Regularity of the diffusion coefficient matrix for the lattice gas with energy

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www.elsevier.com/locate/anihpb

Regularity of the diffusion coefficient matrix for the lattice gas with energy

Yukio Nagahata

Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531, Japan Received 5 April 2003; received in revised form 4 November 2003; accepted 30 March 2004

Available online 11 September 2004

Abstract

In this paper we obtain the smoothness of the diffusion coefficient matrix for the lattice gas with energy. Furthermore we also obtain the smoothness of the central limit theorem variances for certain functions.

Résumé

Dans cet article, on montre la régularité de la matrice des coefficients de diffusion pour le gaz sur réseau avec énergie. On obtient également la régularité des variances associées à certaines fonctions par le théorème limite central.

1. Introduction

In our previous paper [6] we have introduced a lattice gas with energy and derived the fluctuation dissipation equation for it. In this paper we prove that the diffusion coefficient matrix appearing in the equation is smooth.

In the derivation of hydrodynamic limit, uniqueness of the Cauchy problem of the weak solution of limiting diffusion equation is needed. It seems unsolved in the existing literatures. But once smoothness and uniform ellip- ticity of the diffusion coefficient matrix is established and if there exists a Lipschitz continuous solution, then the uniqueness question is resolved.

The smoothness of the self-diffusion coefficient of the symmetric simple exclusion process is proved by Landim, Olla, and Varadhan [4], and the smoothness of the diffusion coefficient for a lattice gas reversible under the Bernoulli measures is proved by Bernardin [1].

It seems difficult to adapt to our model the method which introduced in [4] and developed in [1], since we do not have any suitable orthonormal basis (with respect to invariant measure) of functions on the configuration space.

E-mail address: nagahata@sigmath.es.osaka-u.ac.jp (Y. Nagahata).

doi:10.1016/j.anihpb.2004.03.006

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In this paper, we choose a basis of the space of continuous functions on the configuration space which is not orthonormal (with respect to invariant measure). We also introduce a Markov process whose state space is a set of indexes of the basis and which may be regarded as a dual process. By using this process, the diffusion coefficient matrix is given by a finite linear combination of smooth functions whose coefficient is given by expectation of the total occupation time for a certain infinite set. We prove that the expectation of the occupation time for the infinite set converges ifd3, and diverges ifd=1,2. But if we examine the linear combination more carefully, then we find that it is a difference of expectations of the total occupations times of a certain infinite set for the processes starting at two different points. Such difference makes sense as in the same way that makes the potential function of one or two dimensional random walk well defined.

This paper is organized as follows: In Section 2 we state the model and results. In Section 3 we introduce a basis of continuous functions on the configuration space and compute the coefficient ofLf with respect to the basis.

We also introduce a Markov process which may be regarded as a dual process of our lattice gas, and solve the resolvent equation by using the process. In Section 4 we estimate the expectation of occupation time of certain set for the process in dimensionsd3. In Section 5, we estimate an expectation of occupation time for a finite state continuous time stochastic jump process. In Section 6, we estimate an expectation of occupation time for certain set for the process. In Section 7 we prove the main results.

2. Model and results

Our lattice gas is a Markov process on the state spaceX:= {0,1, . . . , k}^Z^d. Let η=(ηx)_x_∈_Zd stands for a generic element ofX, so that for eachx,ηx∈ {0,1,2, . . . , k}. For any local functionf we define operatorsπ^(x,y) andπ^x^→^yby

π^(x,y)f (η):=1_{η_x=0,η_y=0}

f (η^(x,y))−f (η) , π^x^→^yf (η):=1_{η_x2,1η_yk−1}

f (η^x^→^y)−f (η) , whereη^(x,y)andη^x^→^yare the configurations defined by

(η^(x,y))_z:=





η_y, ifz=x, ηx, ifz=y, η_z, otherwise, (η^x^→^y)_z:=





ηx−1, ifz=x, η_y+1, ifz=y, ηz, otherwise.

For any local functionf we define the generator

Lf (η):=

x,y∈Z^d:|x−y|=1

c_ex(η_x)π^(x,y)f (η)+c_ge(η_x)π^x^→^yf (η) ,

where| · |is the Euclidean norm of Z^d andcex(r), cge(r)are positive functions ofr=0,1,2, . . . , k. Furthermore we suppose thatcex(0)=cge(0)=cge(1)=0 andcex(r) >0 for 1rkandcge(r) >0 for 2rk.

The process is thought to model a time evolution of the dynamics of a gas of particles having energy. The values ofηxare interpreted in such a way that ifηx=0, then the sitexis vacant, and ifηx=0, then there exists a particle having (discrete) energyηxat sitex. A particle at sitex moves to a nearest neighbor sitey at ratecex(ηx)ify is vacant. One unit of energy of the particle at sitexis transferred to the particle at a neighboring siteyat ratecge(ηx) if the energy of the particle at the siteyis less thank.

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Consider the family of product measures on the product space{0,1,2, . . . , k}^Z^d with the marginal distribution

ν_p,α

{η: η_x=l} :=









1−p ifl=0,

p_Z¹

α ifl=1,

p_Z¹

α

α^l−1

c_ge(2)c_ge(3)···c_ge(l) if 2lk

for allx∈Z^d, where 0p1, 0α <∞andZα is the normalization constant. Put ρ:=ρ(p, α):=E^ν^p,α[η₀],

thenρis a rational function of two variablesp, α, and for eachp,ρis a strictly increasing function ofα. Therefore there exists an inverse functionα˜= ˜α(p, ρ)say. We definePp,ρ by

Pp,ρ(·):=ν_p,_α(p,ρ)_˜ (·),

for 0p1 andpρ < kp. For any local functionf,E_p,ρ[f]is a smooth function ofpandρ. We can easily check thatLis symmetric with respect toP_p,ρ. One can show that there exists a unique closed extension ofLin the space of continuous functionsC(X)with supremum norm (see [5]). We denote bye^{t L}the semigroup generated by the closed extension.

We define the shift operatorτ_xforx∈Z^d, which acts on allA⊂Z^d, and local functionsf as well as configu- rationsηas follows:

τxA:=x+A, τxf (η):=f (τxη), (τxη)z:=ηz−x.

LetD(p, ρ)˜ =(D˜_i,j(p, ρ))_i,j_∈{_1,2_}be a 2×2 symmetric matrix defined via the variational formula a· ˜D(p, ρ)a

=

i,j

a_iD˜_i,j(p, ρ)a_j

:=inf

u Ep,ρ

cex(η0)

π^(0,e)(a1η0+a21_{η₀=0})+

x

π^(0,e)τxu 2

+cge(η0)

π⁰^→^e(a1η0+a21_{η₀=0})+

x

π⁰^→^eτxu 2

, (1)

whereais any 2-dimensional vector,eis a unit vector and inf_uis taken over all local functions. Put χ (p, ρ):=

Ep,ρ[η₀²] −ρ² (1−p)ρ (1−p)ρ p(1−p)

,

D(p, ρ)= ˜D(p, ρ)χ⁻¹(p, ρ), (2)

for 0< p <1 andp < ρ < kp. (χ⁻¹(p, ρ)denotes the inverse matrix ofχ (p, ρ).) Using these notations we can state our main result.

Theorem 2.1. Let(p, ρ)such that 0< p <1 andp < ρ < kp. Then the diffusion coefficient matrix for this model defined by (2) is a smooth function ofpandρ, and has the smooth extension up to the boundary.

The smoothness of the diffusion coefficient matrix is implied by that of the central limit theorem variance for certain functions.

We define the currentsw_e^E, w^P_e by

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w^E_e(η):=c_ex(η₀)η₀1_{η₀=0,η1=0}(η)−c_ex(η₁)η₁1_{η₁=0,η0=0}(η)

+c_ge(η₀)1_{_η₀_2,1_η₁_k₋₁_}(η)−c_ge(η₁)1_{_η₁_2,1_η₀_k₋₁_}(η), w^P_e(η)=c_ex(η₀)1_{_η₀₌_0,η₁₌₀_}(η)−c_ex(η₁)1_{_η₁₌_0,η₀₌₀_}(η).

Denoted byWthe linear space spanned by currentsw^E_e, w^P_e for all positive unit vectorse.

Theorem 2.2. Supposef ∈W. Then_∞

0

x∈Z^dEp,ρ[fe^Ltτxf]dtis a smooth function ofpandρ.

Remark. If the dimensionsdis greater than or equal to 3, the present method is applicable to generalized exclusion process which is introduced by Kipnis, Landim and Olla [2], namely Theorems 2.1 and 2.2 are valid for it in dimensionsd3.

3. Basis ofC(X)and dual process

In this section we will introduce a basis of the space of continuous functions on the configuration space and compute the coefficient ofLf with respect to the basis for a local functionf.

LetC(X)denote the space of continuous functions on X with supremum norm. It is convenient to define a subspaceC₀(X)ofC(X)as follows: LetC₀(X)be the set of functions such that there exist positive constantsc₁ andc₂with

sup

η,η:ηx=η_xfor|x|∞l

f (η)−f (η)c1e⁻^c²^l for alll0, where| · |_∞is the supremum norm on Z^d.

We defineAby A:=

A=(A1, . . . , Ak): AiZ^d,withAi∩Aj= ∅ifi=j ,

whereAZ^d meansA⊂Z^dand|A|is finite. ForA, B∈A,B⊂AmeansBi ⊂Ai for alli,A\Bmeans that theith component isAi\Bi, andτxAmeans thatith component isτxAi (i=1,2, . . . , k). We define two types of cardinality by

#A:=

i

|A_i|,

#A˜ :=

i

i|Ai|,

and a family of functions{ΨA}A∈Aby Ψ_A(η):=

k i=1

x∈A_i

1_{_η_x₌_i_}(η).

ForA∈Awe define the special configurationη^Aby (η^A)z:=

i ifz∈A_ifor 1ik, 0 ifz /∈

iAi.

The cardinality #Aand#A˜ equals the number of particles and total energy for the special configurationη^A, respectively. The functionΨAis the indicator function of the set of configurationsηfor which each site of

iAi is occupied by a particle and each particle onAihas the common energyi.

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Lemma 3.1. The family of functions{ΨA}A∈Ais a basis of the linear spaceC(X). Furthermore, forf ∈C(X)if we definefˆ: A→R by

f (A)ˆ :=

B⊂A

(−1)^#(A^\^B)f (η^B), then

f (η)=

A∈A

f (A)Ψˆ A(η). (3)

Proof. It is not difficult to see that{Ψ_A}A∈Aare linearly independent. Therefore we have only to prove (3).

FixA∈Aandx∈Z^dsuch thatx /∈

iA_i. For 1ikandB⊂A, defineB^x,i∈Aby (B^x,i)j:=

Bi∪ {x} ifj =i, Bj ifj =i.

Then forf ∈C(X), f (Aˆ ^x,i)=

B⊂A

(−1)^#(A^\^B)⁺¹f (η^B)+

B⊂A

(−1)^#(A^\^B)f (η^B^x,i).

Letf be a local function. Then there existsΛ=Λ(f )Z^dsuch thatf depends only on{ηx:x∈Λ}, and

f (η^A)=f (η^A^∩^Λ) (4)

is valid for allA∈A, whereA∩Λ∈Ais defined by(A∩Λ)_i:=A_i∩Λ. These two equality shows that

f (A)ˆ =0 (5)

if

iA_i∩Λ^c= ∅. Therefore right-hand side of (3) is a finite sum. By (4), we have only to check the truth of f (η^B)=

A∈A

f (A)Ψˆ _A(η^B)

for allB∈Asuch that

B_i⊂Λ. By the binomial expansion

A⊂B

(−1)^#(B^\^A)=(1−1)^#B forB∈A, it holds that

A∈A

f (A)Ψˆ _A(η^B)=

A⊂B

C⊂A

(−1)^#(A^\^C)f (η^C)=f (η^B).

Therefore the equality of (3) is valid for every local function.

For a continuous functionf we can approximate it uniformly by a local functionf_nwhich is defined by f_n(η):=f (η_Λ_n),

where

(ηΛn)x:=

ηx ifx∈Λn, 0 ifx /∈Λ_n,

andΛnis a cube centered at origin and of side 2n+1. Sincefnis a local function, we can write fn(η)=

A∈A

fˆn(A)ΨA(η).

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It is not difficult to check that if

iAi⊂Λn thenfˆn(A)= ˆf (A). Therefore (3) holds for every continuous func- tion. 2

From now on, we regard the sequence{ ˆf (A)}Aas the coefficient of the continuous function off in the expansion (3).

We expand currentsw^E_e, w^P_e in the expansion (3). Then we have w^E_e =

k i=1

cex(i)i

Ψ_∅0,i−

k j=1

Ψ_∅0,i;e,j

− k i=1

cex(i)i

Ψ_∅e,i−

k j=1

Ψ_∅0,j;e,i

+ k i=2

k−1

j=1

c_ge(i)Ψ_∅0,i;e,j − k

i=2 k−1

j=1

c_ge(i)Ψ_∅0,i;e,j, (6)

w^P_e = k i=1

cex(i)

Ψ_∅0,i−

k j=1

Ψ_∅0,i;e,j

− k

i=1

cex(i)

Ψ_∅e,i−

k j=1

Ψ_∅0,j;e,i

, where∅^x,i,∅^x,i^;^y,j∈Aforx, y∈Z^d, x=yand 1i, jkare defined respectively by

(∅^x,i)l:=

{x} ifl=i,

∅ otherwise,

(∅^x,i^;^y,j)_l:=









{x} ifl=i=j, {y} ifl=j=i, {x, y} ifl=i=j,

∅ otherwise.

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Denote byθethe reflection operator with respect to 1/2ealong theedirection, namely forx∈Z^d, (θ_ex)_e:=

x_e ife=e,

−x_e+1 ife=e. (8)

We may extendθetoAnaturally. Put

Be:= {∅^0,i: 1ik} ∪ {∅^0,i^;^e,j: 1i, jk} and

˜

w^E_e(A):=











c_ex(i)i ifA= ∅^0,i for 1ik,

−c_ex(i)i+c_ge(i) ifA= ∅^0,i^;^e,j for 2ik, 1jk−1,

−c_ex(1) ifA= ∅^0,1^;^e,jfor 1jk,

−cex(i)i ifA= ∅^0,i^;^e,kfor 2ik,

0 otherwise,

˜

w^P_e(A):=







c_ex(i) ifA= ∅^0,ifor 1ik, cex(i) ifA= ∅^0,i^;^e,jfor 1i, jk, 0 otherwise,

where∅^x,iand∅^x,i^;^y,jare defined by (7). In view of (6), it holds that w^E_e =

A∈Be

˜

w^E_e(A)(Ψ_A−Ψ_θ_e_A), w^P_e =

A∈Bew˜^P_e(A)(ΨA−Ψθ_eA).

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We defineA^x^→^yandA^(x,y)by A^x^→^y:=





(A₁, . . . , A_i₋₁∪ {x}, A_i\ {x}, . . . , A_j\ {y}, A_j₋₁∪ {y}, . . . , A_k), ifx∈Ai, y∈Aj such that 2ik,1j k−1,

A, otherwise, A^(x,y):=

(A₁, . . . , A_i\ {x} ∪ {y}, . . . , A_k), ifx∈A_i, y /∈

jA_jsuch that 1ik, A, otherwise.

Givenx, y∈Z^d, defineΥx,y⊂Xby Υ_x,y:= {η: η_x2,1η_yk−1}.

ThenΨA(η^x^→^y)=ΨA^y→x(η)forη∈Υx,y. Therefore using summation by parts formula, we get π^x^→^yf (η)=1_{_η_x_2,1_η_y_k₋₁_}(η)

A

f (A)ˆ

ΨA^y→x(η)−ΨA(η)

=1_{_η_x_2,1_η_y_k₋₁_}(η)

A

f (Aˆ ^x^→^y)− ˆf (A) Ψ_A(η).

ForA∈A, ifΨA(η)=0 for someη∈Υ_x,y^c , thenA^x^→^y=A. Therefore π^x^→^yf (η)=

A

f (Aˆ ^x^→^y)− ˆf (A) ΨA(η).

Similarly

π^(x,y)f (η)=

A

f (Aˆ ^(x,y))− ˆf (A) Ψ_A(η).

If we definecˆ_ge(x, A)andcˆ_ex(x, A)by ˆ

cge(x, A):=

cge(i) ifx∈Ai for 1ik, 0 otherwise,

ˆ

cex(x, A):=

c_ex(i) ifx∈A_ifor 1ik, 0 otherwise,

then we can rewriteLf in the form Lf (η)=

A∈A

Lˆf (A)Ψˆ A(η),

where

Lˆf (A)ˆ =

x,y∈Z^d,|x−y|=1

cˆ_ge(x, A)f (Aˆ ^x^→^y)− ˆf (A)

+ ˆc_ex(x, A)f (Aˆ ^(x,y))− ˆf (A)

. (10)

LetXsbe a Markov process onAwhose generator isLˆ defined by (10) andPAa distribution of the Markov process starting fromA. Then the process is equivalent to the original process which starts from the configurationη^A. Therefore the Markov process generated byLˆ inherits the conserved quantities from the original process. Namely, if we defineAi,j⊂Aby

Ai,j:= {A∈A: #A=i,#A˜ =j}, (11) thenAi,j becomes the ergodic classes of the Markov process generated byL.ˆ

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LetXs be a Markov process onAwhose generator isLˆ defined by (10) andPAthe distribution of the Markov process starting fromA. Then the process is equivalent to the original process which starts from the configura- tionη^A.

Let the sequence{ ˆgλ(A)}Abe defined by ˆ

gλ(A):=EA

∞ 0

f (Xˆ s)e⁻^λsds, (12)

for a local functionf. Put g_λ(η):=

A

ˆ

g_λ(A)Ψ_A(η).

Lemma 3.2. The functiong_λis well-define and an element ofC₀(X). Furthermore this function is the solution of the resolvent equation

λgλ−Lgλ=f. (13)

Proof. It is enough to prove thatgλis well-define and an element ofC0(X). By the definition ofL, #Xˆ s and#X˜ s

are conserved. By using (5),f (A)ˆ is zero if #Aor#A˜ is large enough, and similarly forgˆλ(A). We define the stopping timeσ_nby

σn:=

inf{s:

i(X_s)_i⊂Λ_n} if{s:

i(X_s)_i ⊂Λ_n} = ∅,

∞ if{s:

i(Xs)i ⊂Λn} = ∅,

whereΛn is a cube centered at origin and of side 2n+1. Pick up a largensuch that if

iAi ∩Λ^c_n= ∅then f (A)ˆ =0. Therefore

ˆ

gλ(A)=EA

_∞

σn

f (Xˆ s)e⁻^λsds

1

λfˆMEA[e⁻^λσⁿ],

wherefˆM =maxAf (A). We defineˆ dn(A)the sum of the supremum distances fromΛn to elements of

iAi: formally if

iAi\Λn= {x¹, . . . , x^m}then d_n(A):=

m i=1

|xⁱ|_∞−n .

Since the jump rate is bounded, it is not difficult to show that there exist constantsC₁, C₂>0, which may depend onλ, such that

EA[e⁻^λσⁿ]C1e⁻^C²^dⁿ^(A).

This shows thatgλ(η)is well-define and an element ofC0(X). 2

We give the reversible measure forL, which we will use in Section 7. We defineˆ m(A, p, ρ)by m(A, p, ρ):=E_p,ρ[Ψ_A],

forA∈A, 0< p <1 andp < ρ < kp. The discrete measure whose mass ofAism(A, p, ρ)is also denoted bym.

Lemma 3.3. For eachA∈A,mis a smooth function ofp, ρ, and for each pair ofp andρ,mis a reversible measure forL.ˆ

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Remark. We decomposeAinto{Ai,j}the ergodic classes of the Markov process. On each ergodic class,m(·, p, ρ) for 0< p <1, p < ρ < kpare essentially the same. That is, for each ergodic class,m(·, p, ρ)andm(·, pρ)are absolutely continuous each other and the Radon–Nikodym derivative is a constant depending only onp, ρ, p, ρ and the ergodic class.

Proof of Lemma 3.3. We have only to prove thatmis a reversible measure forL, or equivalently thatˆ ˆ

cge(x, A)m(A, p, ρ)= ˆcge(y, A^x^→^y)m(A^x^→^y, p, ρ), forA=A^x^→^y, ˆ

c_ex(x, A)m(A, p, ρ)= ˆc_ex(y, A^(x,y))m(A^(x,y), p, ρ), forA=A^(x,y),

for eachp, ρ. But this relation immediately follows from the reversibility of the original process. 2

4. Estimate of the expectation of occupation time in dimensionsd3

In this section we will estimate an expectation of occupation time for a certain set in dimensionsd3.

Firstly we prove a general result on the expectation of occupation time for the Markov process.

Lemma 4.1. LetB⊂A, and suppose that there exist two subsetsB1,B2ofAand functionsf1andf2onAsuch that

B⊂B1⊂B2, (14)

f₁0, (15)

f₁(A)1 ifA∈B1, (16)

Lfˆ 1(A)0 ifA∈B₁^c, (17)

sup

A∈B2^c

f₁(A) <1, (18)

f₂0, (19)

supf₂<∞, (20)

ˆ

Lf₂(A)−1 ifA∈B2. (21)

Then we have sup

A

∞ 0

EA

1_B(Xs)

ds <∞.

Proof. We consider the martingale defined by

M_i,s:=f_i(X_s)−f_i(X₀)− s 0

ˆ

Lf_i(X_u)du,

fori=1,2. SinceM_i,0=0, applying the Doob’s optional sampling time theorem

f1(A)=EA

f1(Xσ_B1∧N)−

σ_B1∧N 0

f1(Xu)du

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for allN >0, whereσ_B₁ is a first hitting time toB1. Substituting the conditions (15)–(17) the right-hand side is greater than

E_A[I_{_σ_B1_<N_}]. Therefore by condition (18)

sup

A∈B₂^cP_A[σ_B₁<∞] sup

A∈B^c₂f₁(A) <1.

Similarly, by using the conditions (19)–(21), we get sup

A∈B2

EA[σ_B^c

2] sup

A∈B2

f2(A) <∞, whereσ_B^c

2 is the first hitting time to B₂^c. On using the Markov property, the supremum of the expectation of occupation time forBis finite. 2

From now to the end of the section, we suppose that the dimensionsd is greater than or equal to 3. In order to apply Lemma 4.1, we prepare some functions. We define functionsh_1,l, h_2,l from Z^d into R₊, which are a small perturbation of the potential function on Z^d, by

h_1,l(x):=

1 if|x|l, √ ₁

1+(|x|−l)²

d−2

if|x|l,

h2,l(x):=

√ 1+l 1+ |x|²

d−2

,

forl0, where| · |is the Euclidean norm of Z^d. Fora=(a₁, a₂, a₃, a₄)such thata₁, a₂, a₃>0 anda₄0, we define a functiongafrom R₊into R₊satisfying the following conditions

• gais a continuously differentiable function on R₊.

• On the interval[0, a1],gais a quadratic function with−a2for the coefficient of the second order term.

• There existsba1such that:

– On the interval[a1, b],gais a quadratic function witha3for the coefficient of the second order term.

– On the interval[b,∞),gais zero.

• The differential coefficient from the right atx=0 is−a4. Formally

g_a(x):=











−a2

x+_2a^a⁴₂2

+^(2a¹^a²₄⁺^a⁴⁾²₁

a₃ +_a¹₂

, if 0xa1, a₃

x−_2a₁_a₂₊_a₄

2a3 +a₁2

, ifa₁xa₁+^2a¹_2a^a²₃⁺^a⁴,

0, ifxa1+^2a¹_2a^a²₃⁺^a⁴.

Let us defineAM by

AM:= {A∈A: #A=M}.

Assume thatΛis a Euclidean ball with radiusl, and defineB:=BM,lby B:=

A∈AM:

A_i∩Λ= ∅or there existx, y∈

i

A_isuch that 1|x−y|∞l

. (22)

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Here| · |∞is the supremum norm of Z^d. ForA∈AM, we define{xi: 1iM} := {xi(A): 1iM} ⊂Z^dby {xi: 1iM} =

i

Ai,

andx_i,j(A)by thejth component ofx_i(A).

Let us define constantsa₁, a₂andc^±by a₁:=

M²+l, 2 c⁻, 1

c⁺M²,0

, (23)

a₂:=

M²+l, 1 c⁻, 1

2c⁺M²,4M²c⁺ c⁻

, (24)

and

c⁻:=min

r c_ex(r), (25)

c⁺:=max

r cex(r). (26)

Lemma 4.2. Define setsB1,B2and functionsf₁, f₂by B1:=B,

B2:=

A∈AM: there existsx∈Asuch that|x|_∞M²+l

or there existsx, y∈Asuch that 1|x−y|_∞M²+l , f1(A):=

M i=1

h1,l(xi)+

1i<jM

h2,l(xi−xj),

f2(A):=

M i=1

d j=1

ga₁

|xi,j|

+

1i<mM

d j=1

ga₂

|xi,j−xm,j| ,

wherea₁, a₂andc^±are defined by (23)–(26). Then the conditions of Lemma 4.1 hold, namely (14)–(21) hold.

Corollary 4.3. Suppose thatd3. Then forBdefined by (22), sup

A

∞ 0

EA

1_B(Xs)

ds <∞.

Proof of Lemma 4.2. We have only to check the truth of the conditions (17) and (21). We defineB0by B0=B0,M:=

A∈AM: xi(A)−xj(A)>1 fori=j .

Then on the setB0,Lˆ behave as a discrete Laplacian on Z^d for eachx_i(A). Formally, if we denote by_d the discrete Laplacian on Z^d, then

Lfˆ ₁(A)= M

i=1

ˆ

c_ex(x_i(A), A)(_dh_1,l)(x_i)+

1i<mM

cˆ_ex

x_i(A), A + ˆc_ex

x_m(A), A

(_dh_2,l)(x_i−x_m), forA∈B0. Sinceh1,l, h2,lare small perturbations of thed-dimensional potential function, each term of the right- hand side is non-positive. SinceB^c₁⊂B0, the condition (17) is valid.

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We also have Lfˆ ₂(A)=

M i=1

d j=1

ˆ c_ex

x_i(A), A

(₁g_a₁)x_i,j(A)

+

1i<mM

d j=1

cˆex

xi(A), A + ˆcex

xm(A), A

1ga₁x_i,j(A)−xm,j(A), forA∈B0. From the choice of the second component ofa1, a2, if|xi,j(A)|M²+l, then

ˆ c_ex

x_i(A), A

(₁g_a₁)x_i,j(A)−2 and if 2|xi,j(A)−xm,j(A)|M²+l, then

cˆ_ex

x_i(A), A + ˆc_ex

x_m(A), A

(₁g_a₁)x_i,j(A)−x_m,j(A)−2.

From the choice of the third component ofa₁, a₂, each terms of the last equality is less than or equal to 1/M². On the setB2∩B0, there existsisuch that|xi,j(A)|M²+lfor allj, or there existsi, msuch that 2|xi,j(A)− xm,j(A)|M²+lfor allj. Therefore on the setB2∩B0, the condition (21) is valid.

On the setB2\B0, without loss of generality we assume that there existn2 and 1jd such thatx_l= x₁+(l−1)e_jfor 1ln, wheree_jis a unit vector for thejth coordinate, and ifn+1iM, then|x_i−x_m|2 for all 1mM,i=m. Furthermore, we can assume that the choice of thex_i(A^(x,y))is similar tox_i(A), namely we assume that

x_i(A^(y,z))=

xi(A) ify=xi(A), z ify=x_i(A), for all 1iM,y∈

iAi andz /∈

iAi. ThenLgˆ a₁(|xi,j(A)|)andLgˆ a₂(|xi,j(A)−xm,j(A)|)make sense for 1i < mM. By elementary computation if 1i < mn, then

Lgˆ _a₂x_i,j(A)−x_m,j(A)0, and

Lgˆ _a₂x_1,j(A)−x_2,j(A)−4c⁺M² c⁻ , Lgˆ a₂x_n₋_1,j(A)−xn,j(A)−4c⁺M²

c⁻ .

The last value^4c⁺_c₋^M² is given by the choice of the fourth component ofa₂. We also compute that Lˆ

1in

g_a₁x_i,j(A)= ˆc_ex

x₁(A), A

g_a₁x_1,j(A)−1−g_a₁x_1,j(A) + ˆcex

xn(A), A

ga₁x_n,j(A)+1−ga₁x_n,j(A).

Since we assume thatx_i,j(A)=x_i₋_1,j(A)+1 for 2in, the right-hand side of the last equality is equal to ˆ

c_ex

x₁(A), A

1in

(₁g_a₁)x_i,j(A) +

ˆ c_ex

x_n(A), A

− ˆc_ex

x₁(A), A

g_a₁x_n,j(A)+1−g_a₁x_n,j(A).

From the choice of the third component ofa₁, the first line is less than or equal tom/M². Since the maximal value ofg_a₁(x)−g_a₁(x+1)is given by the maximal value of differential coefficient ofg_a₁, from the choice ofa₁the second line is less than or equal to 4c⁺(M²+l)/c⁻. From the assumption of thexi forin+1, the other terms forLfˆ 2is less than or equal to 1/M². Thus we have checked the truth of the condition (21). 2

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5. Occupation time for some stochastic processes

In this section, we consider finite state continuous time stochastic jump processes, which may not be Markovian.

LetSbe a finite set, which is a state space of our stochastic process. For eachi∈S, there is given a probability measureµⁱ onS×R₊. Let(Yî,n, σî,n), i∈S, n=0,1,2, . . .be a system of independent random variables taking values inS×R₊such that for eachithe joint distribution of(Yî,n, σî,n)is given byµⁱ for everyn.

Then our process is described as follows. Suppose that the process starts ati₀∈S. Then one takes the random variable(Yⁱ⁰^,0, σⁱ⁰^,0) and let the process stay at i₀ up to the time σⁱ⁰^,0 and jump to Yⁱ⁰^,0 at time σⁱ⁰^,0. The procedure is repeated over by letting it start ati₁=Yⁱ⁰^,0 and taking up(Yⁱ¹^,1, σⁱ¹^,1)place of(Yⁱ⁰^,0, σⁱ⁰^,0). We denote the distribution of the process which starts fromi∈SbyPi. We suppose several conditions for the joint distributionµⁱ. Let us define

Fi,j(λ):=E^µⁱ

1_{_Yi=j}e⁻^λσⁱ . (A.1) It holds that

F_i,i(λ)=0 for alli∈S.

(A.2) There exist constantsp_i,j, f_i,j,and a functiong(λ)such that F_i,j(λ)=p_i,j−f_i,jg(λ)+o

g(λ) , wherep_i,j 0 and for alli∈S

j∈S

p_i,j=1;

f_i,j is non-negative; andg(λ)is a positive and increasing function from R₊to R₊which vanishes atλ=0. Here o(·)is Landau’s symbol asλtends to 0. In this paperg(λ)will be√

λor−1/logλaccordingly asd=1 or 2.

(A.3) The stochastic matrixpi,j in (A.2) is irreducible.

(A.4) There exists a pair(i, j )such thatfi,j is strictly positive.

(A.5) There existsi₀such thatf_i₀_,j=0 for allj. Furthermore, there exists a positive constantf_i₀ such that

j

F_i₀_,j(λ)=1−f_i₀λ+o(λ).

Lemma 5.1. Suppose that (A.1)–(A.5) hold. Then for alli∈S, the Laplace transform of occupation time fori₀is asymptotically equivalent to 1/g(λ)(asλtends to 0), namely there exists a constantC1which depends onisuch that

E_i ∞ 0

1_{_i₀_}(X_s)e⁻^λsds= C₁ g(λ)+o

1 g(λ)

.

Furthermore for alli, j ∈S, there exists a limit of the difference of Laplace transform of occupation time fori₀, namely there exists a constantC₂which depends oniandj such that

lim

λ→0

E_i

∞ 0

1_{_i₀_}(X_s)e⁻^λsds−E_j ∞ 0

1_{_i₀_}(X_s)e⁻^λsds

=C₂.

Proof. LetQ_i,i₀(λ)be the Laplace transform of the occupation time fori₀with starting pointi∈S. Letσ₁be the first jump time. Then on using the conditional independence of the process after the timeσ₁and the jump timeσ₁ givenXσ₁,