1.1 Transportation-information inequalities W

in PROBABILITY

TRANSPORTATION-INFORMATION INEQUALITIES FOR CONTIN- UUM GIBBS MEASURES

YUTAO MA¹

School of Mathematical Sciences&Lab. Math. Com. Sys., Beijing Normal University, 100875, Beijing China.

email: [email protected] RAN WANG²

School of Mathematics and Statistics, Wuhan University, 430072 Hubei, China.

email: [email protected] LIMING WU³

Institute of Applied Math., Chinese Academy of Sciences, 100190 Beijing, China and Laboratoire de Math. CNRS-UMR 6620, Université Blaise Pascal, 63177 Aubière, France.

email: [email protected]

SubmittedMarch 14, 2011, accepted in final formJuly 25, 2011 AMS 2000 Subject classification: 60E15. 60K35.

Keywords: transportation-information inequality, concentration inequality, Gibbs measure, Lya- punov function method.

Abstract

The objective of this paper is to establish explicit concentration inequalities for the Glauber dy- namics related with continuum or discrete Gibbs measures. At first we establish the optimal transportation-information W₁I-inequality for the M/M/∞-queue associated with the Poisson measure, which improves several previous known results. Under the Dobrushin’s uniqueness con- dition, we obtain some explicitW₁I-inequalities for Gibbs measures both in the continuum and in the discrete lattice. Our method is a combination of Lipschitzian spectral gap, the Lyapunov test function approach and the tensorization technique.

1 Introduction

1.1 Transportation-information inequalities W

I

LetX be a Polish space equipped with the Borelσ-fieldB, and letdbe a lower semi-continuous metric on the product spaceX × X (which does not necessarily generate the topology ofX). Let

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M1(X)be the space of all probability measures onX. Givenp≥1 and two probability measures µandνonX, we define the quantity

W_p,d(µ,ν) =inf Z Z

d(x,y)^pdπ(x,y) 1/p

,

where the infimum is taken over all probability measuresπon the product spaceX × X with marginal distributionsµandν(say coupling of(µ,ν)). This infimum is finite onceµandνbelong toM₁^p(X,d):={ν∈ M1(X); R

d^p(x,x₀)dν <+∞}, where x₀is some fixed point ofX. This quantity is commonly referred to be as the L^p-Wasserstein distance between µ andν. When d(x,y) =1_x6=y (the trivial metric), it is known that 2W_1,d(µ,ν) =kµ−νkT V, the total variation of the measureµ−ν.

Given a Dirichlet form E on L²(µ) := L²(X,µ) with domain D(E), let I(ν|µ) be the Fisher- Donsker-Varadhan information ofνwith respect toµ

I(ν|µ) = (E(p

f,p

f) if ν= fµ,p

f ∈D(E);

+∞ otherwise. (1)

Suppose that((X_t)t≥0,Pµ)is anX −valued reversible Markov process associated with the Dirichlet form(E,D(E)). We always assume that it is ergodic, i.e., ifh∈D(E)satisfiesE(h,h) =0, then h=0,µ−a.s..

Motivated by the concentration inequality for the empirical mean ¹_tRt

0g(X_s)ds for a familyA of bounded observables g, Guillinet al. [8]introduced the following transportation-information inequality

α

‚ sup

g∈A

ν(g)−µ(g)

Œ

≤I(ν|µ), ∀ν∈ M₁¹(X), (2) where α : R→ [0,+∞) is some non-decreasing and left-continuous function with α(0) = 0.

WhenA is the family of all bounded measurable andd-Lipschitzian functionsgwithkgk_Lip(d):= sup_x,y∈X ^|g(_d^x)−₍ ^g(y)|

x,y) ≤1, the previous inequality becomes by the Kantorovitch-Rubinstein duality, α(W_1,d(ν,µ))≤I(ν|µ),∀ν∈ M₁¹(X). (3) More precisely Guillinet al.[8]obtained

Theorem 1.1. ([8, Theorem 2.4] or [5, Theorem 2.2]) Let α : R → [0,+∞) be some non- decreasing and left-continuous function with α(0) =0. Given a family A of bounded measurable functions g (say g∈bB), the following properties are equivalent:

(a) The transportation-information inequality (2) holds.

(b) The following concentration inequality holds for each g∈ Aand any initial distributionνµ, Pν

‚1 t

Zt

0

g(X_s)ds> µ(g) +r

Œ

≤ kdν

dµk2e^−tα(r), ∀t,r>0. (4) Herek · k2is the norm of L²(µ).

In particular, the W₁I -inequality (3) is equivalent to Pν

‚1 t

Z t

0

g(X_s)ds> µ(g) +r

Œ

≤ kdν

dµk2e^−tα(r), ∀t,r>0 (5) for all g∈bBwithkgk_Lip(d)≤1.

Recently, Gao and the third named author [6] proved a tensorization result for the Wasserstein distance (see Lemma 4.2 below) and established the “dimension-free" transportation-information inequalitiesW_pI(p≥1)for the discrete Gibbs measure, under the Dobrushin’s uniqueness condi- tion ([3,4]).

1.2 Continuum Gibbs measure and generator of the Glauber dynamic

LetB(R^d)be the Borelσ−algebra onR^d(d≥1). We denote byBb(R^d)⊂ B(R^d)the collection of all bounded Borel sets. For each A∈ Bb(R^d), |A| denotes the Lebesgue measure of A. We consider, as configuration space, the setΩof all locally finite point measures onR^d, i.e.,

Ω:= (

ω=X

i

δx_i:ω(A)<∞for allA∈ Bb(R^d) )

.

with theσ−algebraF generated by the counting variablesN_A:ω→ω(A), whereA∈ Bb(R^d). Given theactivity z>0 (the name “activity" comes from Ruelle[18]), letPbe the law of Poisson point process onR^d with intensity measurezd x.

LettingΛbe a bounded open subset ofR^d, we consider also the finite volume configuration space

Ω_Λ:={ω∈Ω: supp(ω)⊂Λ} (6)

withσ−algebraF_Λgenerated by the functionN_A, whereAruns over the Borelσ−field ofΛand ω_Λ=P

x∈suppω∩Λδx. The image measureP_ΛofPbyω→ω_Λis the law of Poisson point process onΛwith intensity measurezd x. The configuration spaceΩ_Λ under the Prohorov metric, with the weak convergence topology, is a Polish space.

We say that an elementηofΩis a boundary condition onΛ^c, if η=

X+∞

k=1

δy_k, y_k∈Λ^c,k∈N.

Let ϕ:R^d →R⁺∪ {+∞}be a nonnegative measurable even function, representing a repulsive pair interaction. The finite volume Gibbs measure inΛfor a given boundary conditionη, at inverse temperatureβ >0, is given by

µ^η_Λ(dωΛ):= (Z_Λ^η)⁻¹exp¦

−βH_Λ^η(ωΛ)©

P_Λ(dωΛ) (7)

whereZ_Λ^ηis the normalization constant and H_Λ^η(ωΛ):=1

2 Z

Λ²

ϕ(x−y)ωΛ(d x)ωΛ(d y) + Z

Λ

ωΛ(d x) Z

Λ^c

ϕ(x−y)η(d y)

is the Hamiltonian inΛ. This is the mathematical model for continuous gas in statistical physics, see the book of Ruelle[18].

LetrF be the space of realF −measurable functions, andbFbe the space of thoseF∈rF which are moreover bounded. For any f ∈rF, following Picard[16], consider the difference operators

D⁺_xf(ω):=f(ω+δx)−f(ω),

D⁻_xf(ω):=1_x∈supp(ω)[f(ω−δx)−f(ω)]. (8) Recall thatD⁺_x plays the same role in the Malliavin calculus over the Poisson space as the Malliavin derivative on the Wiener space ([16,19]and references therein).

We shall work on the Glauber dynamic, which is formally generated by the pre-generator (see [1,12,20])

L_Λ^ηf(ωΛ) = Z

Λ

D⁻_xf(ωΛ)ωΛ(d x) +z Z

Λ

e^{−βD+xHηΛ(ωΛ)}D⁺_xf(ωΛ)d x, f ∈bFΛ. (9) It is easily checked that for all f,g∈bF_Λ

〈f,−L_Λ^ηg〉_µ^η_Λ= Z

ΩΛ

dµ^η_Λ(ωΛ) Z

Λ

D⁻_xf(ωΛ)D⁻_xg(ωΛ)ωΛ(d x)

= Z

ΩΛ

dµ^η_Λ(ωΛ) Z

Λ

e^{−βD+xHΛη(ωΛ)}D⁺_xf(ωΛ)D⁺_xg(ωΛ)zd x

=:E_Λ^η(f,g).

(10)

Then(−L_Λ^η,bFΛ)is a nonnegative definite, symmetric operator on L²(µ^η_Λ)(indeed it is essen- tially self-adjoint by Kondratiev and Lytvynov[12]). HenceE_Λ^ηis a closable form and its closure (E_Λ^η,D(E_Λ^η))is a Dirichlet form onL²(µ^η_Λ), generating a symmetric Markov semigroup(P_t^Λ,η)t≥0on L²(µ^η_Λ)such thatP_t^Λ,η1=1,µ^η_Λ−a.s., associated with a reversible Markov process((X^Λ,η_t )t≥0,P_µ^η_Λ) such that its sample paths arePµ^η_Λ−càdlàg. (P_t^Λ,η)t≥0is a strongly continuous semigroup of con- tractions onL²(µ^η_Λ), whose generator will be denoted by(L_Λ^η,D(L_Λ^η))(D(L_Λ^η)being its domain inL²(µ^η_Λ)).

This dynamic, as a classical probabilistic model in statistical mechanics, was first introduced and studied by Preston in[17]. Bertiniet al. [1]established the existence of a spectral gap, which is uniformly positive in the volume and boundary conditions, for the Glauber dynamic in the high temperature-low activity regime. The third named author[20]improved their work and extended to the hard core case by Poissonian approximation and Liggett’s M−εtheorem for lattice gas.

Kondratiev and Lytvynov[12]also obtained independently the spectral gap estimate in[20], by a different and simpler method.

In this paper we will always work on finite volume case for two reasons: 1) ourW₁I-inequality explodes in the infinite volume case even in the free case; 2) all interesting physical quantities (such as mean number of particles per unit volume) in the infinite volume case are calculated by approximation via finite volume ([18]).

Objective and organization.The objective of this paper is to establish some explicit transportation- information inequalityW₁I for the Glauber dynamic above related with the continuum Gibbs mea- sureµ^η_Λ, under the Dobrushin’s uniqueness condition (cf.[20])

D:=z Z

R^d

1−e^−βϕ(y)d y<1. (11)

As an interesting prelude to this end, we begin with theM/M/∞queue system in §2 (the jumps counterpart of the Ornstein-Uhlenbeck process), for which the optimal transportation-information inequality is obtained by means of the Lipschitzian spectral gap and Lyapunov test function method, improving some previous known results. In section 3, by generalizing the arguments of section 2, we obtain explicit W₁I inequality for the continuum Gibbs measureµ^η_Λ, under the Dobrushin’s uniqueness condition. Section 4 is devoted to the discrete spin system. For this model we establishW₁I-inequality by the tensorization technique in Gao and Wu[6].

2 M / M /∞ queue system

For the simplicity and the clarity of our presentation we begin with a simple model: M/M/∞

queue system. Letµbe the Poisson measure with meanλ >0 onNequipped with the Euclidean distanceρ. For each bounded measurable function f onN, consider the Dirichlet form

E(f, f) =λX

n∈N

(f(n+1)−f(n))²µ(n) (12) and the corresponding generator (with the convention f(−1):=f(0))

Lf(n) =λ(f(n+1)−f(n)) +n(f(n−1)−f(n)),∀n∈N.

It is an ideal model for a queue system with a number of servers much larger than the number of clients (such as in an automatic computer service center). It is well known that the Poincaré constantc_P equals 1, but the log-Sobolev inequality does not hold(see[19]).

Theorem 2.1. With respect to the Euclidean metricρ(x, y) =|x−y|onN, for the Poisson measure µwith meanλ >0, the following W₁I -inequality holds true:

W_1,ρ(ν,µ)≤2p

λI+I, ∀ν∈ M₁¹(N), (13) where I=I(ν|µ).This inequality is of the form(3)withα(r) = (p

λ+r−p

λ)²,which is optimal.

Remark 2.2. By Theorem 1.1 theW₁Iinequality (13) is equivalent to the following concentration inequality of Bernstein type: for any g:N→RwithkgkLip(ρ)=1 andµ(g) =0,

Pν

‚1 t

Zt

0

g(X_s)ds>2p λx+x

Œ

≤ kdν

dµk2e^{−t x}, ∀t,x>0

for any initial measureνµ. For the functiong₀(n):=n−λ, Gaoet al. [5]showed that ν(g₀)−µ(g₀)≤2p

λI+I, I:=I(ν|µ), ∀ν∈ M₁¹(N)

is optimal (our result is motivated by this fact, of course). A different but direct way to see the optimality of (13) is to take ν as the Poisson measure with parameter aλ where a > 1 : W_1,ρ(ν,µ) =λ(a−1)andI :=I(ν|µ) =λ[p

a−1]². Then (13) becomes equality for suchν. Remark 2.3. The optimal transportation-information inequality (13) is a definite improvement on the existing results on this model obtained by Gaoet al.[5], Gao and Wu[6]. However our proof is largely inspired by those general works. For other known concentration inequalities on this model, see Joulin[10], Liu and Ma[13], Joulin and Ollivier[11](for numerous other interesting models too). Chafaï [2]obtained theΦ-Sobolev inequalities (including the L¹-log-Sobolev inequalities) for theM/M/∞queue.

Proof of Theorem 2.1. Step 1. Lipschitzian spectral gap.First of all, we claim that k(−L)⁻¹k_Lip(ρ):= sup

kgkLip(ρ)=1k(−L)⁻¹gk_Lip(ρ)=1 (14)

for this model. The simplest way to see this known fact is to remark the following commutation relation between the generatorL and the difference operatorDG(n):= G(n+1)−G(n)(for a functionGonN):

DLG=LDG−DG.

Given any g:N→Rwithkgk_Lip(ρ)=1 andµ(g) =0, if−LG= g, then(1− L)DG=D g. By the resolvent of the infinitesimal generatorL, for any f withkfk_∞=1

k(1− L)⁻¹fk_∞=k Z∞

0

e^−sP_sf dsk_∞≤ Z∞

0

e^−sds=1, where it follows by taking f ≡1

k(1− L)⁻¹k_∞:= sup

kfk∞=1k(1− L)⁻¹fk_∞=1.

Hence

kGkLip(ρ)=kDGk∞≤ k −(1− L)⁻¹k∞· kD gk∞=1

and this inequality becomes equality ifD g=1 (i.e. g(n) =g₀(n) =n−λ). That shows the fact.

Step 2. Lyapunov function method.For (13) we may assume thatν= fµwithp

f ∈D(E)and I:=I(ν|µ) =E(p

f,p f)>0.

Given any function gonNwithµ(g) =0 andkgk_Lip(ρ)=1, letGbe the solution to the Poisson equation−LG=gwithµ(G) =0. For anyδ >0, we have (these few lines are the starting point of our approach)

ν(g)−µ(g) =〈g,f〉_µ=E(G,f)

= X∞

n=0

λµ(n)(G(n+1)−G(n))(f(n+1)−f(n))

≤ s∞

X

n=0

λµ(n)(p

f(n+1)−p f(n))²

· s∞

X

n=0

λµ(n)(G(n+1)−G(n))²(p

f(n+1) +p f(n))²

≤p I

s∞

X

n=0

λµ(n)

(1+δ)f(n+1) + 1+ 1 δ

f(n)

.

where the last inequality relies on the fact thatk(−L)⁻¹k_Lip(ρ)=1 in Step 1,kgk_Lip(ρ)=1 and the elementary inequality(x+y)²≤(1+δ)x²+ (1+δ⁻¹)y²for any x,y∈R,δ >0. The last term in the square root above, denoted byB, is (usingλµ(n) = (n+1)µ(n+1))

B= (1+1 δ)λ

X∞

n=0

µ(n)f(n) + (1+δ) X∞

n=0

(n+1)µ(n+1)f(n+1)

= X∞

n=0

µ(n)f(n)

(1+δ)n+ 1+ 1 δ

λ

.

We now employ the method of Lyapunov test function developed in Guillinet al.[8]for bounding the last term. The basic fact behind this approach is : for any functionV ≥1, if−^L_V^V is bounded from below, then

Z

−LV

V dν≤I(ν|µ), ∀ν∈ M₁¹(N). (15) That was proved in[8, Lemma 5.6]for general reversible Markov processes. Our task now is to find a good functionV such that

(1+δ)n+ 1+1 δ

λ≤ −aLV

V (n) +b (16)

for two positive constantsa,b, and (15) will imply B≤aI+b.

TakingV(n) =κⁿfor some constantκ >1, the previous inequality holds witha= (1+δ)κ/(κ−1) andb= (1+δ)κ+ (1+_δ¹)λby simple algebra. Asδ >0,κ >1 are arbitrary, we get

ν(g)−µ(g)≤p I inf

κ>1,δ>0

r

I(1+δ)κ/(κ−1) + (1+δ)κ+ (1+1 δ)

λ

=I+2p λI where the equality is attained atκ=1+p

I/λandδ=κ⁻¹. Therefore the desired transportation- information inequality (22) follows by taking the supremum over all functionsgsuch thatµ(g) = 0 andkgk_Lip(ρ)=1.

Remark 2.4. Given an increasing function w on N which induces a metricρw as ρw(x,y) =

|w(x)−w(y)|, the Lipschitzian norm of the Poisson operator k(−L)⁻¹k_Lip(ρw) is known for a general birth-death process (i.e. Lf(n) =b_n(f(n+1)−f(n))+a_n(f(n−1)−f(n))with the birth rateb_n>0 for anyn≥0 and the death ratea₀=0,a_n>0 for anyn≥1), due to Liu and the first named author[13]. In fact, consider the corresponding Poisson equation

− Lϕ=w−µ(w), (17)

which admits a unique and explicit solutionϕwith zero mean ([13]). Theorem 2.1 in[13]says that

k(−L)⁻¹k_Lip(ρw)=kϕkLip(ρw). (18)

This fact together with the Lyapunov test function method above can produce theW₁I inequality for quite general birth-death processes. Notice also that for w(n) = g₀(n) =n−λ, the previous identification (18) ofk(−L)⁻¹kLip(ρw)gives the result of Step 1 above, forϕ=g₀.

3 W

I -inequality for continuum Gibbs measure

In this section we generalize the arguments in §2 to study theW₁I-inequality for the continuum Gibbs measureµ^η_Λ.

3.1 Lipschitzian norm of (−L

)

We consider the total variation metricdonΩ_Λ: for anyω,ω⁰∈Ω_Λ,

d(ω,ω⁰) =kω−ω⁰kTV. (19)

Given any functionalF∈rF_Λ, we callF is Lipschitzian with respect todif kFk_Lip(d):= sup

ω6=ω⁰

|F(ω)−F(ω⁰)|

d(ω,ω⁰) <∞.

By Lemma 2.2 in[14],

kFk_Lip(d)= sup

x∈Λ,ωΛ∈ΩΛ

|D⁺_xF(ω_Λ)|. (20) Denote byC⁰_Lipthe set of functionalsF∈rF_ΛwithkFk_Lip(d)<∞andµ^η_Λ(F) =0.

Recall the usual Lipschitzian norm of(−L_Λ^η)⁻¹onC_Lip⁰ : k(−L_Λ^η)⁻¹k_Lip(d)= sup

kgkLip(d)≤1k(−L_Λ^η)⁻¹gk_Lip(d). (21)

First we give a key lemma which provides a sharp estimate of the Lipschitzian norm of(−L_Λ^η)⁻¹ and which is essentially due to the third named author[20].

Lemma 3.1. Suppose that the Dobrushin’s uniqueness condition holds, i.e., D=z

Z

R^d

(1−e^−βϕ(x))d x<1.

We have

k(−L_Λ^η)⁻¹kLip(d)≤ 1 1−D. Proof. By Theorem 5.1 in[20], for any functionalF∈bFΛ∩C_Lip⁰ ,

kP_t^Λ,ηFkLip(d)≤e^−(1−D)tkFkLip(d). Hence

k(−L_Λ^η)⁻¹Fk_Lip(d)= sup

x∈Λ,ωΛ∈ΩΛ

|D⁺_x(−L_Λ^η)⁻¹F(ω_Λ)|

= sup

x∈Λ,ωΛ∈ΩΛ

|D⁺_x Z∞

0

P_t^Λ,ηF(ωΛ)d t|

≤ Z∞

0

sup

x∈Λ,ωΛ∈ΩΛ

|D⁺_xP_t^Λ,ηF(ω_Λ)|d t

≤ Z∞

0

e^−(1−D)td tkFkLip(d)= 1

1−DkFkLip(d).

For general F ∈C_Lip⁰ , let F_n= (F∧n)∨(−n), we can approximate F byF_n−µ^η_Λ(F_n), then the desired result follows.

3.2 W

I -inequality

The main result of this paper is the following theorem

Theorem 3.2. For the continuum Gibbs measureµ^η_Λ given in (7) with the nonnegative even pair interactionϕ, suppose that the Dobrushin’s uniqueness condition holds, i.e.,

D=z Z

R^d

(1−e^−βϕ(x))d x <1.

Then the transportation-information inequality below holds W_1,d(ν,µ^η_Λ)≤ 1

1−D

I+2p z|Λ|I

, ∀ν∈ M₁¹(ΩΛ) (22)

where I = I(ν|µ^η_Λ)is the Fisher-Donsker-Varadhan’s information related withE_Λ^ηgiven in (10) and the metric d is the total variation metric defined in(19).

Remark 3.3. Whenϕ=0 (no interaction case), the inequality (22) is optimal. Since in this case D=0 andN_Λ(X_t)is just theM/M/∞queue withλ=z|Λ|and then Theorem 2.1 guarantees its optimality.

Remark 3.4. Since the Lipschitzian norm w.r.t. d of F(ω) = _|Λ|¹N_Λ(ω) (the mean number of particles per unit volume ofω) is 1/|Λ|, hence by (22) and Theorem 1.1 we have for allt,r>0 and initial distributionνµ^η_Λ,

Pν

‚ 1 t|Λ|

Z t

0

N_Λ(X_s)ds−µ^η_Λ(N_Λ)

|Λ| >r

Œ

≤ kdν dµk2exp



−t|Λ|hp

z+ (1−D)r−p zi2‹

. This concentration inequality shows that the Glauber dynamics here is a very efficient tool for estimatingµ^η_Λ(N_Λ)/|Λ|.

The same argument as Theorem 2.1, namely estimatingk(−L_Λ^η)⁻¹kLip(d)plus Lyapunov condition (16), works for proving Theorem 3.2. Then with Lemma 3.1, it remains to find some good function V such that Lyapunov condition is verified. For this aim, we begin by introducing the generalized domainDe(L_Λ^η).

A continuous function his said to be in the µ^η_Λ−extended domain De(L_Λ^η)of the generator of the Markov process((X^Λ,η_t ),µ^η_Λ)if there is some measurable functiongsuch thatRt

0|g|(X_s^Λ,η)ds<

+∞,µ^η_Λ−a.s., and

M_t:=h(X^Λ,η_t )−h(X₀^Λ,η)− Zt

0

g(X_s^Λ,η)ds

is a local µ^η_Λ-martingale. It is obvious that g is uniquely determined up toµ^η_Λ−equivalence. In such case one writesh∈De(L_Λ^η)andL_Λ^ηh=g.

Lemma 3.5. There exists a function V:ΩΛ→[1,∞)inDe(L_Λ^η)such that for anyδ >0, (1+δ)N_Λ(ωΛ) + (1+ 1

δ)z|Λ| ≤ −aL_Λ^ηV(ωΛ)

V(ωΛ) +b, ωΛ∈ΩΛ

a= (1+δ) κ

κ−1, b= (1+δ)κ+ (1+ 1 δ)

z|Λ|.

(23)

Proof.For a constantκ >1, takeV(ωΛ) =κ^NΛ(ωΛ). Then

−L_Λ^ηV(ωΛ)

V(ω_Λ) = (1−κ⁻¹)N_Λ(ωΛ)−(κ−1)z Z

Λ

e^{−βD+xHηΛ(ωΛ)}d x. Asϕ≥0, we see that (23) holds.

Proof of Theorem 3.2In order to establish (22), we may assume thatν= fµ^η_Λwithp

f ∈D(E) andI=I(ν|µ^η_Λ)>0.

Given anyg∈C_Lip⁰ withkgk_Lip(d)=1, letG= (−L_Λ^η)⁻¹g. By Cauchy-Schwarz inequality and (10), we have

ν(g)−µ^η_Λ(g) =〈g,f〉_µ^η_Λ=〈−L_Λ^ηG,f〉_µ^η_Λ=E_Λ^η(G,f)

= Z

ΩΛ

dµ^η_Λ Z

Λ

e^{−βD+xHηΛ(ωΛ)}D⁺_xG(ωΛ)D⁺_xf(ωΛ)zd x

≤p I

sZ

ΩΛ

dµ^η_Λ Z

Λ

e^{−βD+xHηΛ(ωΛ)}(D⁺_xG(ωΛ))² p

f(ωΛ+δx) +p

f(ωΛ)2

zd x.

We treat the term in the last square root as in the proof of Theorem 2.1, Z

ΩΛ

dµ^η_Λ Z

Λ

e^{−βD+xHΛη(ωΛ)}(D⁺_xG(ωΛ))² p

f(ωΛ+δx) +p

f(ωΛ)2

zd x

≤ Z

ΩΛ

dµ^η_Λ Z

Λ

e^{−βD+xHΛη(ωΛ)}(D⁺_xG(ω_Λ))² (1+δ)f(ω_Λ+δx) + (1+1

δ)f(ω_Λ) zd x

≤ kGk²_Lip(d) Z

ΩΛ

dµ^η_Λ Z

Λ

e^{−βD+xHηΛ(ωΛ)}

(1+δ)f(ω_Λ+δx) + (1+1 δ)f(ω_Λ)

zd x

=kGk²_Lip(d) Z

ΩΛ

f(ω_Λ)dµ^η_Λ

(1+δ) Z

Λ

e^{−βD+xHΛη(ωΛ−δx)}ω_Λ(d x) + (1+ 1 δ)

Z

Λ

e^{−βD+xHηΛ(ωΛ)}zd x

≤ 1

(1−D)² Z

ΩΛ

(1+δ)N_Λ(ω_Λ) + (1+ 1 δ)z|Λ|

ν(dω_Λ)

≤ 1

(1−D)² Z

ΩΛ

−aL_Λ^ηV(ωΛ) V(ωΛ) +b

ν(dωΛ)

≤ 1

(1−D)²

aI+b

,

whereδ >0 is arbitrary, the third crucial equality is due to the duality formula in the Malliavin calculus on the Poisson space ([16]) saying for any measurable functionalF:ΩΛ×Λ7→[0,+∞],

Z

ΩΛ

dµ^Λ_η Z

Λ

ωΛ(d x)F(ωΛ,x) = Z

ΩΛ

dµ^η_Λ Z

Λ

exp{−βE(x,ωΛ)}F(ωΛ+δx,x)zd x with

E(x,ω_Λ):= (R

Λϕ(x−y)ωΛ(d y), ifR

Λ|ϕ(x−y)|ωΛ(d y)<∞;

+∞, otherwise ,

the fourth inequality is true by the Lipschitzian spectral gap estimate in Lemma 3.1, the last but second inequality is an application of (23) with constantsa,bgiven there and the last one follows by[8, Lemma 5.6]as recalled in (15).

Now by the same optimization procedure over κ >1,δ >0 as in the proof of Theorem 2.1, we obtain

ν(g)−µ^η_Λ(g)≤ 1 1−D

I+2p

z|Λ|I

where the desired result (22) follows, sinceginC_Lip⁰ withkgkLip(d)=1 is arbitrary.

4 W

I -inequality for the discrete spin system

The discrete spin system and the Dobrushin’s interdependence coefficient. Let T be a finite subset ofZ^dandγ:Z^d×Z^d→R⁺be a nonnegative interaction function satisfyingγi j=γji and γii=0 for alli,j∈Z^d. The Gibbs measure onN^T with boundary condition(x_k)k∈T^c is defined by

µT(d x_T|x) = e^{−12P{i,j}∩T6=;γi jxixj}

Z(x_Tc) Πi∈Tσ_λi(d x_i) (24) where¦

σλi(·)©

i∈Z^d are the given Poisson measures onNwith meansλi>0 _i∈Zd, andZ(x_Tc)is the normalization factor. When T ={i},µT(d x_T|x)is simply denoted byµi:=µi(d x_i|x), which is the conditional distribution ofx_i knowing(x_j)j6=i. In the present case,µi(d x_i|x)is the Poisson distributionP(λie^{−Pj6=iγi jxj})with parameterλie^{−Pj6=iγi jxj}.

The purpose of this section is to propose another approach : tensorization technique, to establish theW₁I-inequality for the discrete Gibbs measureµT(d x_T|x)from (13) for Poisson measure. For this dependent tensorization, the key tool is the Dobrushin’s interdependence matrixC:= (c_{i j})i,j∈T

w.r.t. the Euclidean metricρonN, defined by

c_{i j}= sup

x=x⁰offj

W_1,ρ

µi(d x_i|x),µi(d x⁰_i|x⁰)

|x_j−x⁰_j| , ∀i,j∈Z^d (25) (obviouslyc_ii=0). Then the Dobrushin’s uniqueness condition[3,4]is

D:=sup

j∈T

X

i∈T

c_{i j}<1. (26)

The Dobrushin’s interdependence coefficientc_{i j} can be easily identified for this model.

Lemma 4.1. ([14, Lemma 3.1])For i6= j inZ^d,

c_{i j}=λi(1−e^{−γi j}). (27)

The transportation-information inequality W₁I for the discrete spin system. Consider the metric

d_l1(x,y):=X

i∈T

|x_i−y_i|, ∀x,y∈N^T (28) onN^T. The following disintegration ofW₁-metric is our starting point.

Lemma 4.2. (Gao-Wu[6, Theorem 3.1])LetµTbe the discrete Gibbs measure given in (24). Assume the Dobrushin’s uniqueness condition

D=sup

j∈T

X

i∈T

λi(1−e^{−γi j})<1.

Then for allνT∈ M₁¹(N^T), W_1,d

l1(νT,µT)≤ 1

1−DE^νTX

i∈T

W_1,ρ(νi,µi) (29)

whereνiis the conditional distribution of x_iknowing(x_j)j6=i.

We now introduce the Glauber dynamic. For eachi∈T andˆx_i:=x_T\{i}fixed, consider the site’s Dirichlet form associated with the Poisson measureµi(d x_i|x):

Ei(f,f):=λie⁻

P

j6=iγi jx_j X

x_i∈N

(f(x_i+1)−f(x_i))²µi(x_i|x), D(Ei):={f ∈L²(µi);Ei(f,f)<+∞}.

which corresponds to the M/M/∞queue with parameterλ = λie⁻

P

j6=iγi jx_j

. Define the global Dirichlet formETonT by

D(ET):= (

g∈L²(µT): g_i∈D(Ei), forµT−a.e.ˆx_iand Z

N^T

X

i∈T

Ei(g_i,g_i)dµT<+∞

) , ET(g,g):=

Z

N^T

X

i∈T

Ei(g_i,g_i)dµT, g∈D(ET) (30) whereg_i(x_i):=g(x_i,xˆ_i)withˆx_i:=x_T\{i}fixed.

The following additivity property of the Fisher information will be needed.

Lemma 4.3. (Guillin et al. [8, Lemma 2.12]) Let νT,µT be probability measures on N^T such that I_T(νT|µT)< +∞, and let µi,νi be the conditional distributions of x_i knowing xˆ_i underµ,ν respectively. Then

I_T(νT|µT) =E^νTX

i∈T

I_i(νi|µi) (31)

where I_i(νi|µi)is the Fisher-Donsker-Varadhan information related to the Dirichlet form(Ei,D(Ei)). Proof. For the completeness we reproduce the proof. Let f = dνT/dµT. Then dνi/dµi =

f/µi(f) = f_i/µi(f_i),νT−a.s. wheref_i(x_i) = f(x_i,ˆx_i). ForµT−a.e. ˆx_i fixed, I_i(νi|µi) =Ei



 È f_i

µi(f_i), È f_i

µi(f_i)



= 1 µi(f_i)Ei(p

f_i,p f_i) (forµi(f_i)is constant withxˆ_ifixed). We obtain

E^νTX

i∈T

I_i(νi|µi) =E^µTfX

i∈T

1 µi(f_i)Ei(p

f_i,p

f_i) =E^µTX

i∈T

Ei(p f_i,p

f_i), which completes the proof.

We can now state the main result of this section.

Theorem 4.4. Let µT be the Gibbs measure given in (24). Assume the Dobrushin’s uniqueness condition

D=sup

j∈T

X

i∈T

λi(1−e^{−γi j})<1. (32) Then for anyνT∈ M₁¹(N^T,d_l1), it holds that

W_1,d

l1(νT,µT)≤ 1

1−D







 2

v u u t

X

i∈T

λi

! I+I









(33)

where I=I_T(νT|µT).

Proof of Theorem Gibbs By Theorem 2.1, we know that for eachµi=µi(·|x), it holds that W_1,ρ(νi,µi)≤2p

λiI_i(νi|µi) +I_i(νi|µi),∀νi∈ M₁¹(N). (34) Under the Dobrushin’s uniqueness condition (32), by Lemma 4.2 and (34), we have by Cauchy- Schwarz inequality,

(1−D)W_1,d

l1(νT,µT) ≤ E^νTX

i∈T

W_1,ρ(νi,µi)

≤ 2E^νTX

i∈T

pλiI_i(νi|µi) +E^νTX

i∈T

I_i(νi|µi)

≤ 2 rX

i∈T

λi·E^νTX

i∈T

I_i(νi|µi) +E^νTX

i∈T

I_i(νi|µi) where the desired inequality follows by Lemma 4.3.

Remark 4.5. The inequality (33) in the free case is again sharp. Indeed ifγi j =0 (no interaction case) it is optimal, as seen by applying Theorem 2.1 to the functionP

i∈Tx_i.

References

[1] L. Bertini, N. Cancrini and F. Cesi. The spectral gap for a Glauber-type dynamic in a continuous gas.Ann. Inst. Henri Poincaré Probab. Stat.,38(1): 91-108, 2002.MR1899231

[2] D. Chafaï. Binomial-Poisson inequalities andM/M/∞queue,ESAIMS Probab. Stat.,10: 317- 339, 2006.MR2247924

[3] R. L. Dobrushin. The description of a random field by means of conditional probabilities and condition of its regularity.Theory Probab. Appl.,13: 197-224, 1968.MR0231434

[4] R. L. Dobrushin. Prescribing a system of random variables by conditional distributions.Theory Probab. Appl.,15: 458-486, 1970.

[5] F. Q. Gao, A. Guillin and L. M. Wu. Berstein type’s concentration inequalities for symmetric Markov processes. To appear inSIAM: Theory of Probability and its Applications.2011.

[6] F. Q. Gao and L. M. Wu. Transportation information inequalities for Gibbs measures. Preprint, 2007.

[7] A. Guillin., A. Joulin, C. Leonard and L. M. Wu. Transportation-information inequalities for Markov processes (III): processes with jumps. Preprint, 2008.

[8] A. Guillin, C. Léonard, L. M. Wu and N. Yao. Transportation-information inequalities for Markov processes. Probab. Theory Relat. Fields, 144: 669-695, 2009.MR2496446

[9] A. Guillin, C. Léonard, F.-Y. Wang and L. M. Wu. Transportation-information inequalities for Markov processes (II): relations with other functional inequalities. Available online at http://arxiv.org/abs/0902.2101.

[10] A. Joulin. A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature.Bernoulli,15(2): 532–549, 2009.MR2543873

[11] A. Joulin and Y. Ollivier. Curvature, concentration, and error estimates for Markov chain Monte Carlo.Ann. Probab.38(6): 2418-2442, 2010.MR2683634

[12] Yu. Kondratiev and E. Lytvynov. Glauber dynamics of continuous particle systems.Ann. Inst.

Henri Poincaré Probab. Stat.,41: 685-702, 2005.MR2144229

[13] W. Liu and Y.T. Ma. Spectral gap and convex concentration inequalities for birth-death pro- cesses.Ann. Inst. Henri Poincaré Probab. Stat.,45(1): 58–69 (2009).MR2500228

[14] Y. T. Ma, S. Shen, X. Y. Wang and L. M. Wu. Transportation inequalities: from Poisson to Gibbs measures.Bernoulli. 17(1): 155-169, 2011.MR2797986

[15] Y. Ollivier. Ricci curvature of Markov chains on metric spaces.J. Funct. Anal.,256(3): 810–

864, 2009.MR2484937

[16] J. Picard. Formules de dualité sur l’espace de Poisson.Ann. Inst. Henri Poincaré Probab. Stat., 32(4): 509–548, 1996.MR1411270

[17] C. Preston. Spatial birth-and-death processes, in: Proceedings of the 40th Session of the International Statistical Institute (Warsaw), vol. 2, inBull. Inst. Internat. Statist., 46: 371-391, 1975.MR0474532

[18] D. Ruelle,Statistical Mechanics: Rigorous Results, Benjamin, 1969MR0289084

[19] L. M. Wu. A new modified logarithmic Sobolev inequality for Poisson processes and several applications.Probab. Theory Relat. Fields,118: 427-438, 2000.MR1800540

[20] L. M. Wu. Estimate of the spectral gap for continuous gas.Ann. Inst. Henri Poincaré Probab.

Stat.,40: 387-409, 2004.MR2070332