Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment

We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level-1 contractions in both settings. Motivation for this work comes from variational descriptions of the quenched free energy of directed polymer models where the same Markov process entropy appears.


Introduction
After surveying the background of the present work, this introductory section describes the random walk in a dynamic random environment (RWDRE) model and then some general notions such as large deviation principles and the point of view of the particle.The section concludes with an overview of the rest of the paper.
1.1.Background.This paper studies an entropy function for Markov processes that appears in random medium models.We give here some background motivation.A much-studied model is the random path in a random potential model, also called the polymer model.The random environment ω comes from a probability space (Ω, S, P) with an ergodic group action {T x } x∈Z d .The random path is a random walk X k on Z d .The potential V (ω, z) is a function of ω and a step z of the random walk.A key quantity is the limiting quenched free energy where E 0 is the expectation of the random walk and ω is fixed.The limit exists for P-almost every ω under hypotheses on the moments of V and the degree of mixing of P.
The limit g(V ) can be calculated only in a handful of exactly solvable models that exist only in 1 + 1 dimension.More generally, properties of g(V ) have remained an insurmountable problem.This question is the positive temperature version of the question of understanding limit shapes of stochastic growth models such as first-and last-passage percolation.The latter question has also remained insurmountable since the origins of the subject over 50 years ago, except for a few exactly solvable models in 1 + 1 dimension.For surveys of models of type (1.1), see [9,14].
F. Rassoul-Agha was partially supported by National Science Foundation grant DMS-1407574 and by Simons Foundation grant 306576.
T. Seppäläinen was partially supported by National Science Foundation grants DMS-1306777 and DMS-1602486, by Simons Foundation grant 338287, and by the Wisconsin Alumni Research Foundation.
A. Yilmaz was partially supported by European Union FP7 Marie Curie Career Integration Grant no.322078.
expresses g(V ) as an infimum over the L 1 (P) closure of gradients F (ω, z) = f (T z ω) − f (ω), which we called the space of cocycles.Since this formula is not the topic of the present paper, we refer to [24,38,39] for precise definitions.
The second formula gives g(V ) as the dual of an entropy adapted to the point of view of the particle: The supremum is over probability measures µ on Ω × {steps} with a natural invariance property and with a P-absolutely continuous Ω-marginal µ Ω .The entropy is given by (1.4) Formula (1.3) was proved in [39], and this formulation is Theorem 7.5 in [24].
Article [24] extended these formulas from positive to zero temperature, that is, to last-passage percolation models.The goal is to shed light on g(V ) and limit shapes through the variational formulas.The relationship between formulas (1.2) and (1.3) is well understood presently only for directed polymers in weak disorder (Examples 3.7 and 7.7 in [24]) and in periodic environments (Section 8 in [24]).
Here is a brief overview of the current state of the study of these formulas.The cocycle variational formula (1.2) has been studied in several subsequent papers while the entropy formula (1.3) has received no serious attention before the present paper.[38] shows that (1.2) always has a minimizer and uses the minimizer(s) to characterize weak and strong disorder of directed polymers.[25] proves the existence of Busemann functions for the exactly solvable 1+1 dimensional log-gamma polymer and shows that these provide minimizing cocycles for (1.2) and also a limiting polymer measure for infinite paths.[22,23] construct the minimizing cocycles for the 2-dimensional corner growth model with general i.i.d.weights and use these to investigate Busemann functions, geodesics and the competition interface.These notions have become central in the field of random medium models over the last twenty years, beginning with the work of Newman in the early 1990s on the geodesics of first-passage percolation [30].
In the current paper we begin the study of the entropy (1.4).This entropy is the level-2 projection of an entropy that appears in the rate function of a level-3 quenched large deviation principle (LDP) for RWDRE.(See (2.2) and Theorem 2.2 in Section 2.) We study the entropy in this large deviations context.In particular, we consider its relation to the entropy that serves as the rate function for a level-3 averaged LDP.
The point-to-point version of the quenched free energy (1.1) is defined for ξ in the convex hull of the support of the kernel p(z), and where [nξ] is a lattice point that approximates nξ and is reachable from the origin in n steps.The entropy variational formula now takes the form where Z 1 is the step variable under distribution µ.Formula (1.6) was proved in [36] for a directed walk in an i.i.d.environment and a local potential V ∈ L d+ε (P) for ε > 0. This formulation is Theorem 7.6 in [24].
Minimizing entropy under a mean step condition E µ [Z 1 ] = ξ as in (1.6) is also done in the level-3 to level-1 contraction in large deviation theory.For this reason the main focus of the present paper is to study these contractions, both averaged and quenched.The averaged contraction can be understood completely.Then we seek to characterize when the averaged and quenched contractions lead to the same level-1 rate function and have the same minimizers.
The quenched rate function is hard to study.It begins with an entropy of a familiar type.But this entropy is corrected in a singular manner to account for the environment distribution P, and then regularized again to be lower semicontinuous.The opaqueness of the l.s.c.regularization makes it difficult to analyze examples.By simplifying the situation so that the environment varies only temporally we can describe fully also the quenched contraction.We discover that the connection between the averaged and quenched rate functions can break down rather spectacularly.This part of the paper illuminates earlier large deviation work by Comets [7] and one of the authors [2,42] that appears in the equilibrium statistical mechanics of disordered Gibbs measures.
The present paper studies only random walk in a dynamic random environment while connections to polymer models are left for future work.Our results in Section 3 begin with the level-3 averaged LDP from the point of view of the particle and the existence of the relevant limiting specific relative entropy.After understanding the contraction from the level-3 to level-1 averaged LDP we turn to study the quenched rate functions.
denote the set of probability measures on R. Elements of Ω = P Z×Z d are called space-time environments and they are of the form ω = (ω i,x ) (i,x)∈Z×Z d .Each ω ∈ Ω defines a time-inhomogeneous discrete-time Markov chain (X i ) i≥0 on Z d for which X 0 = 0 and the transition probability from state x to y at time i is If ω is randomly sampled from a probability distribution P on (Ω, S) rather than being deterministic, then (X i ) i≥0 is a random walk (RW) in a dynamic (or space-time) random environment, which we abbreviate as RWDRE.Here, S is the Borel σ-algebra with respect to (w.r.t.) the product topology on Ω. RWDRE (started at the origin) induces a probability measure P 0 (dω, dz) = P(dω)P ω 0 (dz) on the space Ω N = Ω × R N of environments and walks.Here, z = (z i ) i≥1 ∈ R N is a sequence of steps, and P ω 0 is the quenched path measure defined by The marginal of P 0 on R N is called the averaged path measure and also denoted by P 0 whenever no confusion occurs.E, E 0 and E ω 0 stand for expectation under P, P 0 and P ω 0 , respectively.In general, we will write E µ [f ] or f, µ for the integral of a function f against a probability measure µ.
Denote the entire spatial environment at a given time i ∈ Z by ωi = (ω i,x : x ∈ Z d ).Let (T s y ) y∈Z d be the group of spatial translations, defined by (T s y ωi ) x = ω i,x+y for x, y ∈ Z d .Throughout the article, we will make the following underlying assumptions.
• Temporal independence: (ω i ) i∈Z are independent and identically distributed (i.i.d.) under P with a common distribution P s on P Z d , i.e., P = (P s ) ⊗Z .(The subscript of P s stands for "spatial".)• Spatial translation invariance: P s is invariant under (T s y ) y∈Z d .These two conditions are of course satisfied when (ω i,x ) (i,x)∈Z×Z d are i.i.d.However, restricting to that special case would not change the statements or the proofs in this paper.Moreover, it should be relatively straightforward to adapt our results to various discrete-time continuous-space models (such as RWDRE on R d considered in [4,26]) where spatial independence is not applicable.Note in particular that we do not assume ergodicity under spatial translations.
The only condition we impose on the one-step range R of the walk is 2 ≤ |R| < ∞. (|R| is the number of elements in the set R. The case |R| = 1 is trivial.)We will assume without loss of generality that P(ω 0,0 (z) > 0) > 0 for every z ∈ R. (Otherwise, we can replace R by {z ∈ R : P(ω 0,0 (z) > 0) > 0}.)Our quenched results will require various ellipticity conditions which we will indicate as needed in their statements.See also Remark 3.13.
As the name suggests, RWDRE is a variant of the much-studied random walk in a random environment (RWRE) model (see [48] for a survey).In fact, (i, X i ) i≥0 can be viewed as a directed RWRE on Z d+1 because its component in the direction of (1, 0, . . ., 0) is strictly increasing.This directedness simplifies certain aspects of the analysis of the model.Most notably, RWDRE under the averaged measure P 0 is a classical RW on Z d with transition probabilities q(z) = E[ω 0,0 (z)] > 0. In particular, the strong law of large numbers (LLN) and Donsker's invariance principle (IP) hold for the averaged walk.Since any P 0 -almost sure statement holds P ω 0 -almost surely for P-a.e. ω, there is no need for a separate strong LLN for the quenched walk.On the other hand, an averaged IP does not a priori imply a quenched one.Nevertheless, for the i.i.d.case, there is an IP under P ω 0 for P-a.e. ω [33].In stark contrast to these limit theorems, for (undirected) RWRE the validity of even the LLN is an open problem.See [3] for the best sufficient condition in the literature.
1.3.Large deviation principles, the point of view of the particle, and empirical measures.Recall that a sequence (Q n ) n≥1 of Borel probability measures on a topological space X is said to satisfy a large deviation principle (LDP) with (exponential scale n and) rate function I : X → [0, ∞] if I is lower semicontinuous, and for any measurable set G, G o is the topological interior of G and G its topological closure.See [12,13,37] for general background regarding large deviations.
In the context of RWDRE, the LDP for (P 0 (X n /n ∈ • )) n≥1 is nothing but Cramér's theorem for classical multidimensional RW (see, e.g., [37,Chapter 4]), with rate function the convex conjugate of the logarithm of the moment generating function where •, • denotes inner product.This is an averaged LDP, hence the subscript a. (The other subscript of I 1,a stands for level-1 which is explained two paragraphs below.)Establishing the analogous quenched LDP for (P ω 0 (X n /n ∈ • )) n≥1 and identifying the rate function is more arduous.It involves considering certain empirical measures from the point of view (POV) of the particle which we introduce next.
Define space-time translations (T j,y ) (j,y)∈Z×Z d on Ω by (T j,y ω) i,x = ω i+j,x+y .Then, (T i,Xi ω) i≥0 is a discrete-time Markov chain taking values in Ω, and its transition probability from state ω to state ω ′ is given by π(ω Every limit theorem about this so-called environment Markov chain implies a corresponding limit theorem for the walk.This general and robust approach was first introduced in the context of interacting particle systems [27] and was later successfully adapted to RWRE (see for example [31,34,45]).
In light of the previous paragraph, the large deviation behavior of RWDRE can be analyzed via various statistics of either the walk itself or the environment Markov chain.Among these statistics, the empirical velocity X n /n is the coarsest one, and hence its large deviation analysis is referred to as level-1.Finer statistics are provided by the occupation measure which records the environments seen from the POV of the particle.The pair-empirical measure goes one step further by essentially keeping track of the pairs of consecutive environments that the particle sees.(In the Markov chain literature, the pair-empirical measure typically refers to 1 n n−1 i=0 δ Ti,X i ω,Ti+1,X i+1 ω which is measurable w.r.t.our choice of L 2 n .)Pairs can be replaced with ℓ-tuples for any ℓ ≥ 2 to define more detailed empirical measures.Large deviations of each of these empirical measures are called level-2.Finally, level-3 involves the so-called empirical process Here and throughout, Z = (Z i ) i≥1 denotes the sequence of steps Z i = X i −X i−1 of the random path (X i ) i≥0 , and θ is the forward shift on sequences, i.e., (θZ) j = Z j+1 for every j ∈ N.Under the topology of weak convergence of measures, the empirical process contains precisely the same information as all of the empirical measures for ℓ-tuples combined.Level-1,2,3 large deviations for Markov processes were established (under certain conditions) in a series of papers by Donsker and Varadhan [16,17,18].The level terminology was introduced later in [20].
We use this notation to introduce the σ-algebras The reason for the indexing convention is that the distribution of step Z n+1 is part of environment ωn .Note also that A 0,∞ −∞,∞ and A −∞,∞ −∞,∞ are the Borel σ-algebras (w.r.t. the product topology) on Ω N = Ω × R N and Ω Z = Ω × R Z , respectively.For any σ-algebra F , the space of bounded and F -measurable functions is denoted by bF .1.5.Content and organization of the article.Section 2 reviews previous results on large deviations for RWDRE.The new results are in Section 3. The paper is organized so that the results of Section 3.n are proved in Section 3 + n.Section 3 concludes with remarks and open problems.The following list summarizes the results (with proofs in the indicated sections): (i) level-3 averaged LDP for the joint environment-path Markov chain (Section 4); (ii) analysis of the averaged contraction from level-3 to level-1 (Section 5); (iii) alternative formula for the level-3 quenched rate function (Section 6); (iv) relationship of level-3 averaged and quenched rate functions (Section 7); (v) characterizations of the equality of level-1 averaged and quenched rate functions (Section 8); (vi) minimizers of quenched contractions from level-3 to level-1 (Section 9); (vii) spatially constant environments (Section 10).

Summary of previous results on large deviations
Recall from the Introduction that the level-1 averaged LDP, i.e., the LDP for P 0 Xn n ∈ • n≥1 , is simply the multidimensional Cramér theorem with the rate function I 1,a given in (1.8), whereas the statement and the proof of its quenched counterpart is relatively technical.In fact, it is more convenient to first present the level-3 quenched LDP for the environment Markov chain, and we will proceed in this order.
Let S denote the temporal shift operator from the POV of the particle.It acts on Ω N = Ω × R N via S(ω, z) = (T 1,z1 ω, θz), and on Ω Z = Ω × R Z via S(ω, z) = (T 1,z1 ω, θz).On Ω Z S is invertible.We can write S k (ω, z) = (T k,x k ω, θ k z) for all k ∈ Z, with this convention: bi-infinite paths x through the origin and sequences z ∈ R Z are bijectively associated to each other by (2.1) Remark 2.1.The empirical process L ∞ n defined in (1.10) satisfies Thus, L ∞ n is an asymptotically S-invariant element of M 1 (Ω N ) for P-a.e. ω and every realization of Z ∈ R N .For any S-invariant µ ∈ M 1 (Ω N ), let (i) μ be the unique S-invariant extension of µ to Ω Z , (ii) μ− the restriction of μ to A −∞,0 −∞,∞ , and (iii) Projecting this entropy to A 0,1 −∞,∞ and replacing π 0,1 (0, z | ω) with a constant jump kernel p(z) gives the entropy (1.4) discussed in the Introduction.
The rate function of the level-3 quenched LDP is obtained via the following modification of H q .For any µ ∈ M 1 (Ω N ), denote its Ω-marginal by µ Ω , and set (2.3) H S q,P (µ) = H q (µ) if µ is S-invariant and µ Ω ≪ P, ∞ otherwise.
H S q,P is convex but not lower semicontinuous, and the double convex conjugate (H S q,P ) * * of H S q,P gives its lower semicontinuous regularization (see [37,Theorem 4.17]).
Then, for P-a.e. ω, (P ω 0 (L ∞ n ∈ • )) n≥1 satisfies an LDP with rate function I 3,q : M 1 (Ω N ) → [0, ∞] given by I 3,q (µ) = (H S q,P ) * * (µ).This result is a special case of the level-3 quenched LDP we established in [39] for a class of models including both directed and undirected RWRE with a rather general but technical condition on the environment measure.We show in Proposition A.2 in Appendix A that this technical condition holds in our current setting under the ellipticity assumption (2.4).
Since the empirical velocity is a bounded and continuous function of the empirical process, the level-1 quenched LDP follows immediately from Theorem 2.2 via the contraction principle (see, e.g., [37,Chapter 3]).
Originally developed in [43] for undirected RWRE, this second method is less technical and it avoids empirical measures, but it does not provide any formula for the rate function I 1,q .Let D := conv(R) denote the convex hull of R, and ξ * := z∈R q(z)z stand for the LLN velocity of the walk.The following proposition lists some elementary facts regarding the level-1 averaged and quenched rate functions.We provide its proof in Appendix B for the sake of completeness.
(a) I 1,a and I 1,q are convex and continuous on D.
z) for every z ∈ R that is an extreme point of D (unless ω 0,0 (z) is deterministic).
Under additional assumptions, the following further results have been obtained regarding the comparison of the level-1 averaged and quenched rate functions in relation with the spatial dimension.
Then, the following hold at the indicated spatial dimensions. (a Examining the proofs given in the references reveals that the last two conditions in (2.7) can be replaced with somewhat weaker versions.However, the spatial independence of the environment is crucial to the proofs and cannot be relaxed much.
There are other previous results on large deviations for RWDRE such as the ones in [44] regarding the analysis of the averaged and quenched contractions from level-3 to level-1, but we prefer to mention them in later parts of this paper because they will be either covered by our new results or used in the proofs.
Our temporal independence assumption excludes various concrete models such as RW on particle systems.Level-1,2,3 quenched LDPs for such models (which satisfy uniform ellipticity (2.6)) are covered in [39], but averaged LDPs are open in general.See [1] for level-1 averaged and quenched LDPs for RW on onedimensional shift-invariant attractive spin-flip systems.Finally, for previous results on large deviations for RWRE and closely related models, see [46, Section 2], [32] and [39, Section 1.3], and the references therein.

3.1.
Level-3 averaged LDP.For any S-invariant µ ∈ M 1 (Ω N ), the specific relative entropy exists, where is the entropy of µ relative to P 0 on A k,ℓ k,ℓ .The existence of the limit and the identity in (3.1) follow from superadditivity and the independence built into P 0 , and will be justified in Section 4.
Our first result in this paper is the averaged counterpart of Theorem 2.2.Note that it requires only the temporal independence and spatial translation invariance conditions which we assume throughout the paper (see Section 1.2).
), but there is a unique S-invariant probability measure P ∞ 0 on Ω Z that agrees with P 0 on A 0,∞ 0,∞ (see Lemmas 4.2 and 4.3).The LDP of Theorem 3.1 is valid also for the distributions and will in fact be proved first for these.Similar to Corollary 2.3, the contraction principle gives the following (infinite-dimensional) variational formula for the level-1 averaged rate function: Since (1.8) is a much simpler formula than (3.4), the significance of the latter lies not in providing a numerical value for I 1,a (ξ), but in the questions it raises regarding the minimizer(s) of this variational formula, which we pursue next.

3.2.
Minimizer of the averaged contraction.Recall from (1.8) that the level-1 averaged rate function I 1,a is the convex conjugate of the logarithm of the moment generating function φ a defined in (1.9).We have not assumed that D has nonempty interior.Consequently I 1,a is not necessarily differentiable, and instead of its gradient we have to work with the set-valued subdifferential ∂I 1,a (ξ).Facts from convex analysis and some proofs of the claims below are collected in Appendix C. Let ξ ∈ ri(D), the relative interior of D. By basic convex analysis, every ρ ∈ ∂I 1,a (ξ) maximizes in (1.8), that is, is a nonempty affine subset of R d parallel to the orthogonal complement of the affine hull of R. From this last point it follows that any ρ ∈ ∂I 1,a (ξ) can be used below to define a measure Proposition 5.1 in Section 5 provides basic properties of µ ξ , beginning with its well-definedness.
The second result in this paper identifies µ ξ as the unique minimizer of the averaged contraction from level-3 to level-1.
Measure µ ξ was introduced in [44, Definition 1] with different notation and under the stronger assumptions in (2.7).Theorem 3.3 follows from an adaptation of [44, Theorem 1] which roughly says that, conditioned on {X n /n ≈ ξ}, the empirical process L ∞ n converges to µ ξ under P 0 .See Proposition 5.3 for the precise statement.
Next we start analyzing the structure of the averaged contraction minimizer µ ξ ∈ M 1 (Ω N ).First of all, µ ξ is S-invariant (see Proposition 5.1(a)).Using the notation introduced in Section 2, let μξ be the unique S-invariant extension of µ ξ to Ω Z , and for z ∈ R.
Proposition 3.4.For every ξ ∈ ri(D), j ≥ 0, and z ∈ R, Hence, the quenched walk under μξ is Markovian, and its transition kernel We denote the Ω-marginal of µ ξ by µ ξ Ω .The proof of Proposition 3.4 in Section 5 shows that, by martingale convergence, the transition kernel in (3.7) is given by for any ρ ∈ ∂I 1,a (ξ).The following result provides a characterization of the absolute continuity of µ ξ Ω in terms of a structural representation of π ξ 0,1 involving a Doob h-transform.
(i) There exists a function u ∈ L 1 (Ω, S 0,∞ , P) such that P(u > 0) = 1 and then (ii) =⇒ (i).Furthermore, whenever (i) holds, u is equal (up to a multiplicative constant) to The proof that (i) implies (ii) in Theorem 3.5 is adapted from that of [38,Lemma 4.1] which is concerned with disorder regimes of directed random walks in random potentials.The other implication follows from (3.8) under the mild ellipticity condition (3.10) which ensures that µ ξ Ω and P are in fact mutually absolutely continuous on S 0,∞ .For closely related results on directed polymers and ballistic (undirected) RWRE, see [11,Proposition 3.1] and [46,Theorem 3.3], respectively.Remark 3.6.When we choose ξ to be the LLN velocity ξ * = z∈R q(z)z, we can take ρ = 0 because , the original kernel, and in (3.9) we can take u ≡ 1.Thus µ ξ * Ω = P on S 0,∞ , which is also evident directly from the definition of µ ξ * in (3.5).
When d ≥ 3 and the conditions in (2.7) hold, it was shown by one of the authors [44, Theorem 4] that statements (i) and (ii) in Theorem 3.5 are true not only at ξ = ξ * but also for ξ sufficiently close to ξ * , and in this case u ∈ L 2 (Ω, S 0,∞ , P).

3.3.
Modified variational formulas for the quenched rate functions.Recall from (2.3) that the formula given in Theorem 2.2 for the level-3 quenched rate function I 3,q involves absolute continuity w.r.t.P (on S).This formula is valid for a general class of RWRE models.However, in the case of RWDRE, as we have seen in Proposition 3.4 and Theorem 3.5, the relevant σ-algebra is S 0,∞ .Therefore, we next provide appropriately modified formulas for I 3,q and I 1,q which will be central to some of our subsequent results.Define (3.11) H S,+ q,P (µ) = Theorem 3.7.Assume (2.4).Then, for every µ ∈ M 1 (Ω N ), (3.12)I 3,q (µ) = (H S,+ q,P ) * * (µ).Corollary 3.8.Assume (2.4).Then, for every ξ ∈ ri(D), Example 3.9.The need for Theorem 3.7 is justified by the fact that H S q,P (µ) = H S,+ q,P (µ) does not hold in general.The following counterexample is adapted from [5].Assume (2.7) and the following extra condition on the law of the environment: For P-a.e. ω, the quenched walk under this new kernel is deterministic, the law of the environment Markov chain (T i,Xi ω) i≥0 converges weakly to a π ′ -invariant probability measure Q on Ω (see [5,Proposition 1.4]), 3.4.Decomposing the level-3 averaged rate function.The level-3 averaged and quenched LDPs hold with rate functions I 3,a and I 3,q given in (3.3) and (3.12), respectively.Note that I 3,a (µ) ≤ I 3,q (µ) for every µ ∈ M 1 (Ω N ).This follows from Jensen's inequality applied to the convex conjugates of the rate functions, and is shown in Corollary 3.11 for the sake of completeness.How are these two rate functions related beyond this basic inequality?The following theorem provides a partial answer.Additional remarks follow in Section 3.8.
Theorem 3.10.For every S-invariant µ ∈ M 1 (Ω N ), Theorem 3.10 is an application of the chain rule for relative entropy (see [15,Lemma 4.4.7]).It does not require any ellipticity condition.H S0,n (µ Ω | P) is the entropy of µ Ω relative to P on S 0,n , and h S0,∞ (µ Ω | P) is the specific relative entropy whose existence is shown in the proof of Theorem 3.10.
They are all strictly weaker than uniform ellipticity (2.6).
Regarding the equality of the level-3 averaged and quenched rate functions, the following result provides a sufficient condition.It is also noteworthy that under the stronger condition of uniform ellipticity, the entropy H S0,n (µ Ω | P) can grow at most sublinearly for the absolutely continuous marginals of S-invariant measures.
The lower semicontinuity of (H S,+ q,P ) * * and the compactness of {µ ∈ M 1 (Ω N ) : E µ [Z 1 ] = ξ} ensure that the quenched contraction (3.13) always has a minimizer.On the other hand, there is currently no general existence result for minimizers of the quenched contraction (3.14).See Section 3.8 for further remarks.3.7.Spatially constant environments.We illustrate our results in a simplified setting where the spatial variation of the environment is removed.The quenched process Z is now a process of independent but not identically distributed variables.LDPs for such processes were originally established in [2,7,42], motivated in part by their application to the equilibrium statistical mechanics of disordered lattice systems such as the Ising or Curie-Weiss models with random fields or coupling constants.(Some of these large deviation results have been reproduced in Chapter 15 of the textbook [37].)The novelty we provide here is the identification of the averaged and quenched contraction minimizers.We find that many properties such as equality of averaged and quenched rate functions and minimizers fail.
Take a Borel probability measure λ on P (defined in (1.7)).Let (q i ) i∈Z be sampled from P Z according to λ ⊗Z .Define ω ∈ Ω = P Z×Z d by setting (3.18) ω i,x = qi for every i ∈ Z and x ∈ Z d .This induces a probability measure P on (Ω, S).Environments under P are temporally i.i.d. and spatially constant.Hence, P is invariant but not ergodic under the spatial translations (T s y ) y∈Z d .
For ρ ∈ R d and ω ∈ Ω, define ]. Observe that E[W (ρ, ω)] = φ a (ρ).For the sake of eliminating trivial cases where the environment is effectively deterministic, we assume that The condition φ a (ρ) = e ρ,ξ * is the same as ρ ∈ ∂I 1,a (ξ * ) (Proposition C.3 in Appendix C).We start our study by giving a simple formula for the level-1 quenched rate function and showing that it is not equal to the averaged one at any atypical velocity.Proposition 3.17.Assume (2.4) and (3.18).Then, for every ξ ∈ D, If ξ ∈ ri(D) \ {ξ * } and (3.19) holds, then the inequality in (3.20) is strict.
Remark 3.18.In Proposition 3.17, we assume (2.4) in order to apply Corollary 2.3.In fact, when the environment is spatially constant, a weaker ellipticity condition is sufficient for the level-1 quenched LDP, but we do not pursue such technical improvements here.
Since the environments are spatially constant, under the quenched conditioning on ω the Ω-marginal of the empirical process L ∞ n of (1.10) is a deterministic measure that converges to P. Consequently the quenched rate must blow up at measures with the "wrong" Ω-marginal.This was observed in [7, Theorem III.1] and [42,Theorem 3.4].
If (2.4), (3.18) and (3.19) hold, then it follows from Propositions 3.17, 3.20 and 3.21 that all four statements in Theorem 3.12 are false for every ξ ∈ ri(D) \ {ξ * }, which is consistent with their equivalence.Proposition 3.21 shows in a striking way how the alteration of the entropy H q can completely remove the averaged minimizers µ ξ from the effective domain of the quenched rate function.In particular, µ ξ cannot be a minimizer of the quenched contractions (3.13) or (3.14).Our final result identifies the minimizer(s) of these quenched contractions.
When I 1,a (ξ) < I 1,q (ξ), identifying the minimizers (if any) of the quenched contractions (3.13) and (3.14) or saying anything about their structure is an open problem in general.Note that (3.13) always has a minimizer (see Remark 3.16).In contrast, we expect that (3.14) has no minimizers when the environment (ω i,x ) (i,x)∈Z×Z d is i.i.d., but this is yet to be shown.On the other hand, in the case of spatially constant environments, Proposition 3.22 provides the unique minimizer of both of these quenched contractions.
In a recent article [38], we obtained results on the existence and identification of minimizers of variational formula (1.2) and its counterpart for the annealed free energy.This covers the logarithmic moment generating functions log φ a (ρ Xn ] for RWDRE.These functions are the convex conjugates of I 1,a and I 1,q , respectively, by Varadhan's lemma.In future work, we hope to combine these previous results with the current ones and thereby deepen our understanding of the large deviation behavior of RWDRE.

3.8.2.
Connecting the rate functions.How the averaged and quenched rate functions are related to each other is an important question in the study of processes in random environments.For example, at level-1, obtaining an expression for I 1,a in terms of I 1,q would provide us with valuable information regarding how the path and the environment conspire towards the realization of atypical velocities.This question is answered with variational formulas in [8] for one-dimensional nearest-neighbor classical RWRE under the i.i.d.environment assumption and in [21] for the exactly solvable corner growth model with random parameters.It is an open problem for example for RWRE in higher dimensions or under more general conditions.
In the context of RWDRE, Theorem 3.10 provides a partial answer to the aforementioned question at level-3 since it connects I 3,a (µ) = h(µ | P 0 ) with I 3,q (µ) = (H S,+ q,P ) * * (µ) only indirectly via H q (µ).This reduces the original question to understanding the variational expression (H S,+ q,P ) * * (µ) − H q (µ), which is one of our goals for future work.So far, we know that this difference is nonnegative (see Corollary 3.11), and equal to zero at µ ξ if and only if I 1,a (ξ) = I 1,q (ξ) (see Theorem 3.12).

3.8.3.
Equality of the rate functions.When I 1,a (ξ) = I 1,q (ξ) at an atypical velocity ξ ∈ ri(D) \ {ξ * }, the walk is solely responsible (in the exponential scale) for the occurrence of the rare event {X n /n ≈ ξ} under the joint measure P 0 .Theorem 3.12 makes this precise by the statement h S0,∞ (µ ξ Ω | P) = 0. Theorem 2.5 lists the previous results regarding the equality of the level-1 rate functions.The decisive statement for d = 1 is believed to be true also for d = 2.In contrast, recalling Proposition 2.4 (a,d), both C = {ξ ∈ D : I 1,a (ξ) = I 1,q (ξ)} and D \ C have nonempty interiors when d ≥ 3. Hence, there is a phase transition at the boundary of C, and we would like to analyze the structure of µ ξ when ξ ∈ ∂C.The characterizations in Theorem 3.12 can potentially shed light on this problem.Theorem 3.15(b) provides a sufficient condition for I 1,a (ξ) = I 1,q (ξ), namely µ ξ Ω ≪ P on S 0,∞ .Whether this condition is also necessary for I 1,a (ξ) = I 1,q (ξ) is an important open problem which is related to the existence of the critical (i.e., strong but not very strong) disorder regime for directed polymers.See [ decays exponentially to zero as n → ∞ (equivalently I 1,a (ξ) < I 1,q (ξ)).
Here, ξ is a multidimensional analog of inverse temperature, with the LLN velocity ξ * corresponding to infinite temperature.It is tempting to connect this problem of critical disorder with the previous one regarding the structure of µ ξ at the boundary of C, but we refrain from proposing any conjectures.

Level-3 averaged LDP from the point of view of the particle
We start this section with an important point regarding the relative entropies H k,ℓ (µ | P 0 ) defined in (3.2).We will refer to this point below in the proof of Theorem 3.1.
Remark 4.1.It is not necessarily the case that H k,ℓ (µ | P 0 ) = H 0,ℓ−k (µ | P 0 ) for S-invariant µ ∈ M 1 (Ω N ) and 0 < k < ℓ.This is because the distribution of (ω i , Z i+1 ) under P 0 changes with i.Here is an example: The simplest S-invariant probability measure on Ω N is of the product type Take ν(dω i ) = P s (dω i ) and α = δ z for some fixed z ∈ R.Then, Proof of Theorem 3.1.We will transform the problem into a level-3 LDP for an i.i.d.sequence on the space Ω Z = Ω × R Z .First, we define the measure on Ω Z that will give the desired i.i.d.sequence.
Recall the definition (2.1) Note that ϕ ω −n (z −n+1,0 ) is not a probability distribution on vectors z −n+1,0 because it does not sum up to one.Set (4.1) On the σ-algebra A −n,∞ −n,∞ , we define a measure P (−n) by setting for every m ∈ N. Here, c R denotes the counting measure on R.
Lemma 4.2.( P (−n) ) n∈N are consistent probability measures and hence they induce a probability measure P ∞ 0 on Ω Z .
Proof.For every m, n ∈ N and every test function f ∈ bA −n,m −n,∞ , Here, (4.2) uses the temporal independence of the environment, whereas (4.3) follows from exchanging the order of the last integral and sum, recalling the spatial translation invariance assumption, and restoring the order of the last integral and sum.
Recall that (T s y ) y∈Z d are spatial translations defined by (T s y ωj ) x = ω j,x+y for j ∈ Z and x, y ∈ Z d .We use these translations to introduce the so-called slab variables (4.4) This choice of terminology comes from viewing RWDRE in Z d as a directed RWRE in Z d+1 .Note that s j is centered at the point x j on the path.In this sense, the slab variables are adapted to the POV of the particle.Equivalently, they satisfy s j = s 0 • S j for j ∈ Z.For any pair of indices −∞ < k ≤ ℓ < ∞, we write s k,ℓ = (s k , s k+1 , . . ., s ℓ ).
Lemma 4.3.P ∞ 0 is S-invariant and the slab variables (s j ) j∈Z are i.i.d.under P ∞ 0 .
Proof.This is an immediate consequence of the following induction steps.Let E ∞ 0 stand for expectation under by temporal independence and spatial translation invariance, where f and g are test functions on appropriate spaces.Similarly, Denote the full sequence of slab variables by s = (s j ) j∈Z and let τ be the temporal shift on these sequences, defined by (τs) j = s j+1 .With this notation, the empirical process induced by the slab variables is The LDP for L slab n under P ∞ 0 is an instance of the well-known Donsker-Varadhan level-3 LDP for sequences of i.i.d.random variables taking values in Polish spaces, see, e.g., [37,Chapter 6].We state this result as Proposition 4.4 below, after some preparation for providing a formula for the corresponding rate function.
We can glue together the environment components of the slab variables to form an ω ′ ∈ Ω with ω′ j = T s xj ωj for j ∈ Z, and thereby identify the space of slab sequences with Ω Z .(For the sake of convenience, we will write ω instead of ω ′ .)This identification already factors in the POV of the particle, and the shift τ acts on (environment, path) pairs simply by In other words, the sequence s can be thought of as a bijective map on Ω Z .It induces a τ -invariant distribution P ∞ 0 • s−1 on this space.Since the σ-algebras A k,ℓ k,ℓ are now regarded as being generated by the i.i.d.slab variables s j , the problem with shifting relative entropy (cf.Remark 4.1) disappears.For τ -invariant probability measures Q on Ω Z , the specific relative entropy exists by a standard superadditivity argument (see [37,Theorem 6.7]).
given by Next, we transform this LDP into one for the empirical measures The inverse of the map (ω, z) → s is γ : Apply the contraction principle to the LDP in Proposition 4.4 with the map This gives an LDP for (P ∞ 0 (L Z n ∈ • )) n≥1 with rate function Since γ acts bijectively on any collection of adjacent slabs that includes the zeroth slab and hence preserves ).Thus, using (4.5), we can define a specific relative entropy for S-invariant Q by restricting the intervals [k, ℓ) to include 0: The statement of the LDP we have established is simplified as follows.
Proposition 4.5.(P ∞ 0 (L Z n ∈ • )) n≥1 satisfies an LDP with rate function given by As the last step, we transform this LDP into the one we want.Denote the natural Ω Z → Ω N projection by Φ(ω, z) = (ω, z).We can think of the empirical process L ∞ n (introduced in (1.10)) as a function from We drop the projection from the notation since the coordinates of θ i z = z i+1,∞ are defined on Ω Z as well as on Ω N .The contraction principle gives an LDP for Recall that, for any S-invariant µ on Ω N , μ denotes the unique S-invariant extension to Ω Z .By the Sinvariance of both μ and P ∞ 0 , the entropies can be shifted to nonnegative levels so that The first equality above follows from the observation that f → f • S −k and g → g • S k are bijections between bA k,ℓ k,ℓ and bA 0,ℓ−k 0,ℓ−k for k ≤ 0 < ℓ. (However, this is not the case when 0 < k < ℓ, cf.Remark 4.1.)The second equality is valid because μ and µ (resp.P ∞ 0 and P 0 ) agree at nonnegative times.Comparing (3.1) and (4.6), we conclude that I ∞ 3,a is equal to the level-3 averaged rate function I 3,a defined in (3.3).It remains to transfer the LDP from ( separately for lower and upper bounds.This works easily because (i) weak topology on M 1 (Ω N ) is determined by finite-dimensional distributions, (ii) the dependence of L ∞ n on environments at negative times vanishes as n → ∞, and (iii) the measures P 0 and P ∞ 0 agree on environments and steps at nonnegative times.We leave the routine details to the reader.This completes the proof of Theorem 3.1.
where the first and third equalities use (b) and (c), respectively.(See (C.2) in Appendix C for the last equality.)(e) This is immediate from (a) and (d).
The main ingredient in the proof of Theorem 3.3 is the following result.
This result is essentially [44, Theorem 1] which is stated there under the assumptions in (2.7) which are more stringent than our current assumptions.For the sake of completeness and convenience, we provide below a streamlined adaptation of the proof to our setting.
By a standard change-of-measure argument and the level-1 averaged LDP, we see that for any s > 0, lim sup where the last line follows from the exponential Chebyshev inequality.Let G j = g • S j for 0 ≤ j ≤ n − 1 and note that holds by Hölder's inequality under e ρ,Xn −n log φa(ρ) dP 0 .
For 0 The boundedness of f (and, hence, of g) allows us to control the first and last expectations.Therefore, Recalling (5.2), we deduce the following inequality: Note that ζ(0) = 0 and Therefore, ζ(s) = o(s) as s → 0, and it follows from (5.1) that lim sup Combining this inequality with the analogous one for −f (and, hence, −g), we obtain the desired result.
Proof of Theorem 3.3.We checked in Proposition 5.1(e) that µ ξ is a minimizer of (3.4).It remains to rule out other minimizers.For every for every δ > 0, which is an open set.Therefore, by the level-3 and level-1 averaged LDPs.Limit superior as δ → 0 gives −I 3,a (ν) < −I 1,a (ξ) by Proposition 5.3.Thus ν cannot be a minimizer.
Let m, n ≥ 1 and j ≥ 0. The calculation below shows that ).Take a test function f ∈ bA −m,j −m,j+n .Then, by S-invariance of μξ , the definition (3.5) of µ ξ , and two uses of the Markov property of the quenched walk, This verifies (5.3).Let m → ∞ in (5.3).Martingale convergence yields ).For the case j = 0, by the S 0,n -measurability of α n (•, z), In the last expression above μξ can be replaced with µ ξ since the statement does not involve the backward path.Combining the last two displays gives, for j ≥ 0 and n ≥ 1, As n → ∞ martingale convergence yields (3.6).The remainder of Proposition 3.4 follows from this.
For every ξ ∈ ri(D), (5.5) is a positive martingale on (Ω, S 0,∞ , P).Throughout the paper, we will sometimes suppress ρ and simply write u n or u n (ω) whenever it does not lead to any confusion.
Finally, we derive (3.9):For P-a.e. ω and every z ∈ R, Note that the second equality in (5.9) follows from martingale convergence under µ ξ Ω which is mutually absolutely continuous with P since P(u > 0) = 1.

Modified variational formulas for the quenched rate functions
Fix a sequence (f j ) j∈N of test functions f j ∈ bA 0,j −j,∞ that separate M 1 (Ω N ) and satisfy f j ∞ = 1.For every µ ∈ M 1 (Ω N ) and ℓ ∈ N, the set The following result gives the lower bound in Theorem 3.7.Theorem 6.1.Assume that Then, for every ℓ ∈ N and every S-invariant µ ∈ M 1 (Ω N ) such that µ Ω ≪ P on S 0,∞ , (6.2) lim inf Proof.The proof uses a strategy involving a change-of-measure, Jensen's inequality, and the ergodic theorem, which is standard for obtaining LDP lower bounds for Markov chains, and has been successfully carried out in the context of (undirected) RWRE (see [41,45,35] for the level-1,2,3 quenched LDPs).In fact, keeping future applications in mind, the level-3 quenched LDP lower bound was derived in [35,Section 4] in full detail and without using the assumption that the walk is undirected.In particular, the lower bound of the LDP in Theorem 2.2 is covered by [35,Section 4], which readily implies that (6.2) holds for every S-invariant µ ∈ M 1 (Ω N ) such that µ Ω ≪ P (on S).Therefore, to prove Theorem 6.1, we need to replace S with S 0,∞ .Since the walk is directed in time, this modification requires only two minor changes in the proofs in [35,Section 4].Below we go over the whole argument for the sake of completeness, point out the two differences, and provide references for further details.
Step 4. If µ Ω ≪ P on S 0,∞ but (6.3) fails to hold, we introduce a μ ∈ M 1 (Ω N ) of the form μ(dω, dz) = P(dω) ⊗ p ⊗N (dz) for some (deterministic) p ∈ P such that p(z) > 0 for every z ∈ R. Note that (6.5) H q (μ) = E z∈R p(z) log p(z) ω 0,0 (z) < ∞ by (6.1).(In fact, this is the only point in the proof where (6.1) is fully used.)We replace µ with µ ǫ = (1 − ǫ)µ + ǫμ which is an element of the open set G µ,ℓ for sufficiently small ǫ > 0. Since μ is S-invariant, its marginal μΩ ℓ on Ω ℓ is an invariant measure for the transition kernel The Ω ℓ -marginal µ ǫ Ω ℓ of µ ǫ is an invariant measure for a transition kernel π The last inequality follows from the convexity of the relative entropy H q .Finally, we send ǫ to 0, and recall (6.4) to deduce (6.2) as in Step 3.
We are now ready to verify the modified variational formula for the level-3 quenched rate function.
Proof of Corollary 3.8.For every ξ ∈ D, the variational formula I 1,q (ξ) = inf{(H S,+ q,P ) * * (µ) : µ ∈ M 1 (Ω N ), E µ [Z 1 ] = ξ} follows immediately from Theorem 3.7 by the contraction principle.Define Ĩ1,q (ξ) = inf{H S,+ q,P (µ) : which is equal to the RHS of (3.14).Ĩ1,q (ξ) < ∞ because we can choose p have mean ξ in the measure μ in (6.5).Since H S,+ q,P is convex (which readily follows from the convexity of H q ), Ĩ1,q is convex on D and hence continuous on ri(D).For every ξ ∈ ri(D), by the fact that (H S,+ q,P ) * * is the lower semicontinuous regularization of H S,+ q,P (see [37,Theorem 4.17]) and by the chain rule for relative entropy.We can apply the chain rule repeatedly and thereby successively remove all the z-coordinates from the first relative entropy on the RHS.The general step is, for 1 When all z-coordinates have been removed, we end up with this identity: Proof.The relative entropy on the RHS of (7.2) is an upper bound on each term in the sum on the LHS.On the other hand, if simultaneously j ր ∞ and n − j ր ∞, then which implies the desired result.
Recall u n (ρ, ω, x) from definition (5.7).When ρ is understood we can drop it from the notation.The next theorem is adapted from [10, Theorem 3.3] which is concerned with upper bounds for the free energy of directed polymers in random environments.
Proof.This follows immediately from Lemma 8.1 and Theorem 8.2.
We need two additional lemmas before giving the proof of Theorem 3.12.In fact, the second one is part of Theorem 3.15, but we state and prove it separately here to avoid circular reasoning (we will later use Theorem 3.12 in the proof of Theorem 3.15).Lemma 8.4.Let µ and λ be probability measures with finite relative entropy given by with supremum over bounded measurable functions g.Then for any event A, Proof.Assume µ(A) > 0 for otherwise the inequality is trivially true.Then also λ(A) > 0 because finite entropy implies µ ≪ λ.Take g = (− log λ(A)) • 1I A in the variational formula.
Proof.Set V n,i = E i+1 log u n u n,i − E i log u n u n,i .
The desired result follows from plugging these bounds in (D.2).
Continuing with the proof of Theorem D.

3. 6 .
Minimizers of the quenched contractions.Recall from Theorem 3.3 that, for every ξ ∈ ri(D), µ ξ is the unique minimizer of the averaged contraction (3.4) from level-3 to level-1.Finding the minimizers of the quenched contractions (3.13) and (3.14) is more difficult in general.The following result treats the case where the level-1 rate functions are equal.Theorem 3.15.Assume (2.4).For every ξ ∈ ri(D):

µ
follows from a similar argument.If µ Ω ≪ P on S 0,∞ , then we replace f =