The law of large numbers and the law of the iterated ...

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The law of large numbers and the law of the iterated logarithm for infinite dimensional interacting diffusion processes Byron Schmuland and Wei Sun June 22, 2000 Abstract The classical Dirichlet form given by the intrinsic gradient on Γ R d is associated with a Markov process consisting of a countable family of interacting diffusions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion. The quasi-sure analysis of Dirichlet forms is used to find excep- tional sets of configurations for this Markov process. We consider large scale properties of the configuration and show that, for quite general measures, the process never hits those unusual configurations that violate the law of large numbers. Furthermore, for certain Gibbs measures, which model random particles in R d that interact via a po- tential function, we show, for d 3, that the process never hits those unusual configurations that violate the law of the iterated logarithm. AMS (1991) subject classification 60H07, 31C25, 60G57, 60G60 Keywords: configuration space, Dirichlet form, law of large numbers, law of the iterated logarithm 1 Introduction In a pair [3, 4] of fundamental papers in 1998, Albeverio, Kondratiev, and ockner began a study of analysis and geometry on configuration space. 1

Transcript of The law of large numbers and the law of the iterated ...

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The law of large numbers and the law ofthe iterated logarithm for infinite

dimensional interacting diffusion processes

Byron Schmuland and Wei Sun

June 22, 2000

Abstract

The classical Dirichlet form given by the intrinsic gradient on ΓRd

is associated with a Markov process consisting of a countable familyof interacting diffusions. By considering each diffusion as a particlewith unit mass, the randomly evolving configuration can be thoughtof as a Radon measure valued diffusion.

The quasi-sure analysis of Dirichlet forms is used to find excep-tional sets of configurations for this Markov process. We considerlarge scale properties of the configuration and show that, for quitegeneral measures, the process never hits those unusual configurationsthat violate the law of large numbers. Furthermore, for certain Gibbsmeasures, which model random particles in Rd that interact via a po-tential function, we show, for d ≤ 3, that the process never hits thoseunusual configurations that violate the law of the iterated logarithm.

AMS (1991) subject classification 60H07, 31C25, 60G57, 60G60

Keywords: configuration space, Dirichlet form, law of large numbers, lawof the iterated logarithm

1 Introduction

In a pair [3, 4] of fundamental papers in 1998, Albeverio, Kondratiev, andRockner began a study of analysis and geometry on configuration space.

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(This work had been previously announced in [1, 2] and anticipated inthe 1996 papers by Osada [13] and Yoshida [25].) One of the features of[3, 4] was the construction of a configuration-valued Markov process usinga Dirichlet form based on an intrinsic gradient defined on configurationspace.

In this paper, we look at some sample path properties of the Albeverio-Kondratiev-Rockner process in the case where the underlying manifold isd-dimensional Euclidean space. For quite general stationary measures forthe process, we will show that the associated Markov process never hitsconfigurations that violate the law of large numbers. Furthermore, for aRuelle measure µ with small activity parameter z the law of the iteratedlogarithm (Proposition 4) holds for µ-almost every configuration γ ∈ ΓRd .In dimensions d less than or equal to 3, we also show that the associatedMarkov process never hits configurations that violate the law of the iteratedlogarithm. We do so by proving that these sets of unusual configurationsare exceptional sets for the Dirichlet form.

2 Classical Dirichlet forms on configuration space

In this section we recall some of the basic definitions and properties aboutthe classical Dirichlet forms on configuration spaces. For more detaileddefinitions and fuller explanations we refer the reader to [3, 4, 9, 16].

The space of locally finite configurations in Rd is defined by

ΓRd := fγ ⊂ Rd : jγ ∩Kj < ∞ for every compact Kg.

A configuration γ will be identified with the Radon measure∑

x∈γ εx. Thespace ΓRd will be given the topology of vague convergence of measures, andmeasures on ΓRd are defined on the corresponding Borel sets B(ΓRd).

For f ∈ C0(Rd) we let 〈f, γ〉 be the integral of f with respect to themeasure γ, that is, 〈f, γ〉 =

x∈γ f(x). Define

FC∞b := f u : u(γ) = g(〈f1, γ〉, 〈f2, γ〉, . . . , 〈fn, γ〉)

for some fi ∈ C∞0 (Rd) and g ∈ C∞

b (Rn) g .

For u ∈ FC∞b , we define the gradient ∇Γu at the point γ ∈ ΓRd as an

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element of the “tangent space” Tγ(ΓRd) := L2(Rd → Rd; γ) by the formula

(

∇Γu)

(γ;x) :=n∑

i=1

∂g

∂xi

(〈f1, γ〉, 〈f2, γ〉, . . . , 〈fn, γ〉)∇fi(x).

Here ∇ refers to the usual gradient on Rd. It is not hard to prove that ∇Γuis well-defined, even though the representation of u as a cylinder functionis not unique.

Definition 1 For u, v ∈ FC∞b define the square field (u, v)1 as the real-

valued function on ΓRd given by

(u, v)(γ) := 〈∇Γu,∇Γv〉Tγ(ΓRd )

=

Rd

〈(∇Γu)(γ;x), (∇Γv)(γ;x)〉Rd γ(dx).

We will often use the abbreviation (u) := (u, u).In this section we fix a probability measure µ on ΓRd with Radon mean,

that is,∫

ΓRd

γ(K)µ(dγ) < ∞ for all compact K ⊂ Rd. (µ.1)

Furthermore, assume that

∇Γu = ∇Γv µ-a.e. if u, v ∈ FC∞b such that u = v µ-a.e. (µ.2)

Definition 2 For u, v ∈ FC∞b define the pre-Dirichlet form by

E(u, v) := 1

2

ΓRd

(u, v)(γ)µ(dγ).

Let F∞,µb denote the set of µ-equivalence classes determined by FC∞

b .Suppose that (E ,F∞,µ

b ) is closable on L2(ΓRd ;µ) and denote the closure by(E , D(E)). We refer to [3, 4, 9, 16] for concrete examples. By [9], (E , D(E))is then a symmetric, quasi-regular and local Dirichlet form. The quasi-regularity and locality of (E , D(E)) has been proven for certain cases byYoshida [25], and in general by Ma and Rockner [9] but since ΓRd is not

1This notation is based on the Chinese character Tian, which means ‘field’.

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complete with respect to the vague topology it is necessary to use thecompleted state space

ΓRd := fZ+ ∪ f+∞g-valued Radon measures on Rdg.

Since ΓRd ⊂ ΓRd and B(ΓRd) ∩ ΓRd = B(ΓRd), we can consider µ as a mea-sure on (ΓRd ,B(ΓRd)) and correspondingly (E , D(E)) as a Dirichlet form onL2(ΓRd ;µ). The associated strong Markov process (Xt)t≥0 has vaguely con-tinuous sample paths since (E , D(E)) is a local form [8, Chapter V, Theorem1.11]. Note that by [19, 10], if d ≥ 2, the set ΓRd \ ΓRd is E-exceptional fora large class of measures µ on ΓRd .

We recall the following results from Dirichlet form theory. Lemma 1(see [23]) is used to prove that certain sets are E-exceptional while Lemma2 gives us the interpretation in terms of the sample paths of (Xt)t≥0.

Lemma 1 Let un ∈ D(E) be a sequence of E-quasi-continuous functionswith supn E(un, un) < ∞ and un → u pointwise. Then u is an E-quasi-continuous function, in particular, for µ-almost every γ ∈ ΓRd,

Pγ (t → u(Xt) is continuous ) = 1.

If u is µ-square integrable, then u ∈ D(E).

Lemma 2 A set N ∈ B(ΓRd) is E-exceptional if and only if, for µ-almostevery γ ∈ ΓRd,

Pγ (Xt ∈ N for some 0 ≤ t < ∞) = 0.

We sometimes refer to proofs of exceptionality as capacitary since N is E-exceptional if and only if Cap(N) = 0 for a suitably defined capacity Capon ΓRd [8, Chapter III, Theorem 2.11].

3 Law of large numbers on configuration space

Define an Abelian group (Tr)r∈Zd of automorphisms on ΓRd by Trγ(G) =γ(G − r) for any bounded Borel G. In this section we fix a probabilitymeasure µ on ΓRd which is invariant with respect to (Tr)r∈Zd and satisfiesthe conditions (µ.1) and (µ.2) in Section 2.

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For l > 0 let Cl be the cube [−l, l]d, and for every r ∈ Zd let Qr =r + [0, 1]d. Denote the Lebesgue measure of any Borel subset G of Rd byjGj. Define

LLN :=

{

γ ∈ ΓRd : limn∈N

γ(Cn)

jCnjexists

}

=

{

γ ∈ ΓRd : liml∈R+

γ(Cl)

jCljexists

}

.

We first recall in Proposition 1 the well-known fixed time law of largenumbers ([11, Proposition 4.23]), while Proposition 2 gives the full capaci-tary version.

Proposition 1 The law of large numbers holds for µ-almost every γ,that is, µ(LLN c) = 0.

Proof. The family fγ(G) : G ∈ B(Rd), boundedg is trivially an additivecovariant spatial process in the sense of [11]. Therefore, by [11, Proposition4.23],

limn→∞

γ(Cn)

jCnj= Eµ(γ(Q0) j H), µ−a.e.,

where H denotes the σ-algebra of (Tr)r∈Zd-invariant sets in B(ΓRd). tu

Proposition 2 The set LLN c is E-exceptional.

Proof. For every n ≥ 3, let ψn be a smooth function satisfying 1[−n+2,n−2] ≤ψn ≤ 1[−n,n], and jψ′

nj ≤ 1. Define a continuous element of D(E) by

un(γ) := 〈∏d

i=1ψn(xi), γ〉/jCnj.

Taking the limsup through the inequality

jCn−2jjCnj

γ(Cn−2)

jCn−2j≤ un(γ) ≤

γ(Cn)

jCnj,

we find that u(γ) := lim supn un(γ) = lim supn γ(Cn)/jCnj.Bounding the square field gives

(un)(γ) ≤ dγ(Cn)

jCnj2. (1)

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Since Cn is an increasing sequence, there exists a sequence (rm)m∈N in Zd

so Cn = ∪jCnjm=1Qrm for n ∈ N. Then

χCn

jCnj2≤

jCnj∑

m=1

χQrm

m2≤

∞∑

m=1

χQrm

m2,

so that∫

ΓRd

supn∈N

γ(Cn)

jCnj2µ(dγ) =

ΓRd

supn∈N

Rd

χCn(x)

jCnj2γ(dx)µ(dγ)

≤∫

ΓRd

Rd

supn∈N

χCn(x)

jCnj2γ(dx)µ(dγ)

≤∫

ΓRd

Rd

∞∑

m=1

χQrm(x)

m2γ(dx)µ(dγ)

=

ΓRd

∞∑

m=1

γ(Qrm)

m2µ(dγ)

=∞∑

m=1

ΓRd

γ(Qrm)

m2µ(dγ)

=∞∑

m=1

µ(Q0)

m2

< ∞.

Let us denote the random variable X∗(γ) := supn∈N γ(Cn)/jCnj2. Forfixed n ∈ N let Aj = fn, n+ 1, . . . , n+ jg so that

supk≥n

uk(γ) = supj

supk∈Aj

uk(γ).

Now for each j ∈ N, supk∈Ajuk ∈ D(E) and is E-quasi-continuous. Repeated

use of the inequality (u∨ v) ≤ (u)∨ (v), for u, v ∈ D(E) combined withthe bound (1) gives (supk∈Aj

uk) ≤ dX∗, and so

supj

E

(

supk∈Aj

uk, supk∈Aj

uk

)

≤ d

ΓRd

X∗(γ)µ(dγ) < ∞.

Applying Lemma 1, we see that the pointwise limit supk≥n uk belongs toD(E) and is E-quasi-continuous. In addition, the bound for the square

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field also carries over; (supk≥n uk) ≤ dX∗. Applying the same argumentto the decreasing sequence (supk≥n uk)n∈N, we find that the pointwise limitu belongs to D(E) and is E-quasi-continuous.

A parallel argument shows that v(γ) := lim infn→∞ γ(Cn)/jCnj is alsoE-quasi-continuous. Since the two E-quasi-continuous functions u and vagree µ-almost everywhere, they must agree except on an E-exceptional set[8, Chapter IV, Proposition 3.3]. tuRemark. In the above argument we only use the translation invariantproperty of µ to ensure that supr∈Zd Eµ(γ(Qr)) < ∞. So one can expectthat the capacitary version of the law of large numbers also holds for moregeneral measures only if the fixed time law of large numbers holds. Inparticular, this argument also gives the capacitary version of the law of largenumbers for (mixed) Poisson point processes in Rd. This improves the result[22, Proposition 5] by removing the assumption that

R+z log+(z)λ(dz) <

∞.

4 Gibbs measures on configuration space

In this section we give some of the preliminaries on Gibbs measure we willneed to prove our results. Let σ be a measure on Rd that has a density ρwith respect to Lebesgue measure satisfying ρ > 0 almost everywhere, andρ1/2 ∈ H1,2

loc (Rd). Here H1,2

loc (Rd) denotes the local Sobolev space of order 1

in L2loc(R

d;m). The Poisson measure πσ with intensity measure σ is theprobability measure on ΓRd characterized by:

ΓRd

exp(〈f, γ〉)πσ(dγ) = exp

(∫

Rd

(ef(x) − 1)σ(dx)

)

,

for f ∈ C0(Rd). A mixed Poisson measure is given by:

µ :=

R+

πzσ λ(dz),

where λ is a probability measure on R+ with∫

R+z λ(dz) < ∞.

A pair potential is any measurable function φ : Rd → R ∪ f+∞g suchthat φ(−x) = φ(x). For a pair potential φ, a bounded measurable subset

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Λ in Rd, and a configuration γ ∈ ΓRd , the conditional energy of γ in Λ isgiven by the formula

EφΛ(γ) :=

{∑

φ(x− y) if∑

jφ(x− y)j < ∞,+∞ otherwise,

where the summation is taken over all pairs fx, yg ⊂ γ such that fx, yg ∩Λ 6= ∅. We adopt the convention that a sum over the empty set is zero sothat Eφ

Λ(γ) = 0 if either γ(Rd) = 1 or γ(Λ) = 0. We also define

Zz,φΛ (γ) :=

ΓRd

exp[

−EφΛ(γΛc + ωΛ)

]

πzσ(dω).

Here γΛc+ωΛ is the configuration formed by combining the part of γ outsideΛ with the part of ω inside Λ. The parameter z > 0 is called the activity .

Definition 3 A probability measure µ on ΓRd is called a Gibbs measurewith activity z, pair potential φ, and intensity measure σ if, for everybounded measurable Λ ⊂ Rd we have Zz,φ

Λ (γ) < ∞ for µ-almost everyγ ∈ ΓRd and for every ∆ ∈ B(ΓRd),

µ(∆) =

∫∫

ΓRdΓRd

1∆(γΛc + ωΛ)exp

[

−EφΛ(γΛc + ωΛ)

]

Zz,φΛ (γ)

πzσ(dω)µ(dγ). (2)

We recall the definition of the cubes Qr = r + [0, 1]d for r ∈ Zd.

Definition 4(SS) A pair potential φ is called superstable if there existA > 0 and B ≥ 0 so that if Λ = ∪r∈RQr is a finite union of cubes,then

EφΛ(γΛ) ≥

r∈R

[

Aγ(Qr)2 −Bγ(Qr)

]

.

(LR) A pair potential φ is called lower regular if there exists a decreasingpositive function Ψ : N → [0,∞) such that

r∈Zd Ψ(jrj∞) < ∞, and

for any disjoint Λ′and Λ

′′that are finite unions of cubes, then we

have∫∫

Λ′Λ′′

φ(x− y) γ(dx)γ(dy) ≥ −∑

r′,r

′′∈Zd

Ψ(jr′ − r′′j∞)γΛ′ (Qr

′ )γΛ′′ (Qr′′ ),

for all γ ∈ ΓRd. Here j · j∞ refers to the maximum norm on Rd.

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(I) A pair potential φ is called integrable if∫

Rd j exp(−φ(x))−1j dx < ∞.

Definition 5 A measure µ on ΓRd is called tempered if

lim supl→∞

jrj≤l γ(Qr)2

(2l+ 1)d< ∞ for µ-almost every γ ∈ ΓRd .

Definition 6 A probability measure µ on ΓRd is called a Ruelle measure ifµ is a tempered Gibbs measure with activity parameter z > 0, intensityσ equal to Lebesgue measure, and a pair potential φ that is superstable,lower regular, and integrable.

Suppose that µ is a Gibbs measure and Λ a bounded measurable subsetof Rd. Let F(Λ) be the σ-algebra of events ∆ ∈ B(ΓRd) that only dependon the part of the configuration in Λ, that is, 1∆(γ) = 1∆(γΛ) for everyγ ∈ ΓRd . Exchanging the order of integration in (2), we find that µjF(Λ) isabsolutely continuous with respect to πzσjF(Λ) with density

ω 7→∫

ΓRd

[Zz,φΛ (γ)]−1 exp

[

−EφΛ(γΛc + ωΛ)

]

µ(dγ). (3)

In other words, a Gibbs measure is always locally absolutely continuouswith respect to its corresponding Poisson measure. In general we have verylittle information about the density (3), but for Ruelle measures it is knownto be bounded, with a bound that depends on jΛj, the Lebesgue measureof Λ. In particular, for any n ∈ N, the moments satisfy

Eµ(γ(Λ)n) ≤ c(n, jΛj). (4)

Another important tool in studying Ruelle measures is the family of(infinite-volume) correlation functions ρm : (Rd)m → R given by

ρm(x1, . . . , xm)

= zm exp(−∑

i<j

φ(xi − xj))

ΓRd

exp(

−m∑

i=1

〈φ(xi − · ), γ〉)

µ(dγ).

These provide us with useful formulas for Ruelle measures:∫

ΓRd

〈f, γ〉µ(dγ) =∫

Rd

f(x)ρ1(x) dx, (5)

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ΓRd

(〈f, γ〉〈g, γ〉 − 〈fg, γ〉)µ(dγ) =∫

Rd

Rd

f(x) g(y)ρ2(x, y) dx dy. (6)

From now on we will simply assume that the pair potential φ satisfiesthe (mild) additional smoothness and integrability assumptions to ensurethat µ satisfies the conditions in Section 2. By [16, Theorem 6.13 andRemark 7.5] (cf. also [4, Proposition 5.1]) and [9], the classical Dirichletform given by integrating the gradient on ΓRd against such a Ruelle measureis a symmetric, quasi-regular and local Dirichlet form.

For sufficiently small z, by [21, Theorems 5.7 and 5.8], µ is the uniqueGibbs measure for φ and z, and is translation invariant. In particular, thefirst correlation function ρ1(x) = ρ is constant.

Proposition 3 For sufficiently small z, the complement of the set{

γ ∈ ΓRd : limn→∞

γ(Cn)

jCnj= ρ

}

is E-exceptional.

Proof. Since µ is the unique Gibbs measure, it is translation-ergodic by[15, Theorem 4.1], so

Eµ(γ(Q0) j H) = Eµ(γ(Q0)) = ρ, µ-a.e.

which gives the result using Proposition 1 and Proposition 2.

5 Law of the iterated logarithm on configurationspace

In this section we prove a capacitary version of the law of the iteratedlogarithm. This time let Vn = [−n1/d/2, n1/d/2]d be the cube with volumen, and set µn = Eµ(γ(Vn)), σ2

n = Varµ(γ(Vn)), and χn := (2σ2n log log σ

2n)

1/2.Letting Sn = γ(Vn)− µn, we define

LIL :=

{

γ ∈ ΓRd : lim supn

Sn

χn

= 1 and lim infn

Sn

χn

= −1

}

.

As with the law of large numbers, we must begin by proving the fixedtime result, that is, µ(LILc) = 0. The proof is based on the methods

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used in [12, 26], while replacing the strong mixing condition there by theexponential mixing condition in Lemma 3 below.

Even this fixed time result is, as far as we know, the first law of theiterated logarithm for Gibbsian random fields so we’ve included statementsof the crucial lemmas from [12, 26], as Lemmas 3, 4, and 5 below.

To get our results we need to assume that the pair potential is positiveand of finite range R. The positivity in particular allows us to say exactlyhow small z need be for our results to hold. Recall that we have assumedthat the pair potential φ satisfies C :=

[1 − exp(−φ(x))] dx < ∞. Fromnow on, we assume that

z < (3eC)−1. (7)

This bound is more than sufficient, by [21, Theorems 5.7 and 5.8], toguarantee that µ is the unique Gibbs measure for φ and z, and is translationinvariant. Now from (5) and (6) we get

σ2n = ρn+

Vn

Vn

ω2(x, y) dx dy, (8)

where ω2 = ρ2 − ρ2. For z satisfying (7), we have ω2(x, y) = ω2(0, y − x).Also, from (5.18) and (5.19) of [20, Section 4.5], we see that z/(1+Cz) ≤

ρ ≤ z, while (4.37) of [20, Section 4.4] (with B = 0) gives∫

jω2(0, x)j dx ≤ z2eC/(1− zeC)2.

Combining these inequalities we see that∫

jω2(0, x)j dx < ρ if z satisfies(7). From (8) we conclude that there exist constants a, b > 0 so that

0 < a ≤ σ2n

n≤ b < ∞. (9)

In [24, Lemma 4], an exponential L2-mixing for Gibbs fields at low den-sity was proved. We’ve simplified the expression found there by adjustingthe constants c and α. For any two bounded Borel subsets Λ1 and Λ2 ofRd we define d(Λ1,Λ2) := inff‖x − y‖ : x ∈ Λ1, y ∈ Λ2g. We note that thebound (7) also suffices to get Spohn’s result.

Lemma 3 ([24, Lemma 4]) Let Λi be bounded regions (i = 1, 2) andsuppose Ψi ∈ F(Λi) are square integrable. There exist constants α, c >0, depending only on z,φ, such that

jCorrµ(Ψ1,Ψ2)j ≤ minf1, c jΛR1 j e−αd(Λ1,Λ2)g,

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where ΛR := fx ∈ Rd : d(x,Λ) ≤ Rg.

Lemma 4 Let (rj)j∈N be a sequence in Zd so that ri 6= rj if i 6= j.Suppose (ηj)j∈N are mean zero, square integrable random variables sothat ηj ∈ F(Qrj) for all j ∈ N. Suppose also that supj Varµ(ηj) < ∞.Then there is a constant K so that for any n,

Varµ

(

n∑

j=1

ηj

)

≤ Kn,

and any b ≥ 0,

(

max1≤j≤n

(

b+j∑

i=b+1

ηi)2

)

≤ K n (log 2n)2.

Lemma 5 Let Φ(x) = P (Z > x) be the standard normal error function.Then

sup−∞<x<∞

jµ(σ−1n Sn ≥ x)− Φ(x)j = O

(

1

(logn)3

)

. (10)

Also, for any b > 1,

µ

(

max1≤j≤n

jSjj ≥ bχn

)

= O

(

1

(logn)3

)

.

Proposition 4 The law of the iterated logarithm holds for almost everyγ, that is, µ(LILc) = 0.

Proof. The assertion will be proved if we show that for any ε > 0,

µ(jSnj > (1 + ε)χn i.o.) = 0 (11)

andµ(Sn > (1− ε)χn i.o.) = 1. (12)

µ(Sn < −(1− ε)χn i.o.) = 1. (13)

The proof of (11) is almost identical to [12, Theorem 1] and therefore isomitted.

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We proceed to prove (12). For k ∈ N with k ≥ σ1, define mk to be thelargest integer so σmk

≤ k2k−1, and nk the largest integer so σnk≤ k2k. For

k ≥ b+ 1, we have mk ≥ b and so using (9)

amk ≤ k4k−2 ≤ b(mk + 1) ≤ (b+ 1)mk (14)

ank ≤ k4k ≤ b(nk + 1) ≤ (b+ 1)nk. (15)

Since ak4k/(b+ 1) ≤ ank ≤ σ2nk

≤ k4k we have

4k log(k) + log

(

a

b+ 1

)

≤ log(σ2nk) ≤ 4k log(k). (16)

For any λ > 0 put Bk = Bk(λ) = fSnk− Smk

≥ (1− 2λ)χnkg. We want to

show that∑

k µ(Bk) = ∞. We will use the inequality

µ(Snk≥ (1− λ)χnk

) ≤ µ(Bk) + µ(Smk≥ λχnk

). (17)

Using σ2mk

≤ k4k−2 and recalling that χ2nk

≥ σ2nk

≥ (a/(b + 1))k4k, Cheby-shev’s inequality gives us

µ(Smk≥ λχnk

) ≤σ2mk

λ2χ2nk

≤ b+ 1

aλ2k2. (18)

Since this is summable it suffices to show that∑

k µ(Snk≥ (1−λ)χnk

) = ∞.From the Central Limit Theorem (10) and (15) we have

k

jµ(Snk≥ (1− λ)χnk

)− Φ((1− λ)χnk/σnk

)j ≤ c∑

k

(log(nk))−3 < ∞.

Therefore it suffices to show that∑

k

Φ((1− λ)χnk/σnk

) =∑

k

Φ(

(1− λ)√

2 log log(σ2nk))

= ∞.

But this follows in the usual way from the asymptotic relation Φ(x) ∼exp(−x2/2)/x and (16).

Let ζk be the indicator function of Bk. By virtue of the exponentialmixing condition in Lemma 3, and the growth rates (14) and (15) of mk

and nk, you can show that there is c > 0 and k0 ∈ N so that

Covµ(ζk, ζk+j) ≤ k−2 exp(−cj), for all k ≥ k0 and j ≥ 1.

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Therefore

Varµ(n∑

k=k0

ζk) =n∑

k=k0

Varµ(ζk) + 2n−1∑

k=k0

n−k∑

j=1

Cov(ζk, ζj+k)

≤n∑

k=k0

Varµ(ζk) + 2∞∑

k=1

k−2∞∑

j=1

exp(−cj)

≤n∑

k=k0

µ(Bk) +K.

Thus,

µ

(

∞∑

k=k0

ζk ≤ 1

2

n∑

k=k0

µ(Bk)

)

≤ µ

(

n∑

k=k0

ζk ≤ 1

2

n∑

k=k0

µ(Bk)

)

≤ µ

(∣

n∑

k=k0

ζi −n∑

k=k0

µ(Bk)

≥ 1

2

n∑

k=k0

µ(Bk)

)

≤4Varµ(

∑nk=k0

ζk)

(∑n

k=k0µ(Bk))2

≤4(∑n

k=k0µ(Bk) +K)

(∑n

k=k0µ(Bk))2

.

Since∑

k µ(Bk) = ∞, letting n → ∞ gives µ(∑∞

k=k0ζk < ∞) = 0 so

µ(Bk(λ) i.o.) = 1. (19)

Note that Bk(ε/4) ⊂ (Snk≥ (1− ε)χnk

)∪ (−Smk≥ (ε/2)χnk

), so from (19)

1 ≤ µ(Snk≥ (1− ε)χnk

i.o.) + µ(−Smk≥ (ε/2)χnk

i.o.).

But as in (18) we see that∑

k µ(−Smk≥ (ε/2)χnk

) < ∞ so that µ(−Smk≥

(ε/2)χnki.o.) = 0. From this (12) follows and (13) can be proved similarly.

Therefore the theorem is proved.

Proposition 5 In dimensions d ≤ 3, the set LILc is E-exceptional.

Proof. Let n1 = 1 and for k ≥ 1 let nk+1 = [nk +√nk]. For n > 1, choose

k so that nk ≤ n ≤ nk+1 and hence(

χnk

χnk+1

)(

Snk

χnk

)

+

(

µnk− µnk+1

χnk+1

)

<Sn

χn

≤(

χnk+1

χnk

)(

Snk+1

χnk+1

)

+

(

µnk+1 − µnk

χnk

)

.

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Using the formula µnk= ρnk and χnk

=√

2nk log log(nk), the inequalityabove shows that lim supn(Sn/χn) = lim supk(Snk

/χnk).

We now construct a smooth approximation for Snk/χnk

. First we boundbelow the distance between the cubes Vnk+1 and Vnk

:

n1/dk+1/2− n

1/dk /2 ≥ n

1/d−1/2k

16d.

So for every k ≥ 1, we can choose a smooth function ψk satisfying

1[−n

1/dk /2,n

1/dk /2]

≤ ψk ≤ 1[−n

1/dk+1/2,n

1/dk+1/2]

,

and jψ′kj ≤ 17dn

1/2−1/dk . Define a continuous element of D(E) by

uk(γ) := (〈∏d

i=1ψk(xi), γ〉 − µnk)/χnk

.

Taking the limsup through the inequality(

Snk

χnk

)

≤ uk ≤(

χnk+1

χnk

)(

Snk+1

χnk+1

)

+

(

µnk+1 − µnk

χnk

)

.

we find that u(γ) := lim supk uk(γ) = lim supk Snk(γ)/χnk

.Bounding the square field gives

(uk)(γ) ≤

(

17d2n1/2−1/dk

χnk

)2

γ(Vnk+1 \ Vnk) ≤ c

γ(Vnk+1 \ Vnk)

n2/dk

. (20)

Using the definition of nk and since jVnj = n, we get

γ(Vnk+1 \ Vnk)

n2/dk

≤γ(Vnk+1 \ Vnk

)− ρjVnk+1 \ Vnkj

n2/dk

+ρjVnk+1 \ Vnk

jn2/dk

≤(

nk+1

nk

)2/d jγ(Vnk+1)− ρjVnk+1jjnk+1

2/d+

jγ(Vnk)− ρjVnk

jjnk

2/d+ ρ

nk1/2

nk2/d

≤ 4jγ(Vnk+1)− ρjVnk+1jj

n2/dk+1

+jγ(Vnk

)− ρjVnkjj

nk2/d

+ ρn1/2k

n2/dk

. (21)

We need some notation so that we can work with the more convenientcubes Cn. Let k0 be so large that nk0 ≥ 4d and for k ≥ k0 let mk be theinteger that satisfies

[2(mk − 1)]d < nk ≤ (2mk)d.

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Page 16: The law of large numbers and the law of the iterated ...

Then Cmk−1 ⊂ Vnk⊂ Cmk

. The condition on k0 guarantees that mk ≥ 2,then we get

γ(Vnk)− ρjVnk

jnk

2/d≤

(

jCmkj

nk

)2/d jγ(Cmk)− ρjCmk

jjjCmk

j2/d+

ρjjCmkj − jVnk

jjnk

2/d

≤ 4jγ(Cmk

)− ρjCmkjj

jCmkj2/d

+ρ 2d[md

k − (mk − 1)d]

4(mk − 1)2

≤ 4jγ(Cmk

)− ρjCmkjj

jCmkj2/d

+ cmd−3k . (22)

and

ρjVnkj − γ(Vnk

)

nk2/d

≤(

jCmk−1jnk

)2/d jγ(Cmk−1)− ρjCmk−1jjjCmk−1j2/d

+ρjjCmk−1j − jVnk

jjnk

2/d

≤ jγ(Cmk−1)− ρjCmk−1jjjCmk−1j2/d

+ cmd−3k . (23)

Combined with (20), (21), (22), and (23) this gives us, for d ≤ 3,

supk≥k0

(

(uk)(γ))

≤ c

(

1 + supk≥k0

jγ(Cmk)− ρjCmk

jjCmk

j2/d

)

. (24)

As in the proof of Proposition 2, let (rm)m∈N be a sequence in Zd so

Cn = ∪jCnjm=1Qrm for n ∈ N. Denote ηm(γ) = γ(Qrm) − ρ for m ∈ N. By

applying Lemma 4, for d ≤ 3, we have

ΓRd

supn∈N

jγ(Cn)− ρjCnjjjCnj2/d

µ(dγ) =

ΓRd

supn∈N

jη1 + · · ·+ ηjCnjjjCnj2/d

µ(dγ)

≤∫

ΓRd

supm∈N

jη1 + · · ·+ ηmjm2/3

µ(dγ)

= limk→∞

ΓRd

maxm≤2k

jη1 + · · ·+ ηmjm2/3

µ(dγ)

≤ limk→∞

k∑

l=1

ΓRd

max2l−1<m≤2l

jη1 + · · ·+ ηmjm2/3

µ(dγ)

≤∞∑

l=1

1

22(l−1)/3

ΓRd

max2l−1<m≤2l

jη1 + · · ·+ ηmjµ(dγ)

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≤∞∑

l=1

1

22(l−1)/3Eµ

(

maxm≤2l

(η1 + · · ·+ ηm)2

)1/2

≤∞∑

l=1

1

22(l−1)/3(c 2l log(2l+1)2)1/2

≤ c

∞∑

l=1

(l+ 1)2−l/6 < ∞.

Combining this estimate with Proposition 4, the proof is almost identicalwith that of Proposition 2 and therefore is omitted.

Remark. Rockner and Schied [17] have also recently proved a capacitaryversion of the law of large numbers on configuration space using an inter-esting approach based on an intrinsic metric ρ. Indeed, it is easy to seethat LLN c∩ΓRd = fγ ∈ ΓRd : ρ(γ, ω) < ∞ for some ω ∈ LLN c∩ΓRdg so by[17, Corollary 3.2], µ(LLN c) = 0 implies that LLN c ∩ΓRd is E-exceptional.Their technique will also give a capacitary version of the law of the iteratedlogarithm.

However, the Dirichlet form used in [17] extends (E , D(E)), but it isnot known whether the forms coincide. A form with a larger domain hasmore exceptional sets, so proofs of exceptionality are easier in Rockner andSchied’s setting, and do not imply exceptionality in our setting.

References

[1] S. Albeverio, Yu. G. Kondratiev, and M. Rockner: Differential geom-etry of Poisson spaces, C.R. Acad. Sci. Paris, t. 323, Serie I, 1129–1134 (1996).

[2] S. Albeverio, Yu. G. Kondratiev, and M. Rockner: Canonical Dirich-let operator and distorted Brownian motion on Poisson spaces, C.R.Acad. Sci. Paris, t. 323, Serie I, 1179–1184 (1996).

[3] S. Albeverio, Yu. G. Kondratiev, and M. Rockner: Analysis and geom-etry on configuration spaces, J. Funct. Anal. 154, 444–500 (1998).

[4] S. Albeverio, Yu. G. Kondratiev, and M. Rockner: Analysis and ge-ometry on configuration spaces: The Gibbsian case, J. Funct. Anal.157, 242–291 (1998).

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[5] N. H. Bingham: Variants on the law of the iterated logarithm, Bull.London Math. Soc. 18, 433–467 (1986).

[6] Yu. G. Kondratiev, R. A. Minlos, M. Rockner, and G. V. Shchepan’uk:Exponential mixing for classical continuous systems, SFB 343 (Biele-feld) Preprint.

[7] K. Kuwae: Functional calculus for Dirichlet forms, Osaka J. Math.35, 683–715 (1998).

[8] Z. M. Ma and M. Rockner: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer: Berlin, 1992.

[9] Z. M. Ma and M. Rockner: Construction of diffusion processes on con-figuration spaces, SFB 343 (Bielefeld) Preprint 98-056. To appear inOsaka J. Math.

[10] Z. M. Ma and M. Rockner: Diffusions on “simple” configuration spaces,SFB 343 (Bielefeld) Preprint 00-029. To appear in Proc. Leipzig Con-ference.

[11] X. X. Nguyen and H. Zessin: Ergodic theorems for spatial processes,Z. Wahr. verw. Gebiete 48, 133–158 (1979).

[12] H. Oodaira and K. Yoshihara: The law of the iterated logarithm forstationary processes satisfying mixing conditions, Kodai Math. Sem.Rep. 23, 311–334 (1971).

[13] H. Osada: Dirichlet form approach to infinite-dimensional Wiener pro-cesses with singular interactions, Commun. Math. Phys 176, 117–131(1996).

[14] V. V. Petrov: Sums of Independent Random Variables. Akademie-Verlag: Berlin, 1975.

[15] C. Preston: Random Fields. Springer: Berlin, 1976.

[16] M. Rockner: Stochastic analysis on configuration spaces: Basic ideasand recent results, In: New directions in Dirichlet forms. AMS/IPStudies in Advanced Mathematics, Vol. 8, American Mathematical So-ciety 157–231 (1998).

[17] M. Rockner and A. Schied: Rademacher’s theorem on configurationspaces and applications, J. Funct. Anal. 169, 325–356 (1999).

[18] M. Rockner and B. Schmuland: Quasi-regular Dirichlet forms: Exam-ples and counterexamples, Canad. J. Math. 47 (1), 165–200 (1995).

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[19] M. Rockner and B. Schmuland: A support property for infinite-dimensional interacting diffusion processes, C.R. Acad. Sci. Paris,t. 326, Serie I, 359–364 (1998).

[20] D. Ruelle: Statistical Mechanics: Rigorous Results. W.A. Benjamin:New York, 1969.

[21] D. Ruelle: Superstable interactions in classical statistical mechanics,Commun. Math. Phys. 18, 127–159 (1970).

[22] B. Schmuland: Some exceptional configurations, In: Stochastic Modelsed. Luis Gorostiza and Gail Ivanoff, CMS Conference Proceedings,Volume 26, American Mathematical Society, 419–434 (2000).

[23] B. Schmuland: Extended Dirichlet spaces. Comptes Rendus Mathe-matical Reports, Royal Society of Canada 21, 146–152 (1999).

[24] H. Spohn: Equilibrium fluctuations for interacting Brownian particles,Commun. Math. Phys. 103, 1–33 (1986).

[25] M. W. Yoshida: Construction of infinite dimensional interacting diffu-sion processes through Dirichlet forms, Probab. Th. and Rel. Fields106, 265–297 (1996).

[26] K. Yoshihara: The Borel-Cantelli lemma for strong mixing sequencesof events and their applications to LIL, Kodai Math. J. 2, 148–157(1979).

Byron SchmulandDepartment of Mathematical SciencesUniversity of AlbertaEdmonton, Alberta, Canada T6G [email protected]

Wei SunInstitute of Applied MathematicsChinese Academy of SciencesBeijing, China 100080

and

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Department of Mathematical SciencesUniversity of AlbertaEdmonton, Alberta, Canada T6G [email protected]

20