.15). Furthermore,

Rt =O(t−d/2) as tր ∞in P(· ||η∞|>0)-probability.

Proof: The first statement is immediate from Theorem 1.2.1(c). Taking f(x) = δ_x,0in (1.16), we see that

P[X x∈Zd

η2_t,x]_≤C t−d/2|η_0|2 fort>0.

This implies the second statement. ƒ

Fora∈Zd_{, let}ηa

Theorem 1.2.3. Suppose (1.4), (1.13)–(1.14) and that κ2

2G(0)<1. Then,

P[_|ηa_∞||η_∞b _|] =1+κ2G(a−b)

2₋κ₂G(0), a,b∈Z d_.

The proof of Theorem 1.2.3 will be presented in section 3.2. We refer the reader to[11]for similar formulae for discrete time models.

1.3

SDE description of the process

We now give an alternative description of the process in terms of a stochastic differential equation (SDE), which will be used in the proof of Lemma 2.1.1 below. We introduce random measures on [0,_∞)_×[0,_∞)Zd

by

Nz(dsdξ) =X i≥1

1_{(Tz,i,Kz,i)_∈dsdξ_}, N_tz(dsdξ) =1_{s≤t}Nz(dsdξ). (1.21)

Then,Nz,z∈Zd _{are independent Poisson random measures on}[0,∞)_×[0,∞)Zd

with the intensity

ds×P(K∈dξ).

The precise definition of the process (η_t)_t≥0 is then given by the following stochastic differential equation:

η_t,x =η_0,x+ X z∈Zd

Z

N_tz(dsdξ)€ξ_x−z−δ_x,zŠη_s−,z. (1.22)

By (1.4), it is standard to see that (1.22) defines a unique processη_t= (η_t,x), (t≥0) and that(η_t) is Markovian.

Proof of (1.9): Since _|η_t| is obviously nonnegative, we will prove the martingale property. By (1.22), we have

|η_t|=_|η_0|+ X z∈Zd

Z

N_tz(dsdξ) (_|ξ_{| −}1)η_s−,z,

and hence

(1) _|η_t|=_|η_{0| −}κ₁

Z t

0

|η_s|ds+ X z∈Zd

Z

N_tz(dsdξ) (_|ξ_{| −}1)η_s−,z.

We have on the other hand that

κ₁

Z t

0

|η_s|ds= X z∈Zd

Z t

0

ds

Z

P(K∈ξ)(_|ξ_{| −}1)η_s,z.

2

Lemmas

2.1

Markov chain representations for the point functions

We assume (1.4) throughout, but not (1.13)–(1.14) for the moment. To prove the Feynman-Kac formula for two-point function, we introduce some notation.

Forx,y,_ex,_ey_∈Zd_,

Γ_x,ex,y,ey def= P[(Kx−y−δx,y)δex,ey+ (Kex−ey−δex,ey)δx,y]

+P[(Kx−y−δx,y)(Kex−y−δex,y)]δy,ey, (2.1)

V(x) def= X y,ey∈Zd

Γ_x,0,y,ey =2κ₁+ X y∈Zd

P[(Ky−δy,0)(Kx+y−δx+y,0)]. (2.2)

Note that

V(x_−ex) = X y,ey∈Zd

Γ_x,ex,y,ey. (2.3)

Remark: The matrix Γintroded above appears also in [5, page 442, Theorem 3.1], since it is a fundamental tool to deal with the two-point function of the linear system. However, the way we use the matrix will be different from the ones in the existing literature.

We now prove the Feynman-Kac formula for two-point function, which is the basis of the proof of Theorem 1.2.1:

Lemma 2.1.1. Let (X,Xe) = ((X_t,Xe_t)_t≥0,Px,ex

X,Xe) be the continuous-time Markov chain on Z d

×Zd starting from(x,ex), with the generator

L_X,Xef(x,ex) = X y,ey∈Zd

Γ_x,ex,y,ey f(y,ey)₋f(x,ex),

whereΓ_x,ex,y,ey is defined by (2.1). Then, for(t,x,_ex)_∈[0,_∞)_×Zd_×Zd_,

P[η_t,xη_t,ex] =Px,ex X,Xe

–

exp

‚Z t

0

V(X_s−Xe_s)ds

Œ

η_0,X

tη0,Xet

™

, (2.4)

where V is defined by (2.2).

Proof: We first show thatu(t,x,ex)def= P[η_t,xη_t,ex]solves the integral equation

(1) u(t,x,ex)₋u(0,x,ex) = Z t

0

(L_X,Xe+V(x₋xe))u(s,x,ex)ds.

By (1.22), we have

η_t,xη_t,ex−η_0,xη_0,ex = X y∈Zd

Z

where

F_x,ex,y(s,ξ,η)

= (ξ_x−y−δ_x,y)η_s,exη_s,y+ (ξ_ex−y−δ_ex,y)η_s,xη_s,y+ (ξ_x−y−δ_x,y)(ξ_ex−y−δ_ex,y)η2_s,y

Therefore,

u(t,x,ex)₋u(0,x,ex) = X y∈Zd

Z t

0

ds

Z

P[F_x,ex,y(s,ξ,η)]P(K_∈ξ)

= Z t

0

X

y,ey∈Zd

Γ_x,ex,y,eyu(s,y,ey)ds

(2.3) =

Z t

0

 

 X

y,ey∈Zd

Γ_x,ex,y,ey(u(s,y,_ey)₋u(s,x,_ex)) +V(x₋x_e)u(s,x,_ex)   ds

= Z t

0

(L_X,Xe+V(x−xe))u(s,x,_ex)ds.

We next show that

(2) sup

t∈[0,T]

sup x,ex∈Zd

|u(t,x,_ex)_|<_∞for anyT ∈(0,_∞).

We have by (1.4) and (1.22) that, for anyp∈N∗_{, there exists C1∈}(0,∞)such that

P[ηp_t,x]_≤C1

X

y:|x−y|≤rK

Z t

0

P[η_sp_,y]ds, t≥0.

By iteration, we see that there existsC2∈(0,∞)such that

P[ηp_t,x]_≤eC2t

X

y∈Zd

e−|x−y|(1+η_0,p_y), t≥0,

which, via Schwarz inequality, implies (4).

The solution to (1) subject to (2) is unique, for each givenη₀. This can be seen by using Gronwall’s inequality with respect to the normkuk= P_x,ex∈Zde−|x||u(x,ex)|. Moreover, the RHS of (2.4) is a solution to (1) subject to the bound (2). This can be seen by adapting the argument in[8, page 5,Theorem 1.1]. Therefore, we get (2.4). ƒ

Remark: The following Feynman-Kac formula for one-point function can be obtained in the same way as Lemma 2.1.1:

P[η_t,x] =eκ1t_Px

X[η0,Xt], (t,x)∈[0,∞)×Z

d_{, (2.5)}

whereκ₁ is defined by (1.7) and((Xt)t≥0,PXx) is the continuous-time random walk onZ

d _starting

fromx, with the generator

L_Xf(x) = X y∈Zd

Lemma 2.1.2. We have _X

y,ey∈Zd

Γ_x,ex,y,ey = X

y,ey∈Zd

Γ_y,ey,x,ex, (2.6)

if and only if (1.14) holds. In addition, (1.14) implies that

V(x) =2κ₁+κ₂δ_x,0. (2.7)

Proof: We letc(x) =P_y∈ZdP[(K_y−δ_y,0)(K_x+y−δ_x+y,0)]. Then,c(0) =κ₂and,

X

y,ey∈Zd

Γ_x,ex,y,ey = 2κ₁+c(x_−ex), cf. (2.2)–(2.3),

X

y,ey∈Zd

Γ_y,ey,x,ex = 2κ₁+δ_x,ex X

y∈Zd

c(y).

These imply the desired equivalence and (2.7). ƒ

We assume (1.14) from here on. Then, by (2.6), (X,Xe) is stationary with respect to the counting measure onZd_×Zd. We denote the dual process of(X,Xe)by(Y,Ye) = ((Y_t,Ye_t)_t≥0,Px,ex

Y,Ye), that is, the

continuous time Markov chain onZd_×Zd _{starting from(x,ex), with the generator}

L_Y,Yef(x,ex) = X y,ey∈Zd

Γ_y,ey,x,ex f(y,ey)₋f(x,ex). (2.8)

Thanks to (2.6), L_X,Xe andL_Y,Ye are dual operators onℓ2(Zd_×Zd_).

Remark: If we additionally suppose thatP[Kxp] =P[K₋px]forp=1, 2 and x ∈Zd, then,Γx,ex,y,ey = Γ_y,ey,x,xefor all x,ex,y,ey ∈Zd_{. Thus,}(X,Xe)and(Y,Ye)are the same in this case.

The relative motionY_t−Ye_t of the components of(Y,Ye)is nicely identified by:

Lemma 2.1.3. ((Y_t−Ye_t)_t≥0,Px,ex

Y,Ye)and((S2t)t≥0,P

x−ex

S )(cf. (1.12)) have the same law.

Proof: Since (Y,Ye) is shift invariant, in the sense that Γ_{x+v,ex+v,y+v,ey+v} = Γ_x,ex,y,ey for all v ∈Zd,

((Y_t−Ye_t)_t≥0,Px,ex

Y,Ye)is a Markov chain. Moreover, its jump rate is computed as follows. Forx 6= y,

X

z∈Zd

Γ_y+z,z,x,0 = P[K_x−y] +P[K_y−x] +δ_x,0X

z∈Zd

P[(K_y+z−δ_y,z)(K_z−δ_0,z)]

(1.14)

= P[K_x−y] +P[K_y−x].

ƒ

Lemma 2.1.4. For a bounded g:Zd_×Zd_→R_,

Proof: It follows from Lemma 2.1.1 and (2.7) that

(1) LHS of (2.9)= X

We now observe that the operators

f(x,_ex) _7→ Px,ex

are dual to each other with respect to the counting measure onZd_×Zd_{. Therefore,}

RHS of (1)=RHS of (2.9).

Remark: In the case of BCPP, D. Griffeath obtained a Feynman-Kac formula for

X

y∈Zd

P[η_t,xη_t,ex+y]

2.2

Central limit theorems for Markov chains

We prepare central limit theorems for Markov chains, which is obtained by perturbation of random walks.

Lemma 2.2.1. Let((Zt)t≥0,Px) be a continuous-time random walk onZd starting from x, with the

generator

L_Zf(x) = X y∈Zd

a_y−x(f(y)₋f(x)),

where we assume that _X

x∈Zd

|x_|2a_x <_∞.

Then, for any B_∈σ[Z_u; u_∈[0,_∞)], x_∈Zd_{, and f ∈Cb(}Rd_),

lim t→∞P

x_[f((Z

t−mt)/

p

t):B] =Px(B) Z

Rd

f dν,

where m=P_x∈Zdx a_x andν is the Gaussian measure with

Z

Rd

x_idν(x) =0,

Z

Rd

x_ix_jdν(x) = X x∈Zd

x_ix_ja_x, i,j=1, ..,d. (2.11)

Proof: By subtracting a constant, we may assume thatR_Rd f dν =0. We first consider the case that B_{∈ Fs}def= σ[Z_u; u_∈[0,s]]for somes_∈(0,_∞). It is easy to see from the central limit theorem for (Zt)that for any x ∈Zd,

lim t→∞P

x_[f((Z

t−s−mt)/

p

t)] =0.

With this and the bounded convergence theorem, we have

Px[f((Z_t−mt)/pt):B] =Px[PZs_[f((Z

t−s−mt)/

p

t)]:B]_−→0 as t_{ր ∞}.

Next, we takeB _∈σ[Z_u ; u_∈[0,_∞)]. For anyǫ >0, there exists_∈(0,_∞)andBe_{∈ Fs} such that Px[_|1_B−1_eB|]< ǫ. Then, by what we already have seen,

lim t→∞P

x_[f((Z

t−mt)/

p

t):B]_≤ lim t→∞P

x_[f((Z

t−mt)/

p

t):Be] +_kf_kǫ=_kf_kǫ,

where_kfkis the sup norm of f. Similarly,

lim t→∞

Px[f((Z_t−mt)/pt):B]_{≥ −k}f_kǫ.

Sinceǫ >0 is arbitrary, we are done. ƒ

Lemma 2.2.2. Let Z= ((Z_t)_t≥0,Px)be as in Lemma 2.2.1 and and D_⊂Zd _{be transient for Z. On the} other hand, letZe= ((Zet)t≥0,ePx)be the continuous-time Markov chain onZd starting from x, with the

generator

L_Zef(x) = X y∈Zd

e

where we assume that_ea_x,y =a_y−x if x _6∈D_{∪ {}y_}and that D is also transient forZ. Furthermore, wee

whereν is the Gaussian measure such that (2.11) holds.

Thus, letting t_{→ ∞}first, and thens_{→ ∞}, in (2.12), we get the lemma. ƒ

2.3

A Nash type upper bound for the Schrödinger semi-group

We will use the following lemma to prove (1.16). The lemma can be generalized to symmetric Markov chains on more general graphs. However, we restrict ourselves to random walks on Zd, since it is enough for our purpose.

Lemma 2.3.1. Let((Zt)t≥0,Px)be continuous-time random walk onZd with the generator:

(1) X

3

Proof of Theorem 1.2.1 and Theorem 1.2.3

3.1

Proof of Theorem 1.2.1

(a)⇔(b): Define

and hence thath(x) =1+κ2

2h(0)G(x). Thus,h(0)<∞implies (a) and that

h(x) =1+ κ2G(x)

2₋κ₂G(0). (3.1)

(a),(b)_⇒(c): Since (b) implies that lim_t→∞|η_t|=_|η_∞|inL2_{(P), it is enough to prove that}

U_tdef.= X x∈Zd

η_t,xf€(x₋mt)/ptŠ_−→0 inL2_{(P)astր ∞}

for f _∈C_b(Rd_{)such that}R

Rdf dν=0. We set ft(x,ex) = f((x−m)/

p

t)f((_ex−m)/pt). Then, by Lemma 2.1.4,

P[U_t2] = X x,ex∈Zd

P[η_t,xη_t,xe]f_t(x,_ex) = X x,ex∈Zd

η_0,xη_0,exPx,ex Y,Ye

”

e_tf_t(Y_t,Ye_t)—,

wheree_t =expκ₂Rt

0δ0(Ys−Yes)ds

. Note that by Lemma 2.1.3 and (a),

(1) Px,ex Y,Ye

e_∞=h(x−ex)_≤h(0)<_∞.

Since_|η_0|<_∞, it is enough to prove that for eachx,_ex_∈Zd

lim t→∞P

x,ex Y,Ye

”

e_tf_t(Y_t,Ye_t)—=0.

To prove this, we apply Lemma 2.2.2 to the Markov chainZe_tdef.= (Y_t,Ye_t)and the random walk(Z_t) onZd_×Zd _{with the generator}

L_Zf(x,ex) = X y,ey∈Zd

a_x,ex,y,ye f(y,ey)₋f(x,ex) with a_x,ex,y,ey=   

P[K_ey−ex] if x= y and_ex₆= _ey, P[Ky−x] if x6= y andex= ey, 0 if otherwise.

LetD=_{(x,ex)_∈Zd_×Zd_{; x= ex}. Then,}

(2) a_x,ex,y,ey = Γ_y,ey,x,ex if(x,_ex)_6∈D_{∪ {}(y,_ey)_},

since

Γ_y,ey,x,ex

= P[(K_y−x−δ_y,x)δ_ey,ex+ (K_{ey−ex −}δ_ey,ex)δ_y,x+ (K_y−x−δ_y,x)(K_ye−x−δ_ey,x)δ_x,ex].

Moreover, by (1.13),

Finally, the Gaussian measureν_⊗ν is the limit law in the central limit theorem for the random walk

We apply Lemma 2.3.1 to the right-hand-side to get (1.16). ƒ

3.2

Proof of Theorem 1.2.3

By the shift-invariance, we may assume that b=0. We have by Lemma 2.1.4 that

P[ηa_t,xη0_t,ex] =Pa,0

and hence by Lemma 2.1.3 that

P[_|ηa_t||η0_t|] =Pa,0

Acknowledgements:The authors thank Shinzo Watanabe for the improvement of Lemma 2.2.1.

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