Estimate of spectral gap for continuous gas

Estimate of spectral gap for continuous gas

Liming Wu

aLaboratoire de Math. Appl. CNRS-UMR 6620, Université Blaise Pascal, 63177 Aubière, France bDepartment of Math., Wuhan University, 430072 Hubei, China

Received 13 February 2003; received in revised form 23 October 2003; accepted 18 November 2003 Available online 25 March 2004

Abstract

Consider the continuous gas in a bounded domainΛofR^d, described by a Gibbsian measureµ^η_Λassociated with a pair interactionϕ, the inverse temperatureβ, the activityz >0, and the boundary conditionη. Whenϕis nonnegative, we show that the spectral gap of a Glauber type dynamic (i.e., some Markov process reversible with respect toµ^η_Λ) inL²(µ^η_Λ)is bounded from below by 1−z

R^d|1−e^−βϕ(y)|dyand from above by 1+z

R^d|1−e^−βϕ(y)|dy, independent ofΛandη. This result improves a previous work by L. Bertini et al. (2002) and is extended also to the hard core case. Our approach consists to approximate the continuous gas model by the discrete spin model and to apply theM-εtheorem of Ligget. Some other results such as uniqueness, exponential convergence of the Glauber dynamic w.r.t. norms of Ligget’s type are also obtained.

2004 Elsevier SAS. All rights reserved.

Résumé

On considère un gaz continu dans un domaine bornéΛdeR^d, décrit par la mesure de Gibbsµ^η_Λassociée à l’interaction paireϕ, la température inverse β, l’activitéz >0 et la condition au bordη. Quandϕ0, nous démontrons que le trou spectralλ₁d’une dynamique du type Glauber (i.e., un processus de Markov réversible par rapport àµ^η_Λ) dansL²(µ^η_Λ)vérifie 1−z

R^d|1−e^−βϕ(y)|dyλ₁1+z

R^d|1−e^−βϕ(y)|dy, indépendamment deΛet deη. Ce résultat améliore le travail de Bertini et al. (2002), et est généralisé au cas de corps durs. Notre méthode consiste à approcher le gaz continu par un modèle de spin discret auquel le théorèmeM-εde Ligget s’applique. Nous établissons également l’unicité de la dynamique de Glauber et sa convergence exponentielle par rapport aux normes du type Ligget.

2004 Elsevier SAS. All rights reserved.

MSC: 60J75; 60G57; 82B20; 82C20

Keywords: Poincaré inequality; Gibbs measures; Birth and death processes

1. Introduction

In statistical mechanics, relations between the mixing properties of the Gibbs measure and the exponential speed at which the associated Glauber dynamics relaxes to equilibrium are a fascinating and important object. For lattice

E-mail address: Li-Ming.Wu@math.univ-bpclermont.fr (L. Wu).

0246-0203/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2003.11.003

spin systems with compact spin space, Stroock and Zegarlinski, in their important and difficult work [18,19] proved the equivalence between the Poincaré inequality, the log-Sobolev inequality, and the Dobrushin–Shlosman mixing condition for the Gibbs measure, both for the Glauber dynamics of pure jumps type or of diffusion type. See Lu and Yau [12] and Martinelli [9] for further development and the recent work by F. Cesi [5] for a simplified proof.

The (partial) extension of their impressive results to unbounded spin case for Glauber dynamics of diffusion type is carried out by Bodineau and Heffler [2,3], Ledoux [10], Yoshida [23] etc.

In this work we are interested in the same question when the discrete latticeZ^d is replaced by the continuum R^d(i.e., gas instead of crystals in physics language). By decompositionR^d:=

k∈Z^dk[0,1)^d, we may regard this continuous gas model as a lattice model with unbounded spin (and with unbounded interaction). For continuous gas, L. Bertini et al. [4] establish the spectral gap existence of a Glauber dynamic for high temperature or low activity. Let us present this interesting work briefly.

LetΛbe a bounded domain inR^d. Given the boundary conditionηoutside ofΛ, consider the Gibbs measureµ^η_Λ inΛassociated with a “stable” pair interactionϕ:R^d→(−∞,+∞], activityz >0 and the inverse temperatureβ (see the next section for precise definition). Under the following assumptions

(H1) ϕ0 andϕis even;

(H2) ϕ is of finite range, i.e.,ϕ(x)=0 if|x|> rfor some finiter >0;

(H3) z

R^d(1−e^−βϕ(y)) dy <_3e¹ (i.e., condition (CE) in [4]);

Bertini, Cancrini and Cesi [4] (Theorem 2.2) establish thatµ^η_Λfor all rectanglesΛsatisfies the Poincaré inequality

Gµ^η_Λ(f, f )E_Λ^η(f, f ), ∀f (1.1)

where the constant G=G(z, β, r) >0 is independent of η andΛ, andE_Λ^η is a quite natural Dirichlet form on L²(µ^η_Λ) generating the Glauber dynamic (which is a birth-death Markov process reversible w.r.t. µ^η_Λ, see Sections 2, 3). Their main idea is:

1) a quasi-factorization of the variance;

2) to establish an exponential decay of correlation between f and g when their “supports” are sufficiently separated by condition (H3) and cluster expansion;

3) the iterative method by doubling the volume and a delicate geometric consideration.

Their result so obtained does not, seems – it however, yield a robust estimate of the spectral gap constantλ₁, like most known results in [18,19,23,12] issued of the iterative method (we emphasize that some explicit spectral gap estimates are given by Bodineau and Heffler [2,3] and Ledoux [10]). The reader can compare their (H3) with the following classical estimate of the convergence radiusR of the cluster expansion of the pressionp=p(z) (thermodynamic limit) in terms of the activityz([17], Theorem 4.5.3):

1 eR

R^d

1−e^−βϕ(y)

dy1. (1.2)

So their result can be roughly read as the spectral gap existence when|z|< R/3.

The main aim of this paper is to improve their result (1.1). Indeed our main result (see Theorem 2.1) says that for nonnegativeϕ, the best constantGfor (1.1), denoted byλ₁(i.e., the spectral gap), satisfies

1−z

R^d

1−e^−βϕ(y)

dyλ11+z

R^d

1−e^−βϕ(y) dy

without the finite range condition (H2). Henceλ₁is uniformly lower bounded once ifz

R^d(1−e^−βϕ(y)) dy <1, a condition weaker than (H3), and sharp in the viewpoint of (1.2). Moreover we extend this result to the hard core case.

The estimate above yieldsλ₁=1 when ϕ=0 (i.e., the free case), a well known result (to all specialists on Malliavin Calculus over the Poisson space). See Ané and Ledoux [1] and the author [21] for modified log-Sobolev inequalities which are stronger than the Poincaré inequality.

Our method will be completely different, and more classical in some sense. Indeed our idea is inspired by the classical Poisson limit theorem, i.e., a Poisson distribution is the weak limit of laws of sums of i.i.d. Bernoulli random variables. Then it is not surprise that we can approximate the Glauber dynamic associated withµ^η_Λ by the spin models on{0,1}^V for which the Ligget’s theorem gives us an explicit exponential rate for the “triple”

norm. Hence the key consists to bound the Ligget’s constant for the spin models on{0,1}^V by means ofϕ, and to transform that convergence rate inL²(µ^η_Λ), and fortunately this is possible.

This paper is organized as follows. The next section is devoted to describe the Gibbs measure, the Glauber dynamic and the main result. In Section 3 we solve some uniqueness problems (which are crucial for approximation) and construct the corresponding Markov process. Section 4 is devoted to the approximation of continuous gas by discrete spin models, which is the crucial part of this paper. As consequence, the spectral gap result is derived in Section 5, together with the exponential convergence in other senses (thanL²).

2. Main result

2.1. Gibbs measure

LetΩ be the space of all point measuresω=

iδ_xi (finite or countable) withx_i different in R^d, which are moreover Radon measures (i.e., finite particles in compact subsets), whereδ_x denotes the Dirac measure at x. LetFA:=σ (ω(B);B (B Borelian)⊂A)for eachA∈B(R^d), the Borelσ-filed ofR^d andF=F_R^d. Given the activityz >0, letP be the law of the Poisson point process onR^d with intensity measurez dx, i.e., a probability measure on(Ω,F)characterized by

(i) P (ω(A)=k)=e^−z|A|(z|_k^A_!^|)k, ∀k∈N for anyA∈B(R^d); here and throughout this paper |A| designs the Lebesgue volume ofA;

(ii) IfAi∈B(R^d),i=1, . . . , n, are disjoint, thenω(Ai),i=1, . . . , n, areP-independent.

Throughout this paper the pair interactionϕ:R^d→(−∞,+∞]will be a Borel-measurable even function which is stable ([17]), i.e.,∃B0 such that

H (ω):=

1i<jn

ϕ(x_i−x_j)−Bn, ∀ω= n

j=1

δ_xj, n1 (stability). (2.1)

We assume often also thatϕis regular [17], i.e.,

R^d

1−e^−βϕ(y)dy <+∞ (2.2)

whereβ=(κT )⁻¹>0 is the inverse temperature. Recall that (see [17]) the stability condition is a necessary and sufficient condition for defining the (free boundary) Gibbs measures on bounded domainsΛ. Moreover for a stable pair interactionϕ,ϕ(x)=H (δx+δ0)−2Bby (2.1), and then the regularity condition (2.2) is equivalent to the integrability ofϕoutside of some finite measure set (e.g.[ϕ1]).

Given a bounded open and non-empty domain Λ⊂R^d and ω∈Ω, let ωΛ =

xi∈Λ supp(ω)δx_i be the restriction of the measureω toΛ, andΩΛ= {ωΛ; ω∈Ω}. The image measurePΛ ofP by ω→ωΛ is the law of Poisson point process onΛwith intensity measurez dx.

We say that an elementηofΩ is inΩ^ϕ, if x→

R^d

ϕ⁻(x−y) dη(y)is locally bounded onR^d. (2.3)

Whenϕ0 or is of finite range, we have of courseΩ^ϕ=Ω.

The Gibbs measure inΛfor a given boundary conditionη∈Ω^ϕ onΛ^c is a probability measure on(Ω_Λ,FΛ) given by

µ^η_Λ(dω_Λ):= 1

Z(Λ, η)e^{−βHΛη(ωΛ)}P_Λ(dω_Λ) (2.4)

where

H_Λ^η(ω_Λ):=H (ω_Λ)+

Λ

ω(dx)

Λ^c

ϕ(x−y) η(dy) is the Hamiltonian (H (ω_Λ)being given in (2.1)), and

Z(Λ, η):=

ΩΛ

e^{−βHΛη(ωΛ)}dP_Λ(ω_Λ)

is the normalization constant. Remark thatH_Λ^η(0)=0 where 0 denotes the zero measure (or the vide state) and by the stability condition,

H_Λ^η(ωΛ)−BNΛ(ω)+

Λ

h^η(x) dω(x)

whereN_Λ(ω)=ω(Λ)is the number of particles in the configurationω_Λ, and h^η(x)=

Λ^c

ϕ(x−y) η_Λ^c(dy) (2.5)

which is bounded from below onΛby (2.3). HenceZ(Λ, η)is a finite positive constant,µ^η_Λis then a well defined probability onΩ_Λ, and moreover

dµ^η_Λ

dP_Λ ∈

1p<+∞

L^p(PΛ). (2.6)

2.2. Generator of the Glauber dynamic

LetrF be the space of realF-measurable functions, andbF the space of thoseF ∈rF which are moreover bounded. For anyr∈rF, according to Picard [16] consider the difference operators

D⁺_xF (ω):=F (ε_x⁺ω)−F (ω), ε_x⁺ω:=ω+1x /∈suppωδ_x; D⁻_xF (ω):=F (ε_x⁻ω)−F (ω), ε_x⁻ω:=ω−1_x∈suppωδ_x;

D_xF (ω):=F (ε_x⁺ω)−F (ε_x⁻ω). (2.7)

Those resulting functions are measurable jointly onR^d×Ω. Recall thatD_x⁺(orD_x) plays the same role in the Malliavin calculus over the Poisson space as the Malliavin derivative on the Wiener space ([16,21] and references therein).

We shall study the Glauber dynamic employed in [4], which is formally generated by the pre-generator

L^η_ΛF (ω_Λ)=

Λ

D⁻_xF (ω_Λ)ω_Λ(dx)+z

Λ

e^{−βD+xHΛη(ωΛ)}D_x⁺F (ω_Λ) dx

= −

Λ

e^{−βDx+HΛη(ωΛ)}D_xF (ω_Λ)

ω_Λ(dx)−z dx

, ∀F ∈bFΛ (2.8)

(recallingD_x⁺H_Λ^η(ωΛ)=0,∀x∈suppωΛ). Its dynamic can be intuitively described as follows: if the configuration of the system isωΛat time t, each particle inωΛwill be killed at ratedt, and a new particle will be born atx∈Λ with rateze^{−βDx+HΛη(ωΛ)}dx dt.

LetFu=sup_ω∈ΩΛ|F (ω)|. Since|

ΛD_x⁻F (ωΛ)ωΛ(dx)|2Fu·NΛ(ω)and D⁺_xH_Λ^η(ωΛ)=

Λ

ϕ(y−x)

ωΛ(dy)+ηΛ^c(dy)

−2BNΛ(ω)+h^η(x), ∀x /∈suppωΛ (2.9)

where h^η(x) given in (2.5) is lower bounded, then it is easy to verify that L^η_ΛF ∈ rFΛ and L^η_ΛF ∈

1p<+∞L^p(µ^η_Λ)for allF∈bFΛ. Moreover ifF₁=F₂, µ^η_Λ-a.s.,L^η_ΛF₁=L^η_ΛF₂,µ^η_Λ-a.s. (by [16], Section 4).

In particularLis a well defined operator onL^p(µ^η_Λ)with domainbFΛ(in which each element represent a class ofµ^η_Λ-equivalent functions, by usual convention). In further by Picard [16] (Proposition 6 and Théorème 2), for all F, G∈bFΛ, we have by a simple calculus

F,−L^η_ΛG

µ^η_Λ =

ΩΛ

dµ^η_Λ(ω_Λ)

x∈suppω

D_x⁻F (ω_Λ)D⁻_xG(ω_Λ)

=

ΩΛ

dµ^η_Λ(ω_Λ)

Λ

e^{−βDx+HΛη(ωΛ)}D_x⁺F (ω_Λ)D⁺_xG(ω_Λ)z dx

=:E_Λ^η(F, G). (2.10)

(L. Bertini et al. [4] (Proposition 2.1) has indicated this fact when ϕ0.) Then (−L^η_Λ, bFΛ)is nonnegative definite, symmetric operator onL²(µ^η_Λ). HenceE_Λ^η is a closable form and its closure(E_Λ^η,D(E_Λ^η))is a Dirichlet form on L²(µ^η_Λ), generating a symmetric Markov semigroup (P_t^Λ,η) on L²(µ^η_Λ) such that P_t^Λ,η1=1, µ^η_Λ- a.s. (since L^η_Λ1=0). The last symmetric Markov semigroup, whose generator is the Friedrichs extension of (L^η_Λ, bFΛ), is the Glauber dynamic used in [4] and in this work. Of course(P_t^Λ,η)is also a strongly continuous semigroup of contractions onL^p(µ^η_Λ), whose generator will be denoted by(L^η_Λ,Dp(L^η_Λ)) (Dp(L^η_Λ)being its domain inL^p(µ^η_Λ)).

Notice that the same kind of model is studied by Olla and Tremoulet [15] (2003).

2.3. Main result

Theorem 2.1. Letϕ:R^d→(−∞,+∞]be an even measurable function onR^d, which is both stable (2.1) and regular (2.2). Assume either

(C1) ϕ0 (nonnegative potential); or

(C2) for somerhc>0,ϕ(x)= +∞if|x|< rhc (hard core) (|x|denotes the Euclidean norm ofx∈R^d) and for someB0,

n

i=1

ϕ(x_i)−2B, (2.11)

for alln1 and all(x1, . . . , xn)∈(R^d)ⁿsuch that|xi−xj|rhcfori=j.

Then for any bounded (non-empty) open domain Λ of R^d and for any η ∈ Ω^ϕ, the spectral gap λ₁ of (−L^η_Λ,D2(L^η_Λ))inL²(µ^η_Λ), i.e. the best constantλ₁0 such that

λ₁µ^η_Λ(F, F )E_Λ^η(F, F ), ∀F ∈D(E_Λ^η) (2.12)

satisfies 1−ze^2βB

R^d

1−e^−βϕ(x)dxλ₁1+ze^βB

R^d

1−e^−βϕ+(x)

dx (2.13)

where the constantBis given by condition (2.11) (of courseB=0 ifϕ0),µ^η_Λ(F, G)denotes the covariance of F, Gunderµ^η_Λ, and(E_Λ^η,D(E_Λ^η)is the closure of the form(E_Λ^η, bFΛ)given in (2.10).

Remarks 2.2. Under the hard core assumption in (C2), condition (2.11) is verified once if there is a nonnegative decreasing functionh:[r_hc,+∞)→R⁺such that

∞

rhc

h(r)r^d−1dr <+∞ and ϕ(x)−h

|x|

, ∀x∈R^d,|x|rhc

(see [17], Section 3.2.5, pp. 37–38). Moreover (C2) implies the stability condition (2.1) with the same constantB.

Remarks 2.3. Letp(z):=β⁻¹lim_Λ↑Rd 1

|Λ|logZ(Λ,0)(thermodynamic limit in the sense of Von Hove, see [17]) be the pressure function. In the nonnegative case (C1), our condition about the existence of spectral gap is sharp in the point of view of the cluster expansion estimate (1.2) (but perhaps not all sharp in reality, becausep(z)may be analytical for realzR). And the spectral gap result above suggests thatR=R₀:=(

R^d(1−e^−βϕ(y)) dy)⁻¹or at leastp(z)is analytical forz∈ [0, R₀), a claim (perhaps known) that I do not know how to prove it.

In the general stable and regular case, the classical estimate of the convergence radius R of the pressure functionalp(z)in terms of the activityzverifies

R·e^2βB

R^d

1−e^−βϕ(x)dx1 e

(see [17], Theorem 4.2.3). And our result suggests that the estimate above may hold with 1/esubstituted by 1 on the right hand side above, in the hard core case (C2) (to which we equally have no answer).

An interesting (open) question is to extend Theorem 2.1 to general stable and regular interactionϕ. Our proof seems working only under (C1) or (C2).

Remarks 2.4. L. Bertini et al. [4] derive, from the spectral gap existence, the exponential decay of correlation µ^η_Λ(F, G)whereF ∈bFAandG∈bFB, when the distance between their “supports”A, Bis large, illustrating the impetus of spectral gap in the understanding of the mixing properties of the underlying Gibbs measure.

Though our theorem above improves their main result [4] Theorem 2.2, but one main contribution of [4] resides in their approach: the exponential decay of correlation in the form of their Corollary 2.5 implies the spectral gap existence, via their quasi-factorization of variance. Indeed F. Cesi [5] utilizes this approach to give a simplified proof of the Stroock–Zegarlinski’s log-Sobolev inequality of the Gibbs measure, but our approach here is valid only for the Poincaré inequality.

Remarks 2.5. The continuous gas model has an essential difference from the lattice spin model with compact spin space: the equivalence between the (uniform) Poincaré inequality and the (uniform) log-Sobolev inequality is lost. In fact it is known since the work of Surgailis [20] on 1984 that even the free Poisson measurePΛdoes not

satisfy the log-Sobolev inequality. And Ané and Ledoux [1] and the author [21] prove the modified log-Sobolev inequalities, one of which is the followingL¹-log-Sobolev inequality [21]

Ent_PΛ(F ):=E^PΛFlog F

E^PΛ(F )E^PΛ

Λ

D⁺_xF D_x⁺logF z dx, ∀0< F∈bFΛ.

This last inequality is equivalent to the exponential decay in the sense of entropy:Ent_PΛ(P_t⁰F )e^−tEnt_PΛ(F ), whereP_t⁰=P_t^Λ,ηwithϕ=0. And it is equivalent to the usual log-Sobolev inequality when the employed Dirichlet form is of diffusion type (but this interesting equivalence fails for jumps processes: the case here). One can then reasonably hope the equivalence of the Poincaré inequality with the L¹-log-Sobolev inequality above for the continuous gas model here.

Notice that Dai Pra, Paganoni and Posta [6] (2002) establish theL¹-log-Sobolev inequality for the lattice gas with unbounded spin.

2.4. A guideline to our proof of Theorem 2.1

For the reader’s convenience, let us outline our approach for proving the lower bound ofλ₁in Theorem 2.1. At first notice that

d dtµ^η_Λ

P_t^Λ,ηF, P_t^Λ,ηF

= −2E_Λ^η

P_t^Λ,ηF, P_t^Λ,ηF

= −2

ΩΛ

dµ^η_Λ(ω_Λ)

Λ

e^{−βDx+HΛη(ωΛ)}

D⁺_xP_t^Λ,ηF (ω_Λ)2

z dx.

If one can prove that for somec >0, for allF belonging aL²(µ^η_Λ)-dense class of test functionsD,

D_x⁺P_t^Λ,ηF (ωΛ)e^−ctD⁺_xF (ωΛ) (2.14)

for some norm · stronger than the norm of L²(Λ×ΩΛ, z dx dµ^η_Λ(ωΛ)), we will get immediately (since D⁺_xH_Λ^η(ωΛ)−2Bby (C1) or (C2)),

µ^η_Λ

P_t^Λ,ηF, P_t^Λ,ηF

= − +∞

t

d dsµ^η_Λ

P_s^Λ,ηF, P_s^Λ,ηF

dsC(F )e^−2ct, ∀F ∈D

for some constantC(F ) >0 depending onF. This implies, by the spectral decomposition,λ₁c(true but not trivial).

For showing (2.14) which is of independent interest, as vaguely said in the introduction, we shall approximate the Poisson measureP_Λ by the product Bernoulli measure on{0,1}^IN,µ^η_Λ by Gibbs measuresµ_N on{0,1}^IN, L^η_Λ byLN on some good test functions spaceD(this is possible ifϕ is continuous and of finite range). A first questions arises:

1) to prove thatP_t^N =e^tLN verifies a relation similar to (2.14) with a constant c >0 independent of N and withD⁺_x substituted by some difference operator∇. This is provided by the LiggetM-εtheorem which says roughly

∇P_t^Nfe^−te^{t ΓN}|∇f|

with the Ligget’s matrixΓN. It remains to boundΓN in some nice norm in such a way that is independent ofN, and fortunately this works w.r.t.∇f∞under (C1) or (C2).

If the story stopped here, it would be simple and lucky. The real story is:

2) to transfer the Ligget’s estimate on(P_t^N)toP_t^Λ,η, thoughLN→L^η_Λ onD, we should prove that ∇P_t^N→ D⁺P_t^Λ,η, orP_t^Nf_n→P_t^Λ,ηF in the sup norm. For the last convergence we should prove at first that(L,D) generates a unique semigroup onbFΛfor applying the Trotter type theorem. The latter is quite difficult, for (P_t^Λ,η)is not defined everywhere onΩ_Λand even so, it is not at all strongly continuous onbFΛand the usual Trotter theorem can not be applied (but the involved techniques work, fortunately).

That is exactly the task of the following section.

3. Uniqueness and the Markov process generated byL^η_Λ

Throughout this paper, for a pair of measure-function(ν, f ),ν(f ):=

f dν. From now on, the bounded open (non-empty) domainΛand the boundary conditionη∈Ω^ϕ, though arbitrary, will be fixed. A genetic element of ΩΛwill be often denoted byωfor simplicity of notation.

In this section we assume the stability (2.1), but not the regularity (2.2). The following duality relation ([16], Remarque 1, p. 518) will be used:

E^PΛ

Λ

F (ε_x⁺ω)G(x, ω)z dx=E^PΛF (ω)

Λ

G(x, ε⁻_xω) dω(x) (3.1)

for allFΛ-measurableF:ΩΛ→R⁺and allB(Λ)⊗FΛ-measurableG:Λ×ΩΛ→R⁺, whereB(Λ)is the Borel σ-field ofΛ.

3.1. Uniqueness problem inL^p(µ^η_Λ)

Proposition 3.1. Assume the stability condition (2.1). Let D1:=

f

ω_Λ(A₁), . . . , ω_Λ(A_n)

1_{maxiωΛ(Ai)1}|n1, A_i∈A, f:Nⁿ→Rbounded

;

(3.2) D2:=

F

ωΛ(h1), . . . , ωΛ(hn)

; hn∈C_b^∞(Λ), F∈C₀^∞(Rⁿ), n1

; whereAis an arbitrary subalgebra ofB(Λ)such thatσ (A)=B(Λ).

(a) For any 1p <+∞,D1andD2are both a core for the generator(L^η_Λ,Dp(L^η_Λ))of(P_t^Λ,η)onL^p(µ^η_Λ). In particular(P_t^Λ,η)is the unique strongly continuous semigroup of bounded operators onL^p(µ^η_Λ)such that its generator extends(L^η_Λ,D1)or(L^η_Λ,D2); and(L^η_Λ,Di),i=1,2, are essentially self-adjoint inL²(µ^η_Λ).

(b) GivenF∈rFΛ L²(µ^η_Λ). ThenF ∈D(E_Λ^η)iff

ΩΛ

dµ^η_Λ(ω_Λ)

Λ

e^{−βD+xHΛη(ωΛ)}(D⁺_xF )²(ω_Λ)z dx <+∞, (3.3) and iff

Ω_Λ

dµ^η_Λ(ω_Λ)

x∈suppωΛ

(D_x⁻F )²(ω_Λ) <+∞. (3.4)

In those cases, the two quantities above coincide withE_Λ^η(F, F ).

Proof. (a) At first the second claim in (a) is a consequence of the first by [22] (Proposition 1.1), and the third claim is another expression of the first forp=2. To show the first claim, notice thatbFΛ, identified withL^∞(µ^η_Λ)up toµ^η_Λ-equivalence, is a core for the generator of the Markov semigroup(P_t^Λ,η)inL^p(µ^η_Λ)(true for any strongly continuous Markov semigroup). Hence it is enough to show that anyF ∈bFΛcan be approximated byFn∈Di in the graph norm topology ofL^η_ΛinL^p(µ^η_Λ), fori=1,2.

In fact we can find Fn∈Di such that Fn →F, µ^η_Λ-a.s. and |Fn|Fu =sup_ω∈ΩΛ|F (ω)| everywhere over ΩΛ. Then Fn−Fp:= Fn−F_L^p_(µ^η

Λ) →0 by the dominated convergence. It remains to prove that L^η_Λ(F_n−F )p→0. By the expression (2.8) and (2.9), we have

L^η_Λ(F_n−F )(ω)

Λ

(F_n−F )(ω)−(F_n−F )(ε⁻_xω) ω(dx)

+z

Λ

e^{−βDx+HΛη(ω)}D⁺_x(F_n−F )(ω) dx |F_n−F|(ω)·N_Λ(ω)+

Λ

|F_n−F|(ε⁻_xω) ω(dx)

+e^2BβNΛ(ω)

Λ

e^−βhη(x)z dx· |Fn−F| +e^2BβNΛ(ω)

Λ

e^−βhη(x)|Fn−F|(ε⁺_xω)z dx.

Let us show that the four terms in the last sum converge all to 0 inL^p(µ^η_Λ), or in probability µ^η_Λ∼PΛ by the dominated convergence (by relation (2.6)).

The first and third terms pose no problem. For the second and fourth terms, it is enough to notice that by the duality formula (3.1), we have

dP_Λ(ω)

Λ

|F_n−F|(ε⁻_xω) ω(dx)=z

dP_Λ(ω)|F_n−F| · |Λ| →0,

z

dP_Λ(ω)

Λ

e^−βhη(x)|F_n−F|(ε⁺_xω) dx=

dP_Λ(ω)|F_n−F|(ω)

Λ

e^−βhη(x)dω(x)→0.

(b) We shall prove only the first “iff” and the second can be proved in the same way. GivenF ∈rFΛ∩L²(µ^η_Λ) satisfying (3.3) and integerL1, letF_L:=(F∧L)∨(−L)∈bFΛ⊂D(E_Λ^η). Then(D_x⁺F_L)²(ω)(D_x⁺F )²(ω) and(D⁺_xF_L)²(ω)↑(D_x⁺F )²(ω)asL↑ ∞, for all(x, ω)∈Λ×Ω_Λ. Then by the expression (2.10) and monotone convergence,E_Λ^η(F_L−F_M, F_L−F_M)→0 asL, M→ ∞and

E_Λ^η(FL, FL)→

Ω_Λ

dµ^η_Λ

Λ

e^{−βDx+HΛη(ωΛ)}(D⁺_xF )²(ωΛ)z dx <+∞,

where it follows thatF∈D(E_Λ^η)and the last quantity coincides withE_Λ^η(F, F ).

Inversely letF ∈D(E_Λ^η). Hence by definition there exists a sequence(Fn)n1⊂bFΛ such thatFn→F in L²(µ^η_Λ)andE_Λ^η(Fn−F, Fn−F )→0 asn→ ∞. By taking a subsequence and a re-definition ofFn, Fif necessary, we may assume without loss of generality thatFn(ω)→F (ω)everywhere inΩΛ. HenceD_x⁺Fn(ω)→D_x⁺F (ω) for all(x, ω)∈Λ×ΩΛ. Thus by the expression (2.10) ofE_Λ^η(Fn, Fn), we see that

E_Λ^η(F_n, F_n)→

ΩΛ

dµ^η_Λ

Λ

e^{−βD+xHΛη(ωΛ)}(D⁺_xF )²(ω_Λ)z dx

the desired result. 2

3.2. Construction of the associated Markov process

As probabilists we always want to construct a good Markov process associated with generatorL^η_Λ. This was done by Holley and Stroock [7] and Picard [16]. Unfortunately the conditions imposed in those known works are not all verified for the model used here.

Proposition 3.2. Assume the stability condition (2.1). There is a strong Markov process((X_t)_t0, (Pω)_ω∈ΩΛ) valued inΩ_Λ∪ {∂}whereΩ_Λis equipped with the weak convergence topology and∂is an extra isolated point to Ω_Λ, defined on some measurable space(W,B), such that

(i) For anyω∈ΩΛ,Pω-a.s.,X₀=ω,Xt=∂ (∀tζ ),t→Xt is right continuous and of pure jumps type on [0, ζ ), i.e., there are a sequence of stopping times 0=T₀< T₁< T₂<· · ·< Tn↑ζ (w.r.t. the natural filtration of(Xt)) such thatXt=XT_nfor allt∈ [Tn, Tn+1).

(ii) LetPtF (ω):=E^ω(F (Xt)1t <ζ):=E^Pω(F (Xt)1t <ζ). It is a semigroup of transition kernels onbFΛand for anyF ∈bFΛand anyω∈ΩΛ,PtF (ω)is continuous differentiable onR⁺and

d

dtP_tF (ω)=(L^η_ΛP_tF )(ω). (3.5)

(iii) Ifνis a nonnegative measure onΩ_Λsuch thatν([N_ΛL]) <+∞, ∀L∈Nandν(L^η_ΛF )=0, ∀F ∈D1(cf.

(3.2)), thenν=Cµ^η_Λfor some constantC.

(iv) Forµ^η_Λ-a.s.ω∈Ω_Λ,Pω(ζ= +∞)=1 andP_tF (ω)=P_t^Λ,ηF (ω),µ^η_Λ-a.s. for allt0,F ∈bFΛ. If we suppose moreover that there are constantsK₁, K₂0 (depending eventually onΛ) such that,

z

Λ

e^{−βni=1ϕ(xi−x)}dxK₁n+K₂, ∀n1,∀x₁, . . . , x_n∈Λ (3.6) thenPω(ζ= +∞)=1, ∀ω∈Ω_Λ, and for anyF∈bFΛand anyω∈Ω_Λ,P_tL^η_ΛF (ω)is finite and continuous on tand

d

dtPtF (ω)=Pt

L^η_ΛF

(ω); (3.7)

moreover if(P˜_t(ω,·)_t0is any semigroup of kernels onbFΛsuch that the Kolmogorov equation (3.5) holds for allF ∈Di (cf. (3.2)) and sup_Fu1 ˜P_tFuis bounded on any bounded intervalt∈ [0, T], thenP˜_t=P_t; where i=1 or 2.

Remarks 3.3. The last uniqueness allows us to say, without ambiguity, that(P_t)is the semigroup of kernels on bFΛgenerated byL^η_Λ. This result satisfies not only our probabilistic desire, but it is also technically crucial in the approximation procedure in the proof of our main results. Notice that this Markov process is constructed for every starting point. It is an easy application of the theory of jumps processes.

Let us also compare (3.5) and (3.7): they are usually known as to be equivalent for strongly continuous semigroup forF in domain of generator. But on the bad space “bFΛ”, (Pt)is not strongly continuous and the domain of its generator has many different definitions. In our case, (3.7) is stronger than (3.5). Moreover (3.7) implies bothPω(ζ = +∞)=Pt1(ω)=1, ∀ω(the non-explosion) and the last uniqueness, but (3.5) does not.

The linear growth condition (3.6) about the birth rate is adopted from [16], Proposition 4 where another condition, which is not satisfied here, is imposed too. It is (fortunately) satisfied for a family of important interaction functions such as those verifying (C1) or hard core condition. But we believe that the conservability (for any starting point) should be true under the only stability condition.