On large deviation rate functions for a continuous-time directed polymer in weak disorder

We show that the endpoint large deviation rate function for a continuous-time directed polymer agrees with the rate function of the underlying random walk near the origin in the whole weak disorder phase.


Introduction
In this article, we study a continuous-time directed polymer model in a Lévy-type random environment. This type of model is known to exhibits a phase transition when space dimension is greater than or equal to three. More precisely, the polymer behaves like the simple random walk when the disorder is weak, whereas it tends to localize when the disorder is strong; see a recent survey [6] for more detail. While most of the research has been devoted to typical behaviors of the polymer, it is also shown in [16,Theorem 6.1] and [6,Exercise 9.1] that deeply inside the weak disorder phase, even the rate function of the large deviation principle for the polymer endpoint coincides with that for the simple random walk near the origin. The aim of this paper is to present a simple argument that extends this result to the whole weak disorder phase for a certain class of continuous-time directed polymer models.
The assumption that ρ has finite mass is not essential, but leads to a simpler presentation. In particular, it implies that the Lévy-processes are of pure jump type and we can therefore write where (B x ) x∈Z d is an i.i.d. collection of Brownian motions and (N x ) x∈Z d is an i.i.d. collection of Poisson point processes on R + × [−1, ∞), which is independent of (B x ) x∈Z d , with intensity measure ds ρ(dr). Given ω, we define a new process (L x ) x∈Z d by with the convention log 0 = −∞, and −∞ being an absorbing state for this process. For a path x : R + → Z d that is right-continuous and has left limits everywhere (cádlág), we define the Hamiltonian by Let (X = (X t ) t≥0 , P κ ) denote the random walk which starts at the origin and jumps to each of the nearest neighbor sites at rate κ > 0. The polymer measure of P κ is the random probability measure µ κ ω,t defined by with the convention e −∞ = 0, where the normalizing constant is given by Z κ ω,t = E κ [e Ht(ω,X) ]. This probability measure gives more weight to paths along which the environment is increasing, and discourages paths that observe a decreasing environment. In particular, we interpret the set {(t, x) : ω x (t) = ω x (t−) − 1} as hard obstacles in space-time and note that the polymer is conditioned to avoid this set.
One may wonder why we introduce the process ω in (1.1) first and transform it to L in (1.2). This is mostly due to historical reasons. We regard Z ω,t as the "partition function" of Gibbs measure µ κ ω,t , but in most of the earlier studies [1,15,19,10,9,14,11], it is regarded as the solution of a stochastic partial differential equation called the parabolic Anderson model. More precisely, the point-to-point partition function is equal to u(t, x), where u denotes the solution to the following initial value problem for a stochastic heat equation with Lévy noise, (1.4) Before reviewing some known results, we introduce more general notation that will be useful later. We use bold symbols to highlight multi-dimensional parameters. Let κ = (κ e ) |e|1=1 ∈ (0, ∞) 2d , and write P κ for law of the Markov process with generator We write Z κ ω,t (resp. Z κ ω,t,x ) for the corresponding partition function (resp. point-to-point partition function). Note that P κ = P κ 1 , where 1 = (1, . . . , 1). Let us first list the known properties of the partition functions: Moreover, the function κ → p(κ 1) is continuous.
In this article, we study the large deviation principle (LDP) for the endpoint distribution under µ κ ω,t . The following abstract existence result holds: Theorem B. For every κ > 0 and every d, the sequence (µ κ ω,t (X t /t ∈ ·)) t∈N satisfies an LDP with deterministic, good, convex rate function J κ (x) := p(κ 1) − p(κ 1, x), P-almost surely.
Theorems A and B are well-known, at least for the related discrete-time polymer model. In the appendix, we briefly outline how to prove them in our continuous time setting referring to some references. Our main result concerns the shape of the rate function for κ ≥ κ cr : Theorem 1.1. Let d ≥ 3 and let I κ denote the large deviation rate function of (P κ (X t /t ∈ ·)) t≥0 .
(i) If κ > κ cr , then J κ and I κ coincide in a neighborhood of the origin.
The same conclusion has been proved in [16,Theorem 6.1] and [6, Exercise 9.1] for the discrete-time directed polymer model under the stronger assumption of L 2 -boundedness, which would correspond to κ > κ L 2 cr in our notation. The continuous-time model has the advantage that the parameter of the model changes from the "inverse temperature" β to the jump rate κ, so that we can use a convolution property of the continuous-time random walk, which allows us to compare the partition functions for different jump rates; see Theorem C below.
Remark 1.2. The coincidence and difference of the quenched and annealed rate functions are studied also in the setting of random walk in random environment: [20,22,21,17,3].
It is an important open problem (see, e.g., [6,Open Problem 9.3]) to prove that the rate function for the directed polymer model is strictly convex near the origin. One of the major reasons is that it is a key to prove the so-called scaling relation, as is proved in [2]. Although Theorem 1.1 gives an affirmative answer, we think it is of limited interest in this aspect (except possibly for κ = κ cr ) since the scaling exponents are known in the weak disorder phase. Apart from the results in weak disorder, the strict convexity is known only for (i) certain exactly solvable models [18] for which the rate function is explicitly known, and (ii) the Brownian polymer model in continuous space [7] for which the rate function agrees with that of the Brownian motion in both strong and weak disorder. The last result is due to a special translation invariance property of the Brownian bridge and Poisson point process.
For the simple random walk model, the bridge loses entropy as the endpoint moves away from the origin, and we expect that the polymer rate function is strictly larger than that of simple random walk in strong disorder. Moreover, it is conjectured that strong disorder holds at the critical value κ = κ cr , so we expect that the conclusion of Theorem 1.1 (i) does not extend to κ ≤ κ cr .

Proof of the main result
The proof of Theorem 1.1 relies on the comparison result from [12]. If (X = (X t ) t≥0 , P ) and (X ′ = (X ′ t ) t≥0 , P ′ ) are two independent processes on the space of cádlág paths in Z d , we write P * P ′ for the law of (X t + X ′ t ) t≥0 . We write P * Q if there exists a probability measure Q ′ such that P = Q * Q ′ . We note P κ 1 +κ2 = P κ 1 * P κ 2 for all κ 1 , κ 2 ∈ (0, ∞) 2d , and therefore both P κ1+κ2 * P κ1 and P κ1+κ2 * P κ2 .
Theorem C ([12, Theorem 1]). Let P and Q be two probability measures on cádlág paths on Z d , and write Z P ω,t := E P [e Ht(ω,X) ] and Z Q ω,t := E Q [e Ht(ω,X) ] for the associated partition functions. Then P * Q implies that for any f : The maximal elements with respect to * are the Dirac measures on a deterministic path. Intuitively, the partition function is smaller (in the concave stochastic order) if there is "less randomness" in the underlying random walk, in the sense that its law is large with respect to * .
Proof. It is easy to see that ((X s ) s∈[0,t] , Q) is a Markov process, so we compute its generator L Q : We now prove the main result: Proof of Theorem 1.1. Let λ ∈ R d . By Lemma 2.1, we have ]. Theorem A (i) shows that almost surely We will show that for all λ small enough, Once we have this identification of the cumulant generating function, we can apply the Gärtner-Ellis theorem to conclude that (µ κ ω,t (X t /t ∈ ·)) t∈N satisfies an LDP near the origin with the rate function I κ . In order to prove (2.1), we introduce so that P κ(λ) = P κ(λ)1 * P δ 1 (λ) , P κ(λ) * P δ 2 (λ) = P κ(λ)1 .

Appendix: Known results
Results similar to Theorem A are well-known for directed polymers in discrete time, see [6]. If we assume that there are no hard obstacles, i.e. ρ([−1, −1 + ε]) = 0 for some ε > 0, all claims can be obtained using the same arguments as in discrete time.
Proof of Theorem A. The existence of p(κ 1) as an L 1 -limit has been shown in [19,Theorem 1.2] and [10,Theorem 3.1], while the existence of the almost sure limit is shown in [9, Theorem 1.1] (under the assumption ρ({−1}) = 0). The existence of the almost sure limit in the hard obstacle case, σ 2 = 0 and ρ = δ {−1} , is presumably well-known, as it is, for example, used in [14]. A proof for the existence of p(κ1) and p(κ 1, x) as almost sure limits in the hard obstacle case, as well as the continuity of κ → p(κ 1), can be found in [11,Propositions 5.5 and 5.6]. Those arguments also apply to general environments and with κ 1 replaced by κ.
The existence of κ cr is shown in [19, Theorem 1.3] for σ 2 = 0, ρ = δ {−1} , while for general environments it follows from the fact that κ → p(κ 1) is increasing, which was shown in [12, display (4.15)]. To show the existence of κ cr , one first follows the arguments from [8, Proposition 3.1] to show that W κ 1 ∞ > 0 is equivalent to L 1 -convergence of (W κ 1 t ) t≥0 , and then applies the same argument as in [12, display (4.8)]. The relation κ cr ≤ κ cr ≤ κ L 2 cr is obvious. Finally, in dimension d ≥ 3, the same argument as in the discrete-time case shows κ L 2 cr (d) < ∞, see [6,Theorem 3.3]. The second statement has been shown for the discrete-time model in [4], and the generalization to continuous time follows along the same lines, using [13,Theorem 7].
The LDP for a discrete-time polymer model is established in [5], but to the best of our knowledge, it is not available for continuous time random walk models in the literature.
Proof of Theorem B. The existence of an LDP for (µ κ ω,t (X t /t ∈ ·)) t∈N with some rate function J ′ follows from the argument of [5, Theorem 1.1]. A concentration inequality corresponding to [5, Proposition 2.3] has been proven for σ = 0, ρ = δ {−1} in [11,Proposition 5.3]. A similar proof applies to the general case, and can also be used to obtain the display following (3.2) in [5]. To identify J ′ with J κ one can follow the arguments from [6, Theorem 9.1]. The convexity of x → J κ (x) can be shown in the same way as in [11,Proposition 5.6]. Finally, to show that J κ is good, it is enough to observe J κ (x) ≥ I κ (x) − (a ∨ 0) − p(κ) → ∞ for x → ∞.