Process-level quenched large deviations for random walk in random environment

www.imstat.org/aihp 2011, Vol. 47, No. 1, 214–242

DOI:10.1214/10-AIHP369

© Association des Publications de l’Institut Henri Poincaré, 2011

Process-level quenched large deviations for random walk in random environment

Firas Rassoul-Agha

and Timo Seppäläinen

aDepartment of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84109, USA.

E-mail:firas@math.utah.edu

bDepartment of Mathematics, University of Wisconsin-Madison, 419 Van Vleck Hall, Madison, WI 53706, USA.

E-mail:seppalai@math.wisc.edu

Received 7 September 2009; revised 9 April 2010; accepted 16 April 2010

Abstract. We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.

Résumé. Nous considérons une marche aléatoire en environment aléatoire ergodique. La marche est elliptique et à pas bornés.

Nous prouvons un principe de grandes déviations au niveau 3, sous presque tout environnement, avec une fonctionnelle d’action liée à une entropie relative.

MSC:60K37; 60F10; 82D30; 82C44

Keywords:Random walk; Random environment; RWRE; Large deviation; Environment process; Relative entropy; Homogenization

1. Introduction

We describe the standard model of random walk in random environment (RWRE) onZ^d. LetΩ be a Polish space andS its Borelσ-algebra. Let{Tz: z∈Z^d}be a group of continuous commuting bijections onΩ: Tx+y=TxTy

andT0 is the identity. LetPbe a{Tz}-invariant probability measure on(Ω,S)that is ergodic under this group. In other words, theσ-algebra of Borel sets invariant under{Tz}is trivial underP.

Denote the space of probability distributions onZ^dbyP= {(p_z)_z∈Zd ∈ [0,1]^Zd:

zp_z=1}and give it the weak topology or, equivalently, the restriction of the product topology. Letω→(p_z(ω))_z∈Zd be a continuous mapping fromΩtoP. Forx, y∈Z^ddefineπ_x,y(ω)=p_y−x(T_xω). We callωand also(π_x,y(ω))_x,y∈Zd an environment because it determines the transition probabilities of a Markov chain.

The set of admissible steps is denoted by R= {z: E[π_0,z]>0}. One can then redefine P = {(pz)z∈R ∈ [0,1]^R:

zpz=1}and transition probabilitiesπx,yare defined only forx, y∈Z^dsuch thaty−x∈R.

Givenωand a starting pointx∈Z^d, letP_x^ω be the law of the Markov chainX_0,∞=(X_n)_n≥0 onZ^d, starting at X₀=x and having transition probabilities(πy,y+z(ω)). That is,

P_x^ω{Xn+1=y+z|Xn=y} =πy,y+z(ω) for ally, z∈Z^d.

1Supported in part by NSF Grant DMS-0747758.

2Supported in part by NSF Grant DMS-0701091 and by the Wisconsin Alumni Research Foundation.

X_0,∞ is called a random walk in environmentωandP_x^ω is called thequencheddistribution. The jointdistribution is P_x(dx0,∞,dω)=P_x^ω(dx0,∞)P(dω). Its marginal on(Z^d)^Z+ is also denoted byP_x and called theaveraged (or annealed) distribution sinceωis averaged out:

P_x(A)=

P_x^ω(A)P(dω) for a measurableA⊂ Z^d_Z+

. The canonical case of the above setting isΩ=P^Zd andpz(ω)=(ω0)z.

Next a quick description of the problem we are interested in. Assume given a sequence of probability measuresQn

on a Polish space(X,B_X)and a lower semicontinuous function I:X → [0,∞]. Then thelarge deviation upper boundholds withrate functionI if

nlim→∞n⁻¹logQ_n(C)≤ −inf

C I for all closed setsC⊂X. Similarly, rate functionI governs thelarge deviation lower boundif

lim

n→∞n⁻¹logQ_n(O)≥ −inf

O I for all open setsO⊂X.

If both hold with the same rate functionI, then thelarge deviation principle(LDP) holds with rateI. We shall use basic, well known features of large deviation theory and relative entropy without citing every instance. The reader can consult references [3–5,15,21].

If the upper bound (resp., lower bound, resp., LDP) holds with some functionI:X → [0,∞], then it also holds with thelower semicontinuous regularizationI_lscofI defined by

Ilsc(x)=sup

infO I:x∈OandOis open

.

Thus the rate function can be required to be lower semicontinuous, and then it is unique.

Large deviations arrange themselves more or less naturally in three levels. Most of the work on quenched large deviations for RWRE has been at level 1, that is, on large deviations for P₀^ω{Xn/n∈ ·}. Greven and den Hol- lander [10] considered the product one-dimensional nearest-neighbor case, Comets, Gantert and Zeitouni [2] the ergodic one-dimensional nearest-neighbor case, Yilmaz [24] the ergodic one-dimensional case with bounded step size, Zerner [25] the multi-dimensional product nestling case, and Varadhan [22] the general ergodic multidimen- sional case with bounded step size. Rosenbluth [17] gave a variational formula for the rate function in [22].Level 2 quenched large deviations appeared in the work of Yilmaz [24] for the distributionsP₀^ω{n⁻¹n−1

k=0δT_Xkω,Z_k+1 ∈ ·}. HereZk=Xk−Xk−1denotes the step of the walk.

Our object of study,level 3orprocess levellarge deviations concerns theempirical process

R^1,_n^∞=n⁻¹

n−1

k=0

δ_{TXkω,Zk+1,∞}, (1.1)

whereZ_k+1,∞=(Z_i)_i≥k+1 denotes the entire sequence of future steps. Quenched distributions P₀^ω{R^1,_n^∞∈ ·}are probability measures on the spaceM1(Ω×R^N). This is the space of Borel probability measures onΩ×R^Nendowed with the weak topology generated by bounded continuous functions.

The levels do form a hierarchy: higher level LDPs can be projected down to give LDPs at lower levels. Such results are calledcontraction principlesin large deviation theory.

The main technical contribution of this work is the extension of a homogenization argument that proves the upper bound to the multivariate level 2 setting. This idea goes back to Kosygina, Rezakhanlou and Varadhan [12] in the context of diffusions with random drift, and was used by both Rosenbluth [17] and Yilmaz [24] to prove their LDPs.

Before turning to specialized assumptions and notation, here are some general conventions.Z+,Z−andNdenote, respectively, the set of non-negative, non-positive and positive integers.| · |denotes the^∞-norm onR^d.{e1, . . . , ed} is the canonical basis ofR^d. In addition toM1(X)for the space of probability measures onX, we writeQ(X)for

the set of Markov transition kernels onX. Our spaces are Polish and theσ-algebras Borel. Givenμ∈M1(X)and q∈Q(X),μ×qis the probability measure onX×X defined by

μ×q(A×B)=

1A(x)q(x, B)μ(dx)

andμq is its second marginal. For a probability measure P,E^P denotes the corresponding expectation operator.

OccasionallyP (f )may replaceE^P[f].

2. Main result

Fix a dimensiond≥1. Following are the hypotheses for the level 3 LDP. In Section3we refine these to state precisely what is used by different parts of the proof.

Ris finite andΩ is a compact metric space. (2.1)

∀x∈Z^d,∃m∈Nandz₁, . . . , z_m∈Rsuch thatx=z₁+ · · · +z_m. (2.2)

∃p > dsuch thatE[|logπ0,z|^p]<∞ ∀z∈R. (2.3)

WhenR is finite the canonical Ω =P^Zd is compact. The commonly used assumption of uniform ellipticity, namely the existence ofκ >0 such that P{π_0,z≥κ} =1 forz∈R andR contains the 2d unit vectors, implies assumptions (2.2) and (2.3).

We need notational apparatus for backward, forward and bi-infinite paths. The increments of a bi-infinite path (x_i)_i∈ZinZ^d withx₀=0 are denoted byz_i=x_i −x_i−1. The sequences(x_i)and(z_i)are in 1–1 correspondence.

Segments of sequences are denoted byz_i,j =(z_i, z_i+1, . . . , z_j), also fori= −∞orj = ∞, and also for random variables:Zi,j=(Zi, Zi+1, . . . , Zj).

In generalη_i,j denotes the pair(ω, z_i,j), but wheniandj are clear from the context we write simplyη. We will also sometimes abbreviateη₋=η_−∞,0. The spaces to which elementsηbelong areΩ₋=Ω×R^Z−,Ω₊=Ω×R^N andΩ=Ω×R^Z. Their relevant shift transformations are

S_z⁻:Ω₋→Ω₋:(ω, z_−∞,0)→(T_zω, z_−∞,0, z), S⁺:Ω₊→Ω₊:(ω, z1,∞)→(Tz₁ω, z2,∞), S:Ω→Ω:(ω, z_−∞,∞)→(T_z1ω,z¯_−∞,∞),

wherez¯i=zi+1. We use the same symbolsS⁻_z,S⁺andSto act onz_−∞,0,z1,∞andz_−∞,∞in the same way.

The empirical process (1.1) lives inΩ₊but the rate function is best defined in terms of backward paths. Invariance allows us to pass conveniently between theses settings. Ifμ∈M1(Ω₊)isS⁺-invariant, it has a uniqueS-invariant extensionμ¯ onΩ. Letμ₋= ¯μ_|Ω₋, the restriction ofμ¯ to its marginal onΩ₋. There is a unique kernelqμonΩ₋ that fixesμ₋(that is,μ₋q_μ=μ₋) and satisfies

qμ

η₋, S_z⁻η₋: z∈R

=1 forμ₋-a.e.η₋. (2.4)

Namely q_μ

η₋, S_z⁻η₋

= ¯μ Z₁=z|(ω, Z_−∞,0)=η₋ .

(Uniqueness here isμ₋-a.s.) Indeed, on the one hand, the aboveq_μdoes leaveμ₋invariant. On the other hand, ifq is a kernel supported on shifts and leavesμ₋invariant, and iff is a bounded measurable function onΩ₋, then

q

η₋, S_z⁻η₋

f (η₋)μ(dη)¯

=

z

q

η₋, S_z⁻η₋

f (T_z−zω, z_−∞,0)1 z=z

μ₋(dη₋)

=

f (T_−zω, z_−∞,−1)1{z₀=z}μ₋(dη₋)

=

f (η₋)1{z₁=z} ¯μ(dη)

=

qμ

η₋, S_z⁻η₋

f (η₋)μ(dη).¯

The RWRE transition gives us the kernelp⁻∈Q(Ω₋)defined by p⁻

η₋, S_z⁻η₋

=π0,z(ω) forη₋=(ω, z_−∞,0)∈Ω₋.

Ifq ∈Q(Ω₋)satisfiesμ₋×qμ₋×p⁻, thenq(η₋,{S_z⁻η₋: z∈R})=1 μ₋-a.s. and their relative entropy is given by

H

μ₋×q|μ₋×p⁻

=

z∈R

q

η₋, S_z⁻η₋

log q(η₋, S_z⁻η₋)

p⁻(η₋, S_z⁻η₋)μ₋(dη₋). (2.5) Letμ0denote the marginal ofμonΩ. Our main theorem is the following.

Theorem 2.1. Let(Ω,S,P,{T_z})be an ergodic system.Assume(2.1), (2.2)and(2.3).Then,forP-a.e.ω,the large deviation principle holds for the lawsP₀^ω{R^1,_n^∞∈ ·},with rate functionH_quen:M1(Ω₊)→ [0,∞]equal to the lower semicontinuous regularization of the convex function

H (μ)= H

μ₋×q_μ|μ₋×p⁻

ifμisS⁺-invariant andμ₀P,

∞ otherwise. (2.6)

We make next some observations about the rate functionH_quen.

Remark 2.1. As is often the case for process level LDPs,the rate function is affine.This follows because we can replaceq_μwith a “universal” kernelq¯whose definition is independent ofμ.Namely,define

U:Ω₋→Ω₋:(ω, z_−∞,0)→(T_−z0ω, z_−∞,−1).

Then,on the event wherelimn→∞1 n

_n−1

k=0δ_Ukη₋exists define

¯ q

η₋, S_z⁻η₋

=q_μ

η₋, S_z⁻η₋

forμ= lim

n→∞

1 n

n−1

k=0

δ_Ukη₋. (2.7)

On the complement,setq(η¯ ₋, S_z⁻η₋)=δz₀(z),for some fixedz0∈R.

Remark 2.2. Let us also recall the convex analytic characterization of l.s.c.regularization.LetCb(X)denote the space of bounded continuous functions on X.Given a function J:M1(X)→ [0,∞],let J^∗:Cb(X)→R be its convex conjugatedefined by

J^∗(f )= sup

μ∈M1(X)

E^μ[f] −J (μ)

and letJ^∗∗:M1(X)→Rbe its convex biconjugate defined by J^∗∗(μ)= sup

f∈Cb(X)

E^μ[f] −J^∗(f ) .

IfJis convex and not identically infinite,J^∗∗is the same as its lower semicontinuous regularizationJlsc;see Propo- sitions3.3and4.1of[8]or Theorem5.18of[15].Thus the rate function in Theorem2.1isHquen=H^∗∗.

As expected, rate functionHhas in fact an alternative representation as a specific relative entropy. For a probability measureνonΩ, define the probability measureν×P₀^·onΩ₊by

Ω₊

fd ν×P₀^·

=

Ω

R^Nf (ω, z_1,∞)P₀^ω(dz1,∞)

ν(dω).

On any of the product spaces of environments and paths, define theσ-algebrasGm,n=σ{ω, z_m,n}. LetH_Gm,n(α|β) denote the relative entropy of the restrictions of the probability measuresαandβto theσ-algebraGm,n. LetΠbe the kernel of the environment chain(T_Xnω), defined asΠf (ω)=E₀^ω[f (T_X1ω)] =

zπ0,z(ω)f (T_zω).

Lemma 2.2. Letμ∈M1(Ω₊)beS⁺-invariant.Then the limit h

μ|μ0×P₀^·

= lim

n→∞

1 nH_G1,n

μ|μ0×P₀^·

(2.8) exists and equalsH (μ₋×q_μ|μ₋×p⁻).

Proof. Fix μ. Let μ^ω,z_i ^1,i−1(·) denote the conditional distribution of Z_i under μ, given G1,i−1. Then by the S- invariance,

¯

μ[Z1=u|G2−i,0](ω, z2−i,0)=μ^T_i^{x1−iω,z2−i,0}(u).

Fori=1 we must interpretG1,0=σ{ω} =Sand(ω, z1,0)simply asω. Observe also that the conditional distribution ofZ_iunderμ₀×P₀^·, givenG1,i−1, isπ_0,(T_xi−1ω).

By two applications of the conditional entropy formula (Lemma 10.3 of [21] or Exercise 6.14 of [15]), H_G1,n

μ|μ₀×P₀^·

= n

i=1

H

μ^ω,z_i ^1,i−1|π_0,(T_xi−1ω)

μ(dω,dz1,∞)

= n

i=1

H

¯

μ[Z₁= ·|G2−i,0](T_xi−1ω, z_1,i−1)|π_0,(T_xi−1ω)

μ(dω,dz1,∞)

= n

i=1

H

¯

μ[Z₁= ·|G2−i,0](ω, z_2−i,0)|π_0,(ω)

μ₋(dω,dz_−∞,0)

= n

i=1

H_G2−i,1

μ₋×q_μ|μ₋×p⁻

. (2.9)

Ask→ ∞, theσ-algebrasG₋k,1generate theσ-algebraG_−∞,1=σ{ω, z_−∞,1}, and consequently H_G2−i,1

μ₋×q_μ|μ₋×p⁻ H

μ₋×q_μ|μ₋×p⁻

asi ∞. (2.10)

We have taken some liberties with notation and regardedμ₋×q_μ andμ₋×p⁻ as measures on the variables (ω, z_−∞,1), instead of on pairs((ω, z_−∞,0), (ω, z_−∞,0)). This is legitimate because the simple structure of the ker- nelsqμandp⁻, namely (2.4) implies thatz_−∞,0=z_−∞,1andω=Tz₁ωalmost surely under these measures.

The claim follows by dividing through (2.9) bynand lettingn→ ∞.

Note that the specific entropy in (2.8) is not an entropy between twoS⁺-invariant measures unless μ₀ is Π- invariant. The next lemma exploits the previous one to say something about the zeros ofHquen.

Lemma 2.3. IfH_quen(μ)=0thenμ(dω,dz1,∞)=μ₀(dω)P₀^ω(dz1,∞)for someΠ-invariantμ₀. Note that it is not necessarily true thatμ0Pin the above lemma.

Remark 2.3. One can show that under(2.1)and(2.2)there is at most oneP_∞∈M1(Ω)that isΠ-invariant and such thatP_∞P;see,for example, [13].In fact,in this caseP_∞∼P.The above lemma shows that the zeros ofH_quen consist ofP₀^∞=P_∞×P₀^·(ifP_∞Pexists)and possibly measures of the formμ₀×P₀^·,withμ₀beingΠ-invariant but such thatμ₀P.

Proof of Lemma2.3. There is a sequence ofS⁺-invariant probability measuresμ^(m)→μsuch thatH (μ^(m))→0 andμ^(m)₀ P. (Ifμ₀Pthen we can takeμ^(m)=μ.) Letμ^(m)₁ denote the marginal distribution on(ω, z_−∞,1)which can be identified withμ^(m)₋ ×q_μ(m) and converges to the corresponding marginalμ₁. By the continuity of the kernel π0,z(ω),μ^(m)₋ ×p⁻→μ₋×p⁻. From these limits and the lower semicontinuity of relative entropy,

H

μ₁|μ₋×p⁻

= lim

m→∞H

μ^(m)₁ |μ^(m)₋ ×p⁻

=0.

This tells us thatμ₋isp⁻-invariant, which in turn implies thatμ₀isΠ-invariant, and together with theS⁺-invariance ofμimplies also thatμ=μ₀×P₀^·. (The last point can also be seen from (2.9) and (2.10).)

We close this section with some examples.

LetΩ=P^Zd withP= {(pz)z∈R∈(0,1)^R:

zpz=1}. Letν∈Ω₊be the law of a classical random walk; i.e.

ν=ν0×P₀^· withν0=δ^⊗Z_α ^d, for someα∈P. ThenH (ν₋×q_ν|ν₋×p⁻)=0. However, if

zzα_z is not in the setN = {E^μ[Z₁]: H_quen(μ)=0}, then,H_quen(ν) >0. Note that by the contraction principle, N is the zero set of the level-1 rate function. Hence if Pis product, by [22]N consists of a singleton or a line segment. Thus we can pickαso that the mean

zα_zdoes not lie inN, and consequently we have measuresν for whichH_quen(ν) >0= H (ν₋×q_ν|ν₋×p⁻). That is, the rateH_quendoes not have to pick up the entropy value.

Lower semicontinuity of relative entropy impliesH_quen(μ)=H (μ)whenμ₀P. This equality can still happen whenμ₀P; i.e. the l.s.c. regularization can bring the rateH_quendown from infinity all the way to the entropy. Here is a somewhat singular example. Assume P{π_0,0(ω) >0} =1 and letζ =(0,0,0, . . .)be the constant sequence of 0-steps inZ^d. For eachω¯∈Ω define the (triviallyS⁺-invariant) probability measureν^ω¯ =δ_{(ω,ζ )¯} onΩ₊. Then, for allω¯ in the (minimal closed) support ofP,

H_quen ν^ω¯

=H

ν^ω₋^¯×q_νω_¯

−|ν₋^ω¯×p⁻

= −logπ_0,0(ω).¯ (2.11)

The caseπ_0,0(ω)¯ =0 is allowed here, which can of course happen if uniform ellipticity is not assumed.

The second equality in (2.11) is clear from definitions, because the kernel is trivial: q_νω_¯

−(η₋, η₋)=1. Since Hquen(ν^ω¯)is defined by l.s.c. regularization and entropy itself is l.s.c., entropy always gives a lower bound forHquen. IfP{ ¯ω}>0 thenν^ω₀^¯ =δω_¯Pand the first equality in (2.11) is true by definition. IfP{ ¯ω} =0 pick a sequence of open neighborhoodsGj ¯ω. The assumption thatω¯ lies in the support ofPimpliesP(Gj) >0. Define a sequence of approximating measures byμ^j=_P(G¹_j)

G_jν^ωP(dω)with entropies H

μ^j×q_μj|μ^j×p⁻

= − 1 P(G_j)

G_j

logπ_0,0(ω)P(dω).

The above entropies converge to−logπ_0,0(ω)¯ by continuity ofπ_0,0(·). We have verified (2.11).

3. Multivariate level 2 and setting the stage for the proofs

The assumptions made for the main result are the union of all the assumptions used in this paper. To facilitate future work, we next list the different assumption that are needed for different parts of the proof.

The lower bounds in Theorem2.1above and Theorem3.1below do not requireΩ compact norR finite. They hold under the assumption thatPis ergodic for{Tz: z∈R}and the following two conditions are satisfied.

P{π0,z>0} ∈ {0,1} for allz∈Z^d. (3.1)

EitherE[|logπ_0,z|]<∞holds for allz∈Ror there exists a

(3.2) probability measureP_∞on(Ω,S)withP_∞Π=P_∞andP_∞P.

Note that (3.1) is a regularity condition that says that either all environments allow the move or all prohibit it.

Our proof of the upper bound uses stricter assumptions. The upper bound holds ifPis ergodic for{T_z: z∈R}, Ris finite,Ω is compact, the moment assumption (2.3) holds, and

∀x∈R,∃m∈N,∃z₁, . . . , z_m∈Rsuch thatx+z₁+ · · · +z_m=0. (3.3) On its own, (3.3) is weaker than (2.2). However, since the additive group generated byR is isomorphic toZ^d for somed≤d, we always assume, without any loss of generality, that

Z^dis the smallest additive group containingR. (3.4)

Then, under (3.4), (3.3) is equivalent to (2.2).

The only place where the conditionp > d(in (2.3)) is needed is for Lemma5.1to hold. See Remark5.3. The only place where (2.2) (or (3.3)) is needed is in the proof of (5.6) in Lemma5.5. This is the only reason that our result does not cover the so-calledforbidden directioncase. A particularly interesting special case is the space–time, or dynamic, environment; i.e. whenR⊂ {z: z·e₁=1}. A level 1 quenched LDP can be proved for space–time RWRE through the subadditive ergodic theorem, as was done for elliptic walks in [22]. Yilmaz [23] has shown that for i.i.d. space–time RWRE in 4 and higher dimensions the quenched and averaged level 1 rate functions coincide in a neighborhood of the limit velocity. In contrast with large deviations, the functional central limit theorem of i.i.d. space–time RWRE is completely understood; see [14], and also [1] for a different proof for steps that have exponential tails.

Next we turn to the strategy of the proof of Theorem2.1. The process level LDP comes by the familiar projective limit argument from large deviation theory. The intermediate steps are multivariate quenched level 2 LDPs. For each ∈Ndefine the multivariate empirical measure

R_n^1,=n⁻¹

n−1

k=0

δ_{TXkω,Zk+1,k+}.

This empirical measure lives on the spaceΩ=Ω×Rwhose generic element is now denoted byη=(ω, z_1,).

We can treatRn^1, as the position level (level 2) empirical measure of a Feller-continuous Markov chain. Denote byPη(with expectationEη) the law of the Markov chain(ηk)k≥0onΩwith initial stateηand transition kernel

p⁺ η, S_z⁺η

=π_x,x+z(ω)=π_0,z(T_xω) forη=(ω, z_1,)∈Ω, where

S_z⁺:Ω→Ω:(ω, z1,)→(T_z1ω, z2,, z).

This Markov chain has empirical measure

L_n=n⁻¹

n−1

k=0

δ_ηk

that satisfies the following LDP. Define an entropyHonM1(Ω)by H(μ)=

inf H

μ×q|μ×p⁺

: q∈Q(Ω)withμq=μ

ifμ0P,

∞ otherwise. (3.5)

His convex by an argument used below at the end of Section4. Recall Remark2.2about l.s.c. regularization.

Theorem 3.1. Same assumptions as in Theorem2.1.For any fixed≥1,for P-a.e.ω,and for allz_1,∈R,the large deviation principle holds for the sequence of probability measures P_η{L_n∈ ·}onM1(Ω)with convex rate functionH^∗∗.

The lower bound in Theorem 3.1follows from a change of measure and the ergodic theorem, and hints at the correct rate function. Donsker and Varadhan’s [6] general Markov chain argument gives the upper bound but without the absolute continuity restriction in (3.5). Thus the main issue is to deal with the case when the rate is infinite. This is nontrivial because the set of measures withμ0Pis dense in the set of probability measures with the same support asP. This is where the homogenization argument from [12,17,24] comes in.

We conclude this section with a lemma that contains the projective limit step.

Lemma 3.2. AssumeP∈M1(Ω)is invariant for the shifts{T_z: z∈R}and satisfies the regularity assumption(3.1).

Assume that for each fixed≥1there exists a rate functionI:M1(Ω)→ [0,∞]that governs the large deviation lower bound for the lawsP_η{L_n∈ ·},forP-almost-everyωand allz_1,∈Ω.Then,forP-a.e.ω,the large deviation lower bound holds forP₀^ω{R_n^1,∞∈ ·}with rate functionI (μ)=sup_≥1I(μ_|Ω),forμ∈M1(Ω₊).

WhenRis finite andΩis compact the same statement holds for the upper bound and the large deviation principle.

Proof. Observe first that P_η is the law of (T_Xkω, Z_k+1,k+)_k≥0 under P₀^ω, conditioned on Z_1, =z_1,. Since P₀^ω{Z_1,=z_1,}>0P-a.s. we have for all open setsO⊂M1(Ω),

lim

n→∞n⁻¹logP₀^ω R_n^1,∈O

≥ lim

n→∞n⁻¹log

P₀^ω{Z_1,=z_1,}P₀^ω R_n^1,∈O|Z_1,=z_1,

= lim

n→∞n⁻¹logP₀^ω R_n^1,∈O|Z_1,=z_1,

≥ −inf

O I.

Similarly, in the case of the upper bound, and whenRis finite, we have for all closed setsC⊂M1(Ω),

nlim→∞n⁻¹logP₀^ω R_n^1,∈C

≤ lim

n→∞ max

z1,∈Rn⁻¹logP₀^ω R_n^1,∈C|Z_1,=z_1,

≤ max

z1,∈R lim

n→∞n⁻¹logP₀^ω R_n^1,∈C|Z_1,=z_1,

≤ −inf

C I.

We conclude that conditioning is immaterial and,P-a.s., the laws ofR^1,_n induced byP₀^ω satisfy a large deviation lower (resp., upper) bound governed byI. The lemma now follows from the Dawson–Gärtner projective limit theorem

(see Theorem 4.6.1 in [3]).

The next two sections prove Theorem 3.1: lower bound in Section 4and upper bound in Section 5. Section6 finishes the proof of the main Theorem2.1.

4. Lower bound

We now prove the large deviation lower bound in Theorem3.1. This section is valid for a general R that can be infinite and a general PolishΩ. Lemmas 4.1and4.2are valid under (3.1) only while the lower bound proof also requires (3.2). Recall that assumption (3.4) entails no loss of generality.

We start with some ergodicity properties of the measures involved in the definition of the functionH. Recall that Ω=Ω×R and that for a measureμ∈M1(Ω),μ0is its marginal onΩ. Denote byP₀⁽⁾the law of(ω, Z1,) underP0.

Lemma 4.1. Let(Ω,S,P,{Tz})be ergodic and assume(3.1)and(3.4)hold.Fix≥1and letμ∈M1(Ω)be such thatμP₀⁽⁾.Letq be a Markov transition kernel onΩsuch that:

(a) μisq-invariant(i.e.,μq=μ);

(b) q(η, S_z⁺η) >0for allz∈Randμ-a.e.η∈Ω; (c)

z∈Rq(η, S_z⁺η)=1,forμ-a.e.η∈Ω.

Then,μ∼P₀⁽⁾and the Markov chain(ηk)k≥0onΩwith kernelq and initial distributionμis ergodic.In particular, we have for allF ∈L¹(μ)

nlim→∞n⁻¹

n−1

k=0

E^Qη F (η_k)

=E^μ[F] forμ-a.e.η. (4.1)

Here,Q_ηis the Markov chain with transition kernelqand initial stateη.

Proof. First, let us prove mutual absolute continuity. Letf= ^dμ

dP₀⁽⁾. Then, by assumptions (a) and (c), 0=

1{f=0}fdP₀⁽⁾=

1{f =0}dμ=

z∈R

q

η, S_z⁺η 1 f

S_z⁺η

=0 μ(dη).

By assumption (b), this implies that forz∈R 0=

1 f

S_z⁺η

=0

μ(dη)=

1 f S⁺_zη

=0

f (η)P₀⁽⁾(dη).

By regularity (3.1) we conclude that1{f (η) >0} ≤1{f (S_z⁺η) >0}, for allz∈R,z_1,∈R, andP-a.e.ω.

By first following the pathz1,, then taking an increment ofz∈R, then following a pathz˜1,∈R, one sees that for allz_1,,z˜_1,∈R, allz∈R, andP-a.e.ω,

1 f (ω, z1,) >0

≤1 f (Tx+zω,z˜1,) >0

. (4.2)

Now pick a finite subsetzˆ1, . . . ,zˆM∈Rthat generatesZ^das an additive group; e.g., take the elements needed for generating the canonical basise₁, . . . , e_d. Note thatM > dcan happen; e.g., taked=1 andR= {2,5}.

Applying (4.2) repeatedly, one can arrange forzto be any point of the formM

i=1kizˆi withki∈Z+. Furthermore, the ergodicity ofPunder shifts{T_z}implies its ergodicity under shifts {T_ˆz1, . . . , T_zˆM}, since the latter generate the former. We can thus average overk=(k1, . . . , kM)∈ [0, n]^M, taken→ ∞, and invoke the multidimensional ergodic theorem (see, for example, Appendix 14.A of [9]). This shows that for allz_1,,z˜_1,∈RandP-a.e.ω

1 f (ω, z1,) >0

≤P ω: f (ω,z˜1,) >0 .

Sincef integrates to 1 there exists az1,∈RwithP{f (ω, z1,) >0}>0. This implies thatP{f (ω,z˜1,) >0} =1 for allz˜_1,∈Rand henceμ∼P₀⁽⁾.

Next, we address the ergodicity issue. By Corollary 2 of Section IV.2 of [16], we have that for anyF ∈L¹(μ)and μ-a.e.η∈Ω,

nlim→∞

1 n

n−1

k=0

E^Qη F (η_k)

=E^μ[F|Iμ,q].

Here,Iμ,qis theσ-algebra ofq-invariant sets:

Ameasurable:

q(η, A)1A^c(η)μ(dη)=

q η, A^c

1A(η)μ(dη)=0

.

Ergodicity would thus follow from showing thatIμ,q isμ-trivial. To this end, letAbeIμ,q-measurable. By assump- tions (b) and (c) and mutual absolute continuity we have that for allz∈R

1A

S_z⁺η

1A^c(η)P₀⁽⁾(dη)=0.

Replacing the set{f >0}byA^c, in the above proof of mutual absolute continuity, one concludes thatP₀⁽⁾(A)∈ {0,1}.

The same holds underμand the lemma is proved.

We are now ready to derive the lower bound. We first prove a slightly weaker version.

Lemma 4.2. Let(Ω,S,P,{T_z})be ergodic and assume(3.1)and(3.4)hold.Fix≥1.Then,forP-a.e.ω,for all z_1,∈R,and for any open setO⊂M1(Ω)

lim

n→∞n⁻¹logP_η{L_n∈O} ≥ −inf H

μ×q|μ×p⁺

: μ∈O, μ₀P, q∈Q(Ω), μq=μ, and∀z∈R, q

η, S_z⁺η

>0, μ-a.s.

.

Proof. Fixμ∈Oandqas in the above display. We can also assume thatH (μ×q|μ×p⁺) <∞. Thenq(η,{S_z⁺η:z∈ R})=1μ-a.s. We can find a weak neighborhood such thatμ∈B⊂O. That is, we can findε >0, a positive integerm, and bounded continuous functionsf_k:Ω→R, such that

B= ν∈M1(Ω): ∀k≤m,E^ν[fk] −E^μ[fk]< ε .

LetFn be theσ-algebra generated byη₀, . . . , η_n. Recall thatQ_ηis the law of the Markov chain with initial stateη and transition kernelq. Then

n⁻¹logP_η{L_n∈O} ≥n⁻¹logP_η{L_n∈B}

≥n⁻¹log

E^Qη[(dQη|Fn−1/dPη|Fn−1)⁻¹1{L_n∈B}]

Q_η{L_n∈B} +n⁻¹logQη{Ln∈B} (by Jensen’s inequality, applied to logx)

≥ −n⁻¹E^Qη[log(dQη|Fn−1/dPη|Fn−1)1{L_n∈B}]

Q_η{L_n∈B} +n⁻¹logQη{Ln∈B}

=−n⁻¹E^Qη[log(dQη|Fn−1/dPη|Fn−1)]

Q_η{L_n∈B} +n⁻¹E^Qη[log(dQη|Fn−1/dPη|Fn−1)1{L_n∈/B}]

Q_η{L_n∈B} +n⁻¹logQη{Ln∈B}

=−n⁻¹H (Qη|Fn−1|Pη|Fn−1)

Q_η{L_n∈B} +n⁻¹Eη[dQη|Fn−1/dPη|_Fn−1log(dQη|Fn−1/dPη|Fn−1)1{Ln∈/B}]

Q_η{L_n∈B} +n⁻¹logQ_η{L_n∈B}

≥ −n⁻¹H (Q_η|F

n−1|P_η|F

n−1)

Q_η{L_n∈B} − n⁻¹e⁻¹

Q_η{L_n∈B}+n⁻¹logQη{Ln∈B}.

In the last inequality we usedxlogx≥ −e⁻¹. Observe next thatμandqsatisfy the assumptions of Lemma4.1. Thus, Q_η{L_n∈B}converges to 1 forμ-a.e.η. Furthermore, if we define

F (η)=

z∈R

q η, S_z⁺η

log q(η, S_z⁺η)

p⁺(η, S_z⁺η)≥0 (by Jensen’s inequality) thenE^μ[F] =H (μ×q|μ×p⁺) <∞and (4.1) implies that forμ-a.e.η

nlim→∞n⁻¹H (Qη|Fn−1|Pη|Fn−1)= lim

n→∞E^Qη

n⁻¹

n−1

k=0

F (ηk)

=E^μ[F] =H

μ×q|μ×p⁺ .