LARGE DEVIATIONS FOR BROWNIAN MOTION IN A RANDOM POTENTIAL

. A quenched large deviation principle for Brownian motion in a stationary potential is proved. As the proofs are based on a method developed by Sznitman [ Comm. Pure Appl. Math. 47 (1994) 1655–1688] for Brownian motion among obstacles with compact support no regularity conditions on the potential is needed. In particular, the suﬃcient conditions are veriﬁed by potentials with polynomially decaying correlations such as the classical potentials studied by Pastur [ Teoret. Mat. Fiz. 32 (1977) 88–95] and Fukushima [ J. Stat. Phys. 133 (2008) 639–657] and the potentials recently introduced by Lacoin [ Ann. Inst. Henri Poincar´e Probab. Stat. 48 (2012) 1010–1028; 1029–1048].


Introduction and the main result
Consider a standard Brownian motion on R d , (Z s ; s ≥ 0), moving in a non-negative stationary ergodic potential.That is, it is assumed that the potential is of the form V (x, ω) := V 0 (τ x ω), x ∈ R d , ω ∈ Ω where V 0 is a real-valued non-negative random variable not identically zero on a probability space (Ω, F, P) and (τ x ; x ∈ R d ) is a family of measurable maps on (Ω, F, P) which satisfies τ x • τ y = τ x+y for all x, y ∈ R d , (x, ω) → τ x ω is measurable on the cartesian product R d × Ω, P is invariant under τ x for all x ∈ R d and is ergodic, that is, if for some A ∈ F, τ x (A) = A for all x ∈ R d then P(A) = 0 or 1.
The quenched path measures are defined by where the normalizing constants S t,ω are the quenched survival functions up to time t S t,ω := E 0 exp(− t 0 V (Z s , ω)ds) , t > 0, ω ∈ Ω. (1.2) Here P x is the Wiener measure on paths starting from x ∈ R d and E x is the expectation with respect to P x .
In ( [36], Thm.0.1) (see also [37], Sect.5.4), Sznitman proved a quenched large deviation principle for the speed of the Brownian motion in a Poissonian potential constructed from obstacles with compact support.Building on this work, Armstrong and Tran ( [1], Cor. 2) proved a quenched LDP for a wide class of Hamiltonians with stationary potentials.However, the homogenization techniques used in [1] require some regularity of the potential.In particular, the sufficient condition given for the LDP involves a finite moment of the Lipschitz norm of the potential.The goal of this paper is to extend the quenched LDP for the speed of the Brownian motion in a random potential to stationary random potentials without imposing a regularity condition.
Our LDP is stated below in Theorem 1.1.Besides stationarity and ergodicity, the sufficient conditions in Theorem 1.1 involve an integrability condition expressed in terms of the Lorentz spaces and the principal eigenvalue of − 1 2 ∆ + V .We first recall these two notions and then we state Theorem 1.1.The Lorentz spaces (see for instance [2, p. 634], or [4]) which appear in our context are defined as In particular, L d+ε P ⊂ L P (d, 1) ⊂ L d P for all ε > 0. The principal Dirichlet eigenvalue of − 1 2 ∆ + V is defined as By ergodicity, λ V is non-random.It is closely related to the asymptotic behavior of the survival function.Indeed, lim t→∞ − 1 t log S t,ω = λ V P -a.s. (1.5) A proof is given in ( [37], Sect.3.1) for non-negative potentials in the Kato class K loc d .These include the stationary potentials which satisfy conditions (1.6) and (1.8) below.
Denote the Lebesgue measure on R d by Leb and the expectation with respect to P by E. The Euclidean ball {x ∈ R d ; |x − y| < R} will be denoted by B(y, R) and B(y) := B(y, 1).Theorem 1.1.Let V be a non-negative, stationary and ergodic potential which satisfies sup x∈B(0) V (x, •) ∈ L P (d, 1). (1.6) and For d = 1 or 2, suppose moreover that there exist positive constants ρ, ε and a measurable function u : Ω → R d such that P -a.s.
Then there is a deterministic, continuous convex rate function I : R d → [0, ∞[ given in (2.10), with level sets {x ∈ R d ; I(x) ≤ c} that are compact for all c ∈ R and such that, P -a.s., for all closed subsets A of R d , lim sup and for all open subsets The expression of the rate function in terms of Lyapunov exponents allows to prove that the change in regime of the Brownian motion with constant drift observed by Sznitman ([36], Thm.0.3) in a Poissonian potential associated to obstacles with compact support actually occurs for a large class of measurable potentials.This phase transition was further studied by Flury [14,15] both in the discrete and the continuous settings.Concurrently, also under some regularity conditions on the potential, Ruess [33] proved the existence of the Lyapunov exponents for Brownian motion in stationary potentials.It does not seem possible to extend his results by approximating a measurable potential by regular potentials.Especially since Ruess [34] gave an example where the Lyapunov exponents are not continuous with respect to the potential.
For random walks in a random potentials there is an extensive literature starting with Varadhan [38] who proved both a quenched and an annealed LDP for the speed of a uniformly elliptic random walk.In his thesis, Rosenbluth [32] proved a quenched large deviation principle for a large class of random walks on Z d with stationary transition probabilities under an integrability condition similar to condition (2.2).In these works, the quenched rate function is expressed as a variational formula in terms of cocycles.
Intensive work to extend both the class of random walks and the class of potentials for which a LDP holds was undertaken by Rassoul-Agha, Seppäläinen, Yilmaz.Some of these results, which include level-3 LDP can be found in [28-31, 42, 43].Yilmaz and Zeitouni [44] studied a class of random walks in a random environment where the annealed and quenched rate functions differ.Sznitman's method, based on Lyapunov exponents, was also used to obtain a LDP for random walks in a random potential in [14,45].Mourrat [25] considered the simple random walk in an i.i.d.potential taking values in [0; +∞] and showed a LDP without assuming a moment condition on V .See also [23].
In this paper, we also follow Sznitman's method.Along the way, we provide sufficient conditions for the intermediate results.The existence of the Lyapunov exponents is shown in Theorem 2.1 for stationary potentials under a weaker integrability condition than (1.6).Then under (1.6), we prove in the shape Theorem 2.2 that the convergence is uniform with respect to the direction.The appropriate tool in this context is provided by Björklund's generalization of the shape Theorem [2].
The main difficulty is in the proof of (1.10) under the additional condition on the principal Dirichlet eigenvalue.We will show how key arguments of ( [36], Sect.2) can be done on a linear scale.This will permit the use of the maximal inequality for cocycles ( [3], Cor. 2) and of a technique introduced in [4].See also [5] for an application in a different context.
The results are stated in Section 2 and the proofs are given in Section 3.
In the last section, we will verify the sufficient conditions for the LDP for long-range Poissonian potentials.These were considered in a previous version of this paper.They are of the form where ω = (ω j ; j ∈ N) is a Poisson cloud in R d , d ≥ 1 and W (x) = |x| −γ ∧ 1 with γ > d.Moreover, for these potentials, it is possible to show that I(x) > 0 for x ∈ R d \ {0}.Potentials constructed in Section 4.1 from a Boolean model also satisfy the sufficient conditions of the LDP.We end the last section with the presentation a model introduced by Ruess [34] which does not have decorrelation properties but still satisfies a large deviation principle.Notations.For y ∈ R d and R > 0, the Euclidean norm of y is denoted by |y| and B(y, R) is the Euclidean ball {x ∈ R d ; |x − y| < R}.B(y) stands for the unit ball B(y, 1) and H(y) = inf{t ≥ 0 : Z t ∈ B(y)} is the hitting time of B(y), the closure of B(y).For an open set D ⊂ R d , T D := inf{t ≥ 0, Z t / ∈ D} and C ∞ c (D) is the space of infinitely differentiable functions with compact support in D. For x ∈ R d , [x] is the element of Z d closest to x, with some fixed rule for ties.
The Lebesgue measure on R d is denoted by Leb and the volume of the unit ball of R d by L d .The principal Dirichlet eigenvalue of − 1 2 ∆ in the unit ball is denoted by λ d .For a random variable X and for A ∈ F, let E[X, A] := E(X1 A ).The constants, whose value may vary from line to line, are denoted by c or C. Some are numbered for subsequent reference.

Overview of the results
In this section, the existence of Lyapunov exponents of a Brownian motion in a stationary potential and the shape theorem will be proved under appropriate moment conditions.Then we will show how Sznitman's method leads to a large deviation principle.
Recently, Ruess [33] considered Brownian motion in a stationary potential.Inspired by Schröder [35], he showed the existence of Lyapunov exponents for a large class of potentials and he expressed them in terms of a variational formula.However, the existence of Lyapunov exponents by itself follows from the subadditive theorem under much weaker assumptions on the potential.
The measurability of e(x, y, ω) can be verified by standard arguments.It rests on the hypothesis that (x, ω) → τ x ω is measurable on R d × Ω.Moreover, under the condition E[sup x∈B(0) V (x, •)] < ∞, the potential V locally belongs to the Kato class K loc d and the probabilities e(x, y, ω) are strictly positive (cf.[37], Sects.1.2 and 5.2).We introduce the Green measure relative to the potential V : where x ∈ R d , ω ∈ Ω and A is a Borel subset of R d .G can be interpreted as the expected occupation time measure of Brownian motion killed at rate V (•, ω).We define g(x, y, ω) as the density function relative to the Green measure and we call it the Green function.The existence of g is proved in ([37], (2.2.3)).
We show in the next theorem that the Green function as well as the probabilities e(x, y, ω) have exponential decay rates which are called Lyapunov exponents.Theorem 2.2 shows that, under a stronger moment condition, the convergence to the Lyapunov exponents is uniform with respect to the directions.Theorem 2.1 (Existence of Lyapunov exponents).Let V be a non-negative, stationary and ergodic potential which satisfies (2.2) For d = 1, 2, assume moreover that (1.8) holds.
Then there is a non-random semi-norm α(•) on R d such that P-a.s. and in α is called the quenched Lyapunov exponent.a(0, x, ω) can be replaced by − log g(0, x, ω) in (2.3).
Björklund [2] extended to a very general context the shape theorem proved in [10] for first-passage percolation with independent passage times and in [4] for stationary passage times.This theorem can be applied in our framework.
For the Proof of Theorems 2.1 and 2.2, we need to define By using the strong Markov property of Brownian motion, it is simple to verifiy that d(•, •, ω) is a semi-norm on R d .By ( [37], Lem.5.2.1),P -a.s.d(•, •, ω) defines a distance on R d which induces the usual topology.These properties still hold in a stationary potential.Theorem 2.2 is first proved for the distance d(0, x, ω).Then Lemmas 2.3 and 2.4 allows to replace d(0, x, ω) by a(0, x, ω) or by − log g(0, x, ω) in equation (2.4).Define The proof of the following lemma can be found in ( [37], Prop.5.2.2).The proof is very general as it requires only basic notions of potential theory.
Then there exists a positive constant C such that for x, y ∈ R d , |x − y| > 4, P -a.s.
Lemma 2.4.Let V be a non-negative, stationary and ergodic potential.
(i) If (2.2), and (1.8) when d = 1 or 2, hold, then for all x ∈ Z d , P -a.s., 6), and (1.8) when d = 1 or 2, hold, then P -a.s., The rate function of large deviation principle will be given in terms of the Lyapunov exponents α λ (x) associated with the potential λ + V where λ ≥ −V where V := inf Ω V .The essential properties of α λ are gathered in the next lemma.The upper bound (2.8) should be compared with ([37], (5.2.31)) and with ([45], (65)) for a random walk in a random potential.
Note that, as in [45, section 6], the results will be stated in terms of V as it highlights the role of the principal eigenvalue λ V and it facilitates the comparison with the results from stochastic homogenization.Lemma 2.5.Let V be a non-negative, stationary and ergodic potential which satisfies (1.6) and (1.8). Then and for all x = 0, In Proposition 2.9 of [37], the lower bound for the Lyapunov exponents has the form for some positive constant C. In particular this implies the non-degeneracy of α 0 (•).But the proof requires specific properties of Poissonian potentials.This lower bound is proved for some long-range potentials in ([23], (2.87)).
The rate function of the LDP will be given by Bounds on the rate function are easily obtained from the estimates on the Lyapunov exponents given in (2.8).
When combined with the convexity properties of the Lyapunov exponents, we obtain the following properties of the rate function.See also ([37], Lem.5.4.1).
Then for all x ∈ R d , Armstrong and Tran [1] obtained a large deviation principle for a diffusion in a stationary convex Hamiltonian with some regularity and under a weak coercivity condition.In the particular case of a Brownian motion in a random potential, the Hamiltonian is given by The rate function of the LDP principle is given in ( [1], Cor. 2) by, where Note that the estimates on H(p) given in ( [1], Lem.3.1) lead to estimates on the rate function.Therefore, as in Lemma 2.6, the rate function of a wide class of Hamiltonians is a non-random convex and continuous function with compact level sets.
Then to see that, under the condition (1.7), the rate function I AT coincide with the rate function given in (2.10), one can proceed as in ( [1], Sect.1.3):For x ∈ R d , by (2.12), With a "gauge theorem" ( [9], Chap.4), one could also give an analogue of H * by describing λ V in terms of the existence of a solution of ∆ 2 u − V u + λu = 0 in the appropriate Sobolev space for an increasing sequence of domains.

Proofs
Proof of Lemma 2.4.For d ≥ 3, there is a positive constant C such that P a.s.for all x = y, g(x, y, ω) ≤ C|x − y| 2−d .Hence Then by the stationarity of the potential and by condition (2.2), (X k ; k ≥ 0) is a stationary sequence of non-negative random variables with finite expectation.Then by Borel-Cantelli lemma, P -a.s., lim k→∞ X k k = 0.It follows that condition (2.6) is satisfied for d ≥ 3.For d = 1 or 2, assume that condition (1.8) is satisfied for some positive numbers ρ, ε and for u : Ω → R d such that E|u| < ∞.
Consider D := B(0, |u| + 2ρ + 1).Construct two increasing sequences of stopping times with respect to the natural right continuous filtration (F t ) on C(R + , R d ).These stopping times describe the successive times of return to B(0) and exit times from D of the Brownian motion Since the Brownian motion is recurrent when d = 1 or 2, the stopping times are a.s.finite and We now have for x ∈ R d , by the strong Markov property, Now, for i ≥ 1, by the strong Markov property and by induction, for all x ∈ B(0), where We now argue that P-a.s., c(ω) < 1.Note that a lower bound on the heat kernel in a region of R d as the one obtained from ( [37], Lem.2.1) (or more generally [11], Thm.3.3.5) is enough to deduce that given ρ > 0 there is η = η(ρ) > 0 such that for all measurable A ⊂ B(0, ρ) and for all x ∈ B(0, ρ) Hence for all x ∈ B(u, ρ), there is a constant C > 0 such that Moreover, by the tubular estimate ( [37], p. 198), there is a positive constant C such that for all t > 0 and x ∈ B(0), Recall here that λ d is the principal Dirichlet eigenvalue of − 1 2 ∆ in the unit ball.Hence, by taking t = |u| + 1 in (3.6), Then by (3.5), (3.7) and by the strong Markov property, for all x ∈ B(0 This provides the following upper bound for c(ω) defined in (3.3). ).
To prove the continuity and concavity properties of α λ one can then proceed as in ( [37], Sect.5.2).

Proof of the upper estimate (1.9)
We follow the arguments of ( [37], (4.6) of Thm.5.4.2).See also ([45], (69) of Thm.19).First assume that A is a compact subset of R d .For each t > 0, it is possible to choose n t points x 1 , x 2 , . . ., x nt in A such that n t grows at most polynomially in t and tA ⊂ B t := ∪ nt k=1 B(x k ).By definitions of S t,ω and Q t,ω , P -a.s. for all λ ≥ 0, exp(−λt Therefore for all λ ≥ 0, by Theorem 2.2, (1.5) and under the assumption that λ Hence lim sup To complete the proof, it remains to interchange the sup and the inf in (3.18).This is done by a classical argument (see for example [13] or [37], p. 250).It does not require additional properties of the potential.Neither does the proof of the general case when A is a closed subset of R d as can be seen from ( [37], p. 250).
Then the arguments given in the proof of ([3], Thm.7) apply to Mh(y, ω).They lead to the following maximal inequality.
Proof.It is possible to generalize the argument used in the proof of (2.8) to any broken line [y, ξ, 0] by combining the tubular estimates ( [37], p. 198) with the strong Markov property as follows.
since M R ≥ 0 and by using the inequalities Then P -a.s. for all δ > 2ε 1−2ε and for all sufficiently large t, there is s ∈]t, (1 + δ)t[ such that τ sv ω ∈ A. Proof.By the ergodic theorem, a.s. and for all t sufficiently large, The principal Dirichlet eigenvalue of − 1 2 ∆ + V in the ball B(0, R), R > 0, is defined as From definitions (1.4) and (3.19), it is clear that Similarly to (1.5), λ V,ω (B(0, R)) is related to the survival time in B(0, R).We will need the following version of ( [37], (3.1.17))where an inf z∈B(0) appears.The argument does not require a Harnack-type inequality.
A close examination of the proof of the following key lemma from [36] or [37], shows that it holds for stationary potentials under the moment condition (2.2). For where θ t , t ≥ 0 is the canonical shift on C(R + , R d ).Note that S m,n,v,s1 is a stopping time and A m,n,v,s1,s2 is the event that Z The strong Markov property implies that {b λ (m, n, v, s 1 , s 2 , ω)} m≥0,n≥0 is a subadditive sequence.A calculation similar to (3.13) shows that Eb λ (0, 1, v, s 1 , s 2 ) < ∞.
Proof of (1.10).Let V := V − V and denote the corresponding Lyapunov exponents by α λ (x).Then α λ (x) = α λ−V (x) and under the assumption that λ V = V , we have that Since for any open set and by the continuity of I(•), to obtain (1.10), it is sufficient to show that for all v ∈ Q d \ {0} and r > 0, lim inf where and r > 0 and consider the following events.For positive numbers t 0 , R, ε, let A 1 (t 0 , R, ε) be the event for all t > t 0 , For the moment, assume that for some positive numbers ε ∈]0, 1/4[, t 0 , R, ε and m, and by Lemma 3.3, for all t sufficiently large there is (3.25) Moreover, let 0 < s 1 < s 2 < 1 and λ > 0 be such that ] Then for η ∈]0, 1 − s 1 [ and for all t sufficiently large so that R + δt < rt, By Lemma 3.5, since ω ∈ Ω and S 0, in contradiction with (3.31).

Application to the Brownian motion with constant drift
From the LDP for the speed of Brownian motion, Varadhan's lemma, one can obtain a LDP for Brownian motion in a random potential with a constant drift as in ( [37], Thm.4.7) by verifying the additional condition (2.1.9) of [12].But since an upper gaussian estimate suffices, it is also verified for a stationary potential.This in turn leads to the observation of a transition from a sub-ballistic to a ballistic regime according to the strength of the drift.A similar phenomenon is proved for the random walk in a random potential in ( [14], Thm.B(a)) and in ( [25], Rem.1.11).
For h ∈ R d , the quenched path measures of the Brownian motion in the random potential V with drift h is given by The transition from a sub-ballistic to a ballistic regime appears clearly when described by the dual norm α * λ (y) := sup x =0 x•y α λ (x) for y ∈ R d and λ ≥ −V .

Examples
In this section, we present some examples of potentials which satisfy the sufficient conditions of Theorem 1.1.

A Poissonian potential: Lacoin's potential
In this section, we show that the potentials introduced by Lacoin in [21], [22] satisfy the conditions of Theorem 1.1.
Their interest stems from the fact that the relations satisfied by their scaling exponents differ substantially from those established by Wüthrich [39][40][41] for a potential of the form (1.11) where W has compact support.
These potentials are constructed from a Poisson Boolean model.Let Ω : whose intensity measure is given by Leb ×ν where ν is a probability measure on [1, ∞[ which depends on a parameter δ > 0 and is defined by ) is an increasing sequence.See ( [24], Sect.1.4) for an alternative description of this model and [19] for results on the percolative properties of the balls B(ω i , r i ).
Given γ > 0, Lacoin's potential V : R d × Ω −→ [0, ∞[ is defined by The behavior of this model depends on the positive parameters δ and γ.For γ + δ > d, the potential is finite a.s. and the survival functions are strictly positive.Additional basic properties of this potential are gathered in the following lemma taken from [23].The volume of the unit ball in R d is denoted by L d .Moreover, the potential is ergodic.
In order to show that Lacoin's potential satisfies the conditions of Theorem 1.1, we first prove a weak independence property similar to ([18], Lem. 6).The method previously used in ( [23], Lem.2.6) lead to a weaker result.We are now ready to verify the hypothesis of Theorem 1.1.
[2]of of Theorem 2.2.By stationarity of the potential and by translation invariance of Brownian motion,d(x, y, τ z ω) = d(x + z, y + z, ω) for x, y, z ∈ R d , ω ∈ Ω.Moreover, by Lemma 5.2.1 of [37], d(•, •, ω) is a.s. a distance on R d .Under the integrability condition (1.6), it follows from (3.13) that d(0, x, ω) is in L P (d, 1) for all x ∈ Z d .Hence the conditions of the shape theorem ([2], Thm.1.2) are satisfied.Therefore, there exists a semi-norm L on R d such that But by Theorem 2.1, α(x) = L(x) for all x ∈ Z d and consequently, for all x ∈ R d .For x ∈ R d , denote by x the nearest neighbor point in Z d of x (with some rule to break ties).Then, |x − x| < √ d and for all x