Averaged large deviations for random walk in a random environment

www.imstat.org/aihp 2010, Vol. 46, No. 3, 853–868

DOI:10.1214/09-AIHP332

© Association des Publications de l’Institut Henri Poincaré, 2010

Averaged large deviations for random walk in a random environment ¹

Atilla Yilmaz

Department of Mathematics, University of California, Berkeley, CA 94720, USA. E-mail:atilla@math.berkeley.edu;

url:http://math.berkeley.edu/~atilla/

Received 20 October 2008; revised 6 May 2009; accepted 16 July 2009

Abstract. In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment onZ^dwithd≥1, and gives a variational formula for the corresponding rate functionI_a. Under Sznitman’s transience condition (T), we show thatI_ais strictly convex and analytic on a non-empty open setA, and that the true velocity of the particle is an element (resp. in the boundary) ofAwhen the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan’s variational formula at any velocity inA.

Résumé. Dans son article de 2003, Varadhan démontre un principe de grandes déviations pour la loi moyennée de la vitesse d’une particule suivant une marche aléatoire au plus proche voisin dans un environnement i.i.d. elliptique surZ^davecd≥1, et donne une formule variationnelle pour la fonction de taux correspondanteIa. Sous la condition (T) de transience de Sznitman, nous montrons queI_aest strictement convexe et analytique dans un ouvert non videA, et que la vraie vitesse de la particule est un élément deA (resp. un élément de la frontière deA) quand la marche est “non-nichée” (resp. nichée). Nous identifions alors l’unique minimisant de la formule variationnelle de Varadhan pour toute velocité deA.

MSC:60K37; 60F10; 82C44

Keywords:Disordered media; Rare events; Rate function; Regeneration times

1. Introduction

1.1. The model

The random motion of a particle onZ^d can be modeled by a discrete time Markov chain. Writeπ(x, x+z)for the transition probability fromx tox+zfor eachx, z∈Z^d, and refer toω_x:=(π(x, x+z))_z∈Zd as the “environment”

atx. If the environmentω:=(ω_x)_x∈Zd is sampled from a probability space(Ω,B,P), then the particle is said to take a “random walk in a random environment” (RWRE). Here, Bis the Borel σ-algebra corresponding to the product topology.

LetU:= {(z₁, . . . , z_d)∈Z^d:|z₁| + · · · + |z_d| =1}. For eachz∈U, define the shiftT_z onΩ by(T_zω)_x=ω_x+z. Assume thatPis stationary and ergodic under(T_z)_z∈U,

P

π(0, z)=0

=1 unlessz∈U(i.e., the walk is nearest-neighbor), and (1.1)

∃κ >0 such thatP

π(0, z)≥κ

=1 for everyz∈U.(This is called uniform ellipticity.) (1.2)

1This research was supported partially by a grant from the National Science Foundation DMS-06-04380.

For anyx∈Z^dandω∈Ω, the Markov chain with transition probabilities given byωinduces a probability measure P_x^ωon the space of paths starting atx. Statements aboutP_x^ωthat hold forP-a.e.ωare referred to as “quenched.” State- ments about the semi-direct productP_x:=P×P_x^ω are referred to as “averaged.” Expectations underP, P_x^ω andP_x are denoted byE, E_x^ωandE_x, respectively.

Because of the extra layer of randomness in the model, the standard questions of recurrence vs. transience, the law of large numbers (LLN), the central limit theorem (CLT) and the large deviation principle (LDP) – which have well- known answers for classical random walk – become hard. However, it is possible by taking the “point of view of the particle” to treat the two layers of randomness as one: If we denote the random path of the particle byX:=(Xn)n≥0, then(TX_nω)n≥0 is a Markov chain (referred to as “the environment Markov chain”) onΩ with transition kernelπ given by

π ω, ω

:=

z:T_zω=ω

π(0, z).

This is a standard approach in the study of random media. See for example [3,7,8,10] or [11].

See [19] or [24] for a general survey of results on RWRE.

1.2. Survey of results on quenched large deviations

Recall that a sequence(Qn)n≥1of probability measures on a topological spaceXsatisfies the LDP with rate function I:X→R⁺∪ {0} ∪ {∞}ifI is lower semicontinuous, and for any measurable setG,

− inf

x∈G^oI (x)≤lim inf

n→∞

1

nlogQ_n(G)≤lim sup

n→∞

1

nlogQ_n(G)≤ −inf

x∈ ¯G

I (x).

Here,G^odenotes the interior ofG, andG¯ its closure. See [4] for general background and definitions regarding large deviations.

In the case of nearest-neighbor RWRE onZ, Greven and den Hollander [6] assume thatPis a product measure, and prove the following theorem.

Theorem 1 (Quenched LDP). ForP-a.e.ω,(P_o^ω(^X_nⁿ ∈ ·))_n≥1satisfies the LDP with a deterministic and convex rate functionI_q.

They provide a formula forI_qand show that its graph typically has flat pieces. Their proof makes use of an auxiliary branching process formed by the excursions of the walk. By a completely different technique, Comets, Gantert and Zeitouni [2] extend the results in [6] to stationary and ergodic environments. Their argument involves first proving a quenched LDP for the passage times of the walk by an application of the Gärtner–Ellis theorem, and then inverting this to get the desired LDP for the mean velocity.

Ford≥1, the first result on quenched large deviations is given by Zerner [25]. He uses a subadditivity argument for certain passage times to prove Theorem1in the case of “nestling” walks in product environments.

Definition 2. RWRE is said to be non-nestling relative to a unit vectoruˆ∈S^d−1if ess inf

P

z∈U

π(0, z)z,uˆ>0. (1.3)

It is said to be nestling if it is not non-nestling relative to any unit vector.In the latter case,the convex hull of the support of the law of

zπ(0, z)zcontains the origin.

By a more direct use of the subadditive ergodic theorem, Varadhan [20] drops the nestling assumption and gener- alizes Zerner’s result to stationary and ergodic environments. The drawback of these approaches is that they do not lead to any formula for the rate function.

Rosenbluth [15] takes the point of view of the particle and gives an alternative proof of Varadhan’s result. Moreover, he provides a variational formula for the rate functionI_q. Using the same techniques, we prove in [22] a quenched LDP for the pair empirical measure of the environment Markov chain. This implies Rosenbluth’s result by an appropriate contraction. In the same work, we also propose an Ansatz for the minimizer of the variational formula forI_q. We then verify this Ansatz for walks onZwith bounded steps.

1.3. Previous results on averaged large deviations

In their aforementioned paper concerning RWRE onZ, Comets et al. [2] prove also

Theorem 3 (Averaged LDP). (P_o(^X_nⁿ ∈ ·))_n≥1satisfies the LDP with a convex rate functionI_a.

They establish this result for a class of environments including the i.i.d. case, and obtain the following variational formula forIa:

I_a(ξ )=inf

Q

I_q^Q(ξ )+ |ξ|h(Q|P) .

Here, the infimum is over all stationary and ergodic probability measures onΩ,I_q^Q(·)denotes the rate function for the quenched LDP when the environment measure isQ, andh(·|·)is specific relative entropy. Similar to the quenched picture, the graph ofI_ais shown to typically have flat pieces. Note that the regularity properties ofI_aare not studied in [2].

Varadhan [20] considers RWRE onZ^d, assumes thatPis a product measure, and proves Theorem3for anyd≥1.

He gives yet another variational formula forI_a. Below, we introduce some notation in order to write down this formula.

An infinite path(x_i)_i≤0with nearest-neighbor stepsx_i+1−x_iis said to be inW_∞^tr ifx_o=0 and limi→−∞|x_i| = ∞. For anyw∈W_∞^tr, letnobe the number of timeswvisits the origin, excluding the last visit. By the transience assump- tion, no is finite. For anyz∈U, letno,z be the number of times wjumps to zafter a visit to the origin. Clearly,

z∈Un_o,z=n_o. If the averaged walk starts from time−∞and its path(X_i)_i≤0up to the present is conditioned to be equal tow, then the probability of the next step being equal tozis

q(w, z):=E[π(0, z)

z∈Uπ(0, z)^no,z] E[

z∈Uπ(0, z)^no,z] (1.4)

by Bayes’ rule. The probability measure that the averaged walk induces on(X_n)_n≥0conditioned on{(X_i)_i≤0=w}is denoted byQ^w. As usual,E^wstands for expectation underQ^w.

Consider the mapT^∗:W_∞^tr →W_∞^tr that takes(xi)i≤0to(xi−x₋1)i≤−1. LetI be the set of probability measures onW_∞^tr that are invariant underT^∗, andE be the set of extremal points ofI. Eachμ∈I (resp.μ∈E)corresponds to a transient process with stationary (resp. stationary and ergodic) increments, and induces a probability measureQ_μ on particle paths(X_i)_i∈Z. The associated “mean drift” ism(μ):= (x_o−x₋₁)dμ=Q_μ(X₁−X_o). Define

Q^w_μ(·):=Q_μ

·|σ (X_i: i≤0)

(w) and q_μ(w, z):=Q^w_μ(X₁=z) (1.5)

forμ-a.e.wandz∈U. Denote expectations underQ_μandQ^w_μbyE_μandE^w_μ, respectively.

With this notation, Ia(ξ )= inf

μ∈E:

m(μ)=ξ

Ia(μ) (1.6)

for everyξ=0, where I_a(μ):=

W_∞^tr

z∈U

q_μ(w, z)logq_μ(w, z) q(w, z)

dμ(w). (1.7)

Aside from showing thatI_ais convex, Varadhan analyzes the set N:=

ξ∈R^d: I_a(ξ )=0 ,

where the rate functionI_avanishes. For non-nestling walks,N consists of a single pointξ_owhich is the LLN velocity.

In the case of nestling walks,Nis a line segment through the origin that can extend in one or both directions. Berger [1]

shows thatN cannot extend in both directions whend≥5.

Rassoul-Agha [14] generalizes Varadhan’s result to a class of mixing environments, and also to some other models of random walk onZ^d.

1.4. Regeneration times Take a unit vectoruˆ∈S^d−1. Let

β=β(u)ˆ :=inf

k≥0:Xk,uˆ<Xo,uˆ .

Recursively define a sequence(τ_m)_m≥1=(τ_m(u))ˆ _m≥1of random times, which are referred to as “regeneration times”

(relative tou), byˆ τ₁:=inf

j >0: X_i,uˆ<X_j,uˆ ≤ X_k,uˆfor alli, kwithi < j < k and τm:=inf

j > τm−1: Xi,uˆ<Xj,uˆ ≤ Xk,uˆfor alli, kwithi < j < k for everym≥2. If the walk is directionally transient relative tou, i.e., ifˆ

P_o

nlim→∞X_n,uˆ = ∞

=1, (1.8)

thenP_o(β= ∞) >0 andP_o(τ_m<∞)=1 for everym≥1. As shown in [16], the significance of(τ_m)_m≥1is due to the fact that

(X_τm+1−X_τm, X_τm+2−X_τm, . . . , X_τm+1−X_τm)_m≥1 is an i.i.d. sequence underP_owhen

ω=(ω_x)_x∈Zd is an i.i.d. collection. (1.9)

Definition 4. RWRE is said to satisfy Sznitman’s transience condition(T)relative to a unit vectoruˆ∈S^d−1if(1.8) holds and

E_o

sup

1≤i≤τ1

exp

c1|X_i|

<∞ for somec1>0. (1.10)

The following theorem lists some of the important facts regarding condition (T).

Theorem 5. Consider RWRE onZ^d.Assume(1.1), (1.2)and(1.9).Take a unit vectoruˆ∈S^d−1.

(a) Ford=1, (1.8)implies(1.10).Hence, (T)is equivalent to(1.8). (See[18],Proposition2.6.)The LLN holds with limiting velocity

ξ_o=E_o[X_τ1|β= ∞]

E_o[τ₁|β= ∞] (1.11)

which can be zero.

(b) Ford≥1,if the walk is non-nestling relative tou,ˆ then Eo

exp{c2τ1}

<∞ (1.12)

for somec2>0.In particular, (T)is satisfied. (See[17],Theorem2.1.)

(c) Ford≥2,if (T)holds relative tou,ˆ then all theP_o-moments ofτ₁are finite.This implies a LLN and an averaged central limit theorem.The LLN velocityξ_ois given by the formula in(1.11),and it satisfiesξ_o,uˆ>0. (See[18], Theorems3.4and3.6.)

1.5. Our results

It follows from Theorem3and Varadhan’s lemma (see [4]) that Λa(θ ):= lim

n→∞

1 nlogEo

exp

θ, Xn

= sup

ξ∈R^d

θ, ξ −Ia(ξ )

(1.13)

for everyθ∈R^d. Hence,Λa=I_a^∗, the convex conjugate ofIa. Withc1andc2as in (1.10) and (1.12), define

C:= θ∈R^d: |θ|< c₂/2

if the walk is non-nestling, θ∈R^d: |θ|< c₁, Λ_a(θ ) >0

if the walk is nestling and condition (T) holds. (1.14) In the latter case, as we will see, Λ_ais a non-negative convex function, andC is nothing but an open ball minus a convex set.

We start Section2by obtaining a series of intermediate results including the following lemma.

Lemma 6. Consider RWRE onZ^d.Assume(1.1), (1.2)and(1.9).If (T)holds relative to someuˆ∈S^d−1,thenΛ_ais analytic onC.Moreover,the HessianHaofΛ_ais positive definite onC.

We then use (1.13) and convex duality to establish the following theorem.

Theorem 7. Under the assumptions of Lemma6,the averaged rate functionIais strictly convex and analytic on the non-empty open set

A:=

∇Λa(θ ): θ∈C

. (1.15)

(a) If the walk is non-nestling,thenAcontainsξ_o,the LLN velocity.

(b) If the walk is nestling andd=1,thenξ_o∈∂A. (c) If the walk is nestling andd≥2,then:

(i) there exists a(d−1)-dimensional smooth surface patchA^bsuch thatξo∈A^b⊂∂A,and

(ii) the unit vectorη_onormal toA^b(and pointing insideA)atξ_osatisfiesη_o, ξ_o>0. (Roughly speaking,Ais facing away from the origin.)

Remark 8. After making this work available online as part of[21],we learned that Peterson[12]independently proved Theorem7for non-nestling walks.His technique is somewhat different from ours since it involves first considering large deviations for the joint process of regeneration times and positions.Later,using that technique,Peterson and Zeitouni[13]reproduced Theorem7in its full generality.Plus,in the nestling case,they showed that

I_a(tξ )=t I_a(ξ ) for everyξ ∈A^bandt∈ [0,1]. (1.16)

Under the assumptions of Theorem7,whend≥4,we recently proved in[23]thatI_a=I_q on a closed set whose interior contains{ξ=0:Ia(ξ )=0}.Also,we gave an alternative proof of(1.16).

In Section3, we identify the unique minimizer in (1.6) for every ξ ∈A. The natural interpretation is that this minimizer gives the distribution of the RWRE path underP_o when the particle is conditioned to escape to infinity with mean velocityξ.

Definition 9. Denote the random steps of the particle by (Z_n)_n≥1:=(X_n−X_n−1)_n≥1.Assume(1.1), (1.2), (1.9) and(T).The HessianHa ofΛ_ais positive definite onC by Lemma6.Hence,for everyξ ∈A,there exists a unique θ∈C satisfying ξ = ∇Λ_a(θ ). For everyK∈N,take any bounded function f:U^N→R such that f ((z_i)_i≥1)is independent of(zi)i>K.Define a probability measureμ¯^∞_ξ onU^Nby setting

fdμ¯^∞_ξ :=E_o[τ1−1

j=0 f ((Z_j+i)_i≥1)exp{θ, X_τK −Λ_a(θ )τ_K}|β= ∞]

E_o[τ₁exp{θ, X_τ1 −Λ_a(θ )τ₁}|β= ∞] . (1.17) Theorem 10. Assume(1.1), (1.2), (1.9) and (T).Recall (1.15)and Definition9.For every ξ ∈A,μ¯^∞_ξ induces a transient process with stationary and ergodic increments via the map

(z1, z2, z3, . . .)→(z1, z1+z2, z1+z2+z3, . . .).

Extend this process to a probability measure on doubly infinite paths(x_i)_i∈Z,and refer to its restriction toW_∞^tr asμ^∞_ξ . With this notation,μ^∞_ξ is the unique minimizer of(1.6).

2. Strict convexity and analyticity

Assume (1.1), (1.2) and (1.9). If the walk is non-nestling, then (1.3) is satisfied for someuˆ∈S^d−1. If the walk is nestling, assume that (T) holds relative to someuˆ∈S^d−1.

2.1. Logarithmic moment generating function Recall (1.13). By Jensen’s inequality,

θ, ξ_o = lim

n→∞

1 nE_o

θ, X_n

≤ lim

n→∞

1 nlogE_o

exp

θ, X_n

=Λ_a(θ )≤ lim

n→∞

1 nlogE_o

e^|θ|n

= |θ|.

Lemma 11. E_o[exp{θ, X_τ1 −Λ_a(θ )τ₁}|β= ∞] ≤1for everyθ∈R^d. Proof. For everyn≥1,θ∈R^d andε >0,

E_o exp

θ, X_τn −

Λ_a(θ )+ε τ_n

= ∞ i=n

E_o exp

θ, X_τn −

Λ_a(θ )+ε τ_n

, τ_n=i

≤^∞

i=n

E_o exp

θ, X_i −

Λ_a(θ )+ε i

= ∞ i=n

e^o(i)−εi≤ ∞ i=n

e^−εi/2=e^−εn/2

1−e^−ε/2₋1

whennis sufficiently large. On the other hand, E_o

exp

θ, X_τn −

Λ_a(θ )+ε τ_n

=Eo

exp

θ, Xτ₁ −

Λa(θ )+ε τ1

Eo

exp

θ, Xτ₁ −

Λa(θ )+ε τ1

|β= ∞n−1

by the renewal structure. Hence,E_o[exp{θ, X_τ1 −(Λ_a(θ )+ε)τ₁}|β= ∞] ≤e^−ε/2. The desired result is obtained

by takingε→0 and applying the monotone convergence theorem.

Recall (1.14). For everyε >0, it is clear thatEo[exp{εu, Xˆ τ₁}|β= ∞]>1. This, in combination with Lemma11, implies thatΛa(εu) >ˆ 0. Therefore,Cis non-empty.

In the nestling case,I_a(0)=0, cf. [20]. It follows from (1.13) and convex duality that 0=I_a(0)= sup

θ∈R^d

θ,0 −Λ_a(θ )

= − inf

θ∈R^dΛ_a(θ ). (2.1)

In other words,Λ_a(θ )≥0 for everyθ∈R^d. The zero-level set{θ∈R^d: Λ_a(θ )=0}of the convex functionΛ_a is convex, andCis an open ball minus this convex set.

Lemma 12. E_o[exp{θ, X_τ1 −Λ_a(θ )τ₁}|β= ∞] =1for everyθ∈C.

Proof. Adopt the convention thatτo=0. For everyn≥1,θ∈Candr∈R, E_o

exp

θ, X_n −rn

= n

m=0

n

i=0

E_o exp

θ, X_n −rn

, τ_m≤n < τ_m+1, n−τ_m=i

= n

m=0

n

i=0

Eo

exp

θ, Xτ_m −rτm

, τm=n−i Eo

exp

θ, Xi −ri

, i < τ1|β= ∞

≤ ∞ m=0

E_o exp

θ, X_τm −rτ_m E_o

sup

0≤i<τ₁

exp

θ, X_i −riβ= ∞

=E_o

sup

0≤i<τ1

exp

θ, X_i −riβ= ∞

×

1+E_o exp

θ, X_τ1 −rτ₁^∞

m=0

E_o exp

θ, X_τ1 −rτ₁

|β= ∞m

<∞ whenever

E_o

sup

0≤i<τ₁

exp

θ, X_i −ri

<∞, E_o exp

θ, X_τ1 −rτ₁

<∞ and (2.2)

Eo

exp

θ, Xτ₁ −rτ1

|β= ∞

<1. (2.3)

Therefore, (2.2) and (2.3) imply that Λ_a(θ )−r= lim

n→∞

1 nlogE_o

exp

θ, X_n −rn

≤0. (2.4)

If the walk is non-nestling, then there exists an ε >0 such that |θ| + |Λ_a(θ )| +ε≤2|θ| +ε < c₂. Take r= Λ_a(θ )−ε. Then, (2.2) follows from Theorem5. Since (2.4) is false, (2.3) is false as well. In other words,

1≤E_o exp

θ, X_τ1 −

Λ_a(θ )−ε τ₁

|β= ∞

<∞. (2.5)

If the walk is nestling, thenΛ_a(θ ) >0 and there exists anε >0 such thatΛ_a(θ )−ε >0. Taker=Λ_a(θ )−ε.

Then, (2.2) follows from (1.10). Since (2.4) is false, (2.5) is true.

Clearly, (2.5) and the monotone convergence theorem imply thatE_o[exp{θ, X_τ1 −Λ_a(θ )τ₁}|β= ∞] ≥1. Com-

bined with Lemma11, this gives the desired result.

Lemma 13. Assume that the walk is nestling.Withc₁as in(1.10),define C^b:=

θ∈∂C: |θ|< c₁

. (2.6)

(a) If|θ|< c₁,thenθ /∈Cif and only ifE_o[exp{θ, X_τ1}|β= ∞] ≤1.

(b) If|θ|< c₁,thenθ∈C^bif and only ifE_o[exp{θ, X_τ1}|β= ∞] =1.

Proof. Recall that Λa(θ )≥0 for every θ∈R^d by (2.1). If |θ|< c1 andθ /∈C, then Λa(θ )=0 and Eo[exp{θ, Xτ₁}|β= ∞] ≤1 by Lemma11. Conversely, if|θ|< c1andEo[exp{θ, Xτ₁}|β= ∞] ≤1, thenΛa(θ ) >0 cannot be true because it would imply that

1=E_o exp

θ, X_τ1 −Λ_a(θ )τ₁

|β= ∞

< E_o exp

θ, X_τ1

|β= ∞

≤1 by Lemma12. Hence,Λa(θ )=0. This proves part (a).

Ifθ∈C^b, thenΛa(θ )=0. Takeθn∈Csuch thatθn→θ. It follows from Lemma12that E_o

exp

θ_n, X_τ1 −Λ_a(θ_n)τ₁

|β= ∞

=1.

SinceΛais continuous atθ,Eo[exp{θ, Xτ₁}|β= ∞] =1 by (1.10) and the dominated convergence theorem.

Λa is a convex function and{θ∈R^d: Λa(θ )=0}is convex. Ifθ is an interior point of this set, thenθ=t θ1+ (1−t )θ₂ for some t∈(0,1)and θ₁, θ₂∈R^d such thatθ₁=θ₂ andE_o[exp{θ_i, X_τ1}|β = ∞] ≤1 for i=1,2.

By Jensen’s inequality,E_o[exp{θ, X_τ1}|β= ∞]<1. The contraposition of this argument concludes the proof of

part (b).

Proof of Lemma6. Consider the functionψ:R^d×R→Rdefined as ψ(θ, r):=E_o

exp

θ, X_τ1 −rτ₁

|β= ∞

. (2.7)

Whenθ∈Cand|r−Λ_a(θ )|is small enough, it follows from (1.10), Theorem5and Lemma12thatψ(θ, r) <∞and ψ(θ, Λa(θ ))=1. Clearly,(θ, r)→ψ(θ, r)is analytic at such(θ, r).

If the walk is non-nestling or if it is nestling but d ≥ 2, then all the Po-moments of τ1 are finite and E_o[τ1exp{θ, X_τ1 −Λ_a(θ )τ1}|β= ∞]<∞by Hölder’s inequality and Theorem5.

If the walk is nestling andd≥1, thenΛ_a(θ ) >0, and (1.10) implies that E_o

τ₁exp

θ, X_τ1 −Λ_a(θ )τ₁

|β= ∞

≤ sup

t≥0

te^{−Λa(θ )t}

E_o exp

θ, X_τ1

|β= ∞

=

eΛa(θ )₋1

E_o exp

θ, X_τ1

|β= ∞

<∞.

In both cases, Lemma12implies that E_o

τ₁exp

θ, X_τ1 −Λ_a(θ )τ₁

|β= ∞

≥E_o exp

θ, X_τ1 −Λ_a(θ )τ₁

|β= ∞

=1.

Therefore,

∂_rψ(θ, r)|r=Λa(θ )= −E_o τ₁exp

θ, X_τ1 −Λ_a(θ )τ₁

|β= ∞

∈(−∞,−1], andΛais analytic onCby the analytic implicit function theorem. (See [9], Theorem 6.1.2.)

Differentiating both sides ofψ(θ, Λa(θ ))=1 with respect toθgives E_o

X_τ1− ∇Λ_a(θ )τ₁ exp

θ, X_τ1 −Λ_a(θ )τ₁

|β= ∞

=0 (2.8)

and

∇Λ_a(θ )=E_o[X_τ1exp{θ, X_τ1 −Λ_a(θ )τ1}|β= ∞]

E_o[τ₁exp{θ, X_τ1 −Λ_a(θ )τ₁}|β= ∞] . (2.9)

Differentiating both sides of (2.8), we see that the HessianHaofΛ_asatisfies v₁,Ha(θ )v₂

=E_o[X_τ1− ∇Λ_a(θ )τ₁, v₁X_τ1− ∇Λ_a(θ )τ₁, v₂exp{θ, X_τ1 −Λ_a(θ )τ₁}|β= ∞]

E_o[τ₁exp{θ, X_τ1 −Λ_a(θ )τ₁}|β= ∞] (2.10) for any two vectorsv₁∈R^dandv₂∈R^d.

We already saw that the denominator of the RHS of (2.10) is finite. A similar argument shows that the numerator is finite as well. Assumption (1.2) ensures that the numerator is positive whenv₁=v₂. Thus,Hais positive definite

onC.

2.2. Rate function

Proof of Theorem7. Λ_ais analytic onC, and the HessianHaofΛ_ais positive definite onC, cf. Lemma6. Therefore, for everyξ ∈A, there exists a uniqueθ=θ (ξ )∈Csuch thatξ = ∇Λ_a(θ ).Ais open since it is the pre-image ofC under the mapξ→θ (ξ )which is analytic by the inverse function theorem. Since

Ia(ξ )= sup

θ∈R^d

θ, ξ

−Λa

θ

= θ (ξ ), ξ

−Λa

θ (ξ )

, (2.11)

we conclude thatI_ais analytic atξ. Differentiating (2.11) twice with respect toξ shows that the Hessian ofI_aatξ is equal toHa(θ (ξ ))⁻¹, a positive definite matrix. Therefore,I_ais strictly convex onA.

If the walk is non-nestling, then 0∈Cand ξ_o=E_o[X_τ1|β= ∞]

E_o[τ₁|β= ∞] = ∇Λ_a(0)∈A by (1.11) and (2.9). This proves part (a).

The rest of this proof focuses on the nestling case. Whend=1, Lemma13implies that 0∈∂C. Take any(θ_n)_n≥1 withθn∈Csuch thatθn→0. Then, any limit point of(∇Λa(θn))n≥1belongs to∂A. (1.11) and (2.9) imply that

lim sup

n→∞ ∇Λ_a(θ_n)=lim sup

n→∞

Eo[Xτ₁exp{θn, Xτ₁ −Λa(θn)τ1}|β= ∞]

E_o[τ₁exp{θ_n, X_τ1 −Λ_a(θ_n)τ₁}|β= ∞] (2.12)

≤ E_o[X_τ1|β= ∞]

Eo[τ1|β= ∞] =ξ_o, (2.13)

where we assume WLOG thatuˆ=1. The numerator in (2.12) converges to the numerator in (2.13) by (1.10) and the dominated convergence theorem. The denominator in (2.13) bounds the liminf of the denominator in (2.12) by Fatou’s lemma.[0, ξo] ∩Ais empty sinceI_ais linear on[0, ξo]. (This only makes sense ifξ_o>0. However, whenξ_o=0, it is clear from (2.9) that 0∈/A.) Therefore, lim infn→∞∇Λ_a(θ_n)≥ξ_o. Hence,ξ_o=limn→∞∇Λ_a(θ_n)∈∂A.

When d ≥2, (2.9), Hölder’s inequality and Theorem 5 imply that ∇Λ_a extends smoothly to C ∪C^b. Re- fer to the extension by ∇Λa. Define A^b := {∇Λa(θ ): θ ∈ C^b}. Note that 0 ∈C^b ⊂∂C by Lemma 13, and ξ_o= ∇Λ_a(0)∈A^b⊂∂A.

The mapθ→ψ(θ,0)=Eo[exp{θ, Xτ₁}|β= ∞]is analytic on{θ∈R^d: |θ|< c1}. For everyθ∈C^b, ∇θψ(θ,0),uˆ

=E_o

X_τ1,uˆexp

θ, X_τ1

|β= ∞

>0.

Lemma13and the implicit function theorem imply thatC^b is the graph of an analytic function. Therefore,A^bis a (d−1)-dimensional smooth surface patch. Note that

∇θψ(θ,0)|θ=0=E_o[X_τ1|β= ∞] =E_o[τ1|β= ∞]ξ_o

is normal toC^bat 0. Refer to the extension ofHa toC∪C^basHa. The unit vectorη_onormal toA^b(and pointing insideA) atξ_oiscHa(0)⁻¹ξ_ofor somec >0 by the chain rule. It is clear from (1.2) and (2.10) that

ηo, ξo =c

ξo,Ha(0)⁻¹ξo

>0.

3. Minimizer of Varadhan’s variational formula

3.1. Existence of the minimizer

Varadhan’s variational formula for the rate functionIaat anyξ=0 is I_a(ξ )= inf

μ∈E:

m(μ)=ξ

I_a(μ). (3.1)

Recall (1.5). There exists a measurable functionqˆ:W_∞^tr ×U → [0,1]such that q(ˆ ·, z)=q_μ(·, z)holds μ-a.s. for everyμ∈Iandz∈U. (See [5], Lemma 3.4.) The formula (1.7) forI_acan be written as

I_a(μ)=

W_∞^tr

z∈U

ˆ

q(w, z)logq(w, z)ˆ q(w, z)

dμ(w). (3.2)

Therefore,I_ais affine linear onI.

Lemma 14.

I_a(ξ )= inf

μ∈I:

m(μ)=ξ

I_a(μ).

Proof. By the definition ofI_ain (3.1), I_a(ξ )≥ inf

μ∈I:

m(μ)=ξ

I_a(μ)

is clear. To establish the reverse inequality, take anyμ∈Iwithm(μ)=ξ. SinceE is the set of extremal points ofI, μcan be expressed as

μ=

Eo

αdμ(α)ˆ +

E\Eo

αdμ(α)ˆ =

Eo

αdμ(α)ˆ +

1− ˆμ(Eo)

˜ μ,

whereEo:= {α∈E: m(α)=0},μˆ is some probability measure onE, andμ˜∈Iwithm(μ)˜ =0. Then, I_a(μ)=

E^oI_a(α)dμ(α)ˆ +

1− ˆμ(Eo)

I_a(μ)˜ (3.3)

≥

E^oI_a m(α)

dμ(α)ˆ +

1− ˆμ(Eo)

I_a(0) (3.4)

≥I_a(ξ ). (3.5)

The equality in (3.3) uses the affine linearity of I_a. (3.4) follows from two facts: (i) I_a(α) ≥I_a(m(α)) and (ii) I_a(μ)˜ ≥I_a(0). The first fact is immediate from the definition ofI_a. See Lemma 7.2 of [20] for the proof of

the second fact. Finally, the convexity ofI_agives (3.5).

Lemma 15. IfIais strictly convex atξ,then the infimum in(3.1)is attained.

Proof. LetW_n:= {(x_i)_−n≤i≤0: x_i+1−x_i ∈U, x_o=0}. The simplest compactification of W:=

nW_n isW_∞:=

{(x_i)_i≤0: x_i+1−x_i ∈U, x_o=0}with the product topology. However, the functions q(·, z) (recall (1.4)) are only defined onW_∞^tr, and even when restricted to it they are not continuous since two walks that are identical in the

immediate past are close to each other in this topology even if one of them visits 0 in the remote past and the other one does not.

Section 5 of [20] introduces a more convenient compactificationWofW. The functionsq(·, z)continuously extend fromW toW. Denote theT^∗-invariant probability measures onW byI, and the extremals ofI byE. Recall that Eo:= {α∈E: m(α)=0}. Then,Eo⊂E⊂E andI⊂I. Note that the domain of the formula forIagiven in (3.2) extends toI.

Take μ_n∈E such that m(μ_n)=ξ and I_a(μ_n)→I_a(ξ ) as n→ ∞. Letμ¯ ∈I be a weak limit point of μ_n. Corollary 6.2 of [20] shows thatμ¯ has a representation

¯ μ=

Eo

αdμˆ1(α)+

1− ˆμ1(Eo)

¯ μ2,

whereμˆ₁is some probability measure onEo, andμ¯₂∈Iwithm(μ₂)=0. Then, I_a(ξ )= lim

n→∞I_a(μ_n)≥I_a(μ)¯ (3.6)

=

Eo

Ia(α)dμˆ1(α)+

1− ˆμ1(Eo)

Ia(μ¯2) (3.7)

≥

Eo

Ia

m(α)

dμˆ1(α)+

1− ˆμ1(Eo)

Ia(0) (3.8)

≥Ia(ξ ). (3.9)

The inequality in (3.6) follows from the lower semicontinuity ofI_a, and the equality in (3.7) is a consequence of the affine linearity ofI_a. (3.8) relies on the fact thatI_a(μ¯₂)≥I_a(0). (See Lemma 7.2 of [20] for the proof.) Finally, the convexity ofIagives (3.9). SinceIais assumed to be strictly convex atξ,μˆ1(α∈Eo: m(α)=ξ,Ia(α)=Ia(ξ ))=1.

Hence, we are done.

3.2. Formula for the unique minimizer

Fix anyξ∈A. Recall Definition9and Theorem10.

Proposition 16. μ¯^∞_ξ is well defined.

Proof. For everyK∈N, take any bounded functionf:U^N→Rsuch thatf ((zi)i≥1)is independent of(zi)i>K. Then, f ((z_i)_i≥1)is independent of (z_i)_i>K for every K> K as well. So, we need to show that (1.17) does not change if we replaceKbyK+1. But, this is clear because

E_{o τ1−1}

j=0

f

(Z_j+i)_i≥1

exp

θ, X_τK+1 −Λ_a(θ )τ_K+1β= ∞

=Eo

_τ

1−1

j=0

f

(Zj+i)i≥1

exp

θ, Xτ_K −Λa(θ )τK

×

e^{θ,XτK+1−XτK−Λa(θ )(τK+1−τK)}β= ∞

=Eo

_τ

1−1

j=0

f

(Zj+i)i≥1

exp

θ, Xτ_K −Λa(θ )τKβ= ∞

.