Asymmetric bridges and KPZ - Semantic Scholar

arXiv:1603.03560v1 [math.PR] 11 Mar 2016

Weakly asymmetric bridges and the KPZ equation Cyril Labb´e Universit´e Paris Dauphine†

Abstract We consider a discrete bridge from (0, 0) to (2N, 0) evolving according to the corner growth dynamics, where the jump rates are subject to an upward asymmetry of order N −α with α > 0. We provide a classification of the static and dynamic behaviour of this model according to the value of the parameter α. Our main results concern the hydrodynamic limit and the fluctuations of the bridge. For α < 1, the hydrodynamic limit is given by the entropy solution of the inviscid Burgers equation with Dirichlet boundary conditions. For α ≤ 1/3, we show that the fluctuations around this hydrodynamic limit are given by the KPZ equation on the line and restricted to the time interval [0, T ): the final time T is infinite when α < 1/3, while in the regime α = 1/3 it equals the finite time needed by the hydrodynamic limit to reach its stationary state.

Contents 1 Introduction

1

2 The invariant measure

11

3 Equilibrium fluctuations

18

4 Hydrodynamic limit

26

5 KPZ fluctuations

46

6 Appendix

61

1 Introduction The simple exclusion process is a statistical physics model that has received much attention from physicists and probabilists over the years. The purpose of the present article is to study the scaling limits of a particular instance of this process, according to the asymmetry imposed on the jump rates. Consider a system of N particles on the linear lattice {1, . . . , 2N }, subject to the † PSL Research University, Ceremade, 75775 Paris Cedex 16, France. Part of this work was carried out while the author was a Research Fellow at the University of Warwick. Date: March 14, 2016 Keywords: exclusion process; height function; discrete bridge; asymmetry; stochastic heat equation; Burgers equation; Kardar-Parisi-Zhang equation. 2010 Mathematical Subject Classification: Primary 60K35; Secondary 60H15; 82C24

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rate pN

0

2N rate 1 − pN

Figure 1: An example of interface. exclusion rule that prevents any two particles from sharing a same site. Each particle, independently of the others, jumps to its left at rate pN and to its right at rate 1 − pN as long as the target site is not occupied. Additionally, we impose a “zero-flux” boundary condition to the system: a particle located at site 1, resp. at site 2N , is not allowed to jump to its left, resp. to its right. At any given time t, let Xi (t) be equal to +1 if the i-th site is occupied, and to −1 otherwise. It is classical to associate to such a particle system a so-called height function, defined by k X Xi , k = 1, . . . , 2N . S(0) = 0 , S(k) = i=1

Necessarily, S(2N ) = 0 so that S is a discrete bridge. The dynamics of the particle system can easily be expressed at the level of the height function: at rate pN , resp. 1 − pN , each downwards corner, resp. upwards corner, flips into its opposite: we refer to Figure 1 for an illustration. The law of the corresponding dynamical interface will be denoted by PN . This dynamics admits a unique reversible probability measure: µN (S) =

1  pN  21 A(S) , ZN 1 − p N

(1.1)

P where A(S) = 2N k=1 S(k) is the area under the discrete bridge S, and ZN is a normalisation constant, usually referred to as the partition function. This observation appears in various forms in the literature, see for instance [JL94, FS10, EL15]. Notice that the dynamics is reversible w.r.t. µN even if the jump rates are asymmetric: this feature of the model is a consequence of our “zero-flux” boundary condition. From now on, we only consider “upwards” asymmetries, that is, pN ≥ 1/2, and we aim at understanding the behaviour of the interface according to the strength of the asymmetry. It is clear that the interface will be pushed higher and higher as the asymmetry increases. On the other hand, the interface is subject to some geometric restrictions: it is bound to 0 at both ends, and it is lower than the deterministic shape k 7→ k ∧ (2N − k). Actually, it is simple to check that under a strong asymmetry, that is, pN = p > 1/2, the interface is essentially stuck to the latter deterministic

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shape. Therefore, to see non-trivial behaviours we need to consider asymmetries that vanish with N . We make the following choice of parametrisation:  4σ  pN = exp , σ>0, α>0, 1 − pN (2N )α

so that

 1  1 σ + + O . 2 (2N )α N 2α The important parameter is α. When it equals +∞, we are in the symmetric regime, while α = 0 corresponds to a strong asymmetry. In the present paper, we investigate the whole range α ∈ (0, ∞). The results of this paper are divided into three parts: first, we characterise the scaling limit of the invariant measure; second, the scaling limit of the fluctuations at equilibrium; and third, we investigate the scaling limit of the dynamics out of equilibrium. As we will see, the model displays a large variety of limiting behaviours, some of them already appear in related contexts of the literature. In the particular case α = 1, let us cite the works of Derrida and his coauthors [ED04, DELO05] on a similar model interacting with reservoirs or of Dobrushin and Hryniv [DH96] on the scaling limit of the static model without the bridge condition. We also refer to the work of Gonc¸alves and Jara [GJ12] on the crossover of the stationary WASEP on the whole line Z to the KPZ equation. We will cite further references on related models below. However, let us emphasise again the reversibility of our model due to the “zeroflux” boundary condition: this specific feature gives rise to limiting behaviours which have not been observed previously in the literature, in particular the KPZ fluctuations on a finite time interval, see below. pN =

From now on, we extend S into a piecewise linear map from [0, 2N ] into R: namely, S is affine on every interval [k, k + 1]. We also let L be the log-Laplace functional associated to the Bernoulli ±1 distribution with parameter 1/2, namely L(h) = log cosh h. 1.1 The invariant measure The main result of this section is a Central Limit Theorem for the interface under µN . To state this result, we need to rescale appropriately the interface according to the strength of the asymmetry. For α ≥ 1, the space variable will be rescaled by 2N so that the rescaled space variable will live in Iα = [0, 1]. On the other hand, for α < 1, we will zoom in a window of order (2N )α around the center of the lattice, hence the rescaled space variable will live in IαN = [−N/(2N )α , N/(2N )α ] for any N ≥ 1, and Iα = R in the limit N → ∞. This being given, we introduce the curve ΣN α around which the fluctuations occur. One would have expected this curve to be defined as the mean of S under µN , but it is actually more convenient to opt for a different definition. However, ΣN α coincides with the mean under µN up to some negligible terms, see Remark 2.5 below. For all k ∈ {0, . . . , 2N }, we set xk = k/2N if α ≥ 1, xk = (k − N )/(2N )α if α < 1, and ΣN α (xk ) =

k X i=1

N L′ (hN i ) , hi =

1 2σ  , i ∈ {1, . . . , 2N } . (1.2) N − i+ (2N )α 2

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0

2N

O(

√

O(

O(N 2−α ) N)

0

√

N)

2N

α

O(N 2 )

N − O(N α )

N

0

2N

O(N α ) Figure 2: Upper left α > 3/2, upper right α ∈ [1, 3/2], bottom α < 1. The red curve is ΣN α : in the first case, it is negligible compared to the fluctuations so we have not drawn it. In between these discrete values xk ’s, ΣN us α is defined by linear interpolation. R x Let N 2−α N ′ mention that Σα (x) ∼ (2N ) σx(1−x) when α > 1, and Σα (x) ∼ 2N 0 L (σ(1− 2y))dy when α = 1. On the other hand, when α < 1, ΣN α differs from the maximal curve k 7→ k ∧ (2N − k) only in a window of order N α around the center of the lattice. We refer to Figure 2 for an illustration and to Equations (2.2) and (2.3) for precise formulae. We are now ready to introduce the rescaling for the fluctuations. For α ≥ 1, we set uN (x) :=

S(x2N ) − ΣN α (x) √ , 2N

x ∈ [0, 1] ,

and for α < 1, we set uN (x) :=

S(N + x(2N )α ) − ΣN α (x) , α (2N ) 2

x ∈ IαN . (d)

Theorem 1.1 Under the invariant measure µN , we have uN =⇒ Bα as N → ∞. The process Bα is a Brownian bridge on [0, 1] when α > 1. For α = 1, resp. α < 1, it is the image of a Brownian bridge on [0, 1] through a deterministic time change that maps [0, 1] onto itself, resp. onto R. Remark 1.2 The covariance of Bα is given by E[Bα (x)Bα (y)] =

qα (0, x) qα (y, 1) , ∀x ≤ y ∈ [0, 1] , qα (0, 1)

(1.3)

for α ≥ 1, and by E[Bα (x)Bα (y)] =

qα (−∞, x) qα (y, +∞) , qα (−∞, +∞)

∀x ≤ y ∈ R ,

(1.4)

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for α < 1, where   Rx ∨ y − x ∧ y x∨y ′′ qα (x, y) = x∧y L (σ(1 − 2u))du  R  x∨y ′′ x∧y L (2σu)du

if α > 1 if α = 1 if α < 1 .

Let us make a few comments on this result. For α > 1, the limiting mean shape, the order of the fluctuations and the limiting law of the fluctuations are “universal”. For α = 1, the limiting mean shape and the covariance of the limiting fluctuations depend on the log-Laplace functional of the step distribution of our static model. This is analogous with the well-known fact that the rate function of moderate deviations does not depend on the step distribution, while the rate function of large deviations does. Notice that the case α = 3/2 is already covered in [EL15]. The case α = 1 can be deduced from previous results of Dobrushin and Hryniv [DH96] on paths of random walks conditioned on having a given large area. We also derive the asymptotics of the partition function ZN . Proposition 1.3 As N → ∞ we have  2 σ 3−2α + O(1 ∨ N 5−4α )   6 (2NR ) ZN 1 log 2N = (2N ) 0 L(σ(1 − 2x))dx + O(1)  2 σ 2−α − 2N log 2 + O(N α ) 2 (2N )

if α > 1 , if α = 1 , if α < 1 .

Finally, let us observe that all the results presented above can be extended to a more general class of static models: namely, to paths of random walks having positive probability of coming back to 0 after 2N steps and whose step distribution admits exponential moments. 1.2 Fluctuations at equilibrium

˙ will denote a space-time white We turn our attention to the dynamics. Below, W noise on [0, ∞) × Iα , that is, a centred Gaussian random distribution such that for ˙ (f )W ˙ (g) = hf, gi. For any two functions f, g ∈ L2 ([0, ∞) × Iα ), we have EW α ≥ 1, we set uN (t, x) :=

S(t(2N )2 , x2N ) − ΣN α (x) √ , 2N

x ∈ [0, 1] , t ≥ 0 ,

while for α < 1, we set S(t(2N )2α , N + x(2N )α ) − ΣN α (x) , x ∈ IαN , t ≥ 0 . α (2N ) 2 Rx ′ For convenience, we set Σ1 (x) = limN →∞ ΣN 1 (x)/(2N ) = 0 L (σ(1 − 2y))dy N α for all R ∞x ∈′ [0, 1], and, for α < 1, Σα (x) = limN →∞ (Σα (x) − N )/(2N ) = x + −x (L (2σy) − 1)dy for all x ∈ R. uN (x) :=

Theorem 1.4 Assume that the process starts from the invariant measure µN . Then, as N → ∞, the process uN converges in distribution to the process u where

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1. For α > 1, u solves ( ˙ , ∂t u = 21 ∂x2 u + W u(t, 0) = u(t, 1) = 0 ,

(1.5)

started from an independent realisation of Bα , 2. For α = 1, u solves ( p ˙ , ∂t u = 12 ∂x2 u − 2σ∂x Σ1 ∂x u + 1 − (∂x Σ1 )2 W u(t, 0) = u(t, 1) = 0 ,

x ∈ (0, 1)

(1.6)

x∈R,

(1.7)

started from an independent realisation of B1 , 3. For α < 1, u solves p 1 ˙ , ∂t u = ∂x2 u − 2σ∂x Σα ∂x u + 1 − (∂x Σα )2 W 2

started from an independent realisation of Bα .

In all cases, convergence holds in the Skorohod space D([0, ∞), C(Iα )). Once again, notice the specific behaviour when α ≤ 1. The proof of this result relies on classical techniques: an important ingredient is the Boltzmann-Gibbs principle, which is adapted to the present setting in Proposition 3.4. 1.3 Hydrodynamic limit The subsequent question we address concerns the convergence to equilibrium: suppose we start from some initial profile S0 at time 0, how does the interface reach its stationary state ? Informally speaking, we aim at understanding how the macroscopic shape of the interface reaches its stationary state given by the curve ΣN α, without paying attention to the fluctuations. Therefore, we restrict ourselves to α < 3/2, since in the complementary cases and under the stationary measure, the mean shape is negligible compared to the fluctuations. We first treat the case α ∈ [1, 3/2). We set mN (t, x) :=

S(t(2N )2 , x2N ) , t≥0, (2N )2−α

x ∈ [0, 1] .

Notice that the scaling in height and space is consistent with the scaling of the mean shape under the invariant measure. Theorem 1.5 Let α ∈ [1, 3/2). We assume that the initial profile mN (0, ·) is deterministic and converges uniformly to some continuous profile m0 (·). In the case α ∈ (1, 3/2), we consider the further requirement that there exists δi > 0 such that the δi -H¨older norm of mN (0, ·) is uniformly bounded over all N ≥ 1. Then, the process mN converges in probability, in the Skorohod space D([0, ∞), C([0, 1])), to the deterministic process m where: 1. If α ∈ (1, 3/2), m is the unique solution of the linear heat equation ( ∂t m = 21 ∂x2 m + σ , m(t, 0) = m(t, 1) = 0 , m(0, ·) = m0 (·) .

(1.8)

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2. If α = 1, m is the solution of the following non-linear heat equation ( ∂t m = 21 ∂x2 m + σ(1 − (∂x m)2 ) , m(t, 0) = m(t, 1) = 0 , m(0, ·) = m0 (·) .

(1.9)

Notice that (1.8) and (1.9) are well-posed parabolic PDEs. The case α = 1 is similar to the results of De Masi, Presutti and Scacciatelli [DMPS89] and of G¨artner [G¨ar88] on the hydrodynamic limit of the WASEP on the line Z: the main difference comes from the boundedness of our lattice. Remark 1.6 The H¨older regularity that we impose on the initial condition when α ∈ (1, 3/2) does not play an important rˆole: it ensures that the process is tight in a space of continuous functions from time 0. For α = 1, the profile mN (t, ·) is 1-Lipschitz for all t ≥ 0 so we do not need to impose this further condition. In the case α < 1, we set mN (t, x) :=

S(t(2N )α+1 , x2N ) , t≥0, 2N

x ∈ [0, 1] .

Roughly speaking, our next result shows that mN converges to the solution m of ( ∂t m = σ(1 − (∂x m)2 ) , (1.10) m(t, 0) = m(t, 1) = 0 , m(0, ·) = m0 (·) . Compare (1.8), (1.9) and (1.10) and observe that, as α decreases, the asymmetric term becomes predominant. Actually, the last equation is not well-defined and we need to work at the level of the derivative of the interface (equivalently, at the level of the density of particles) in order to establish a rigorous result. Furthermore, the proof of the convergence is sensitive to the choice of initial condition. For the sake of clarity, we only consider two initial conditions: • flat: S(t = 0, k) = k mod 2, for all k ∈ {0, . . . , 2N }, • wedge: S(t = 0, k) = (−k) ∨ (k − 2N ), for all k ∈ {0, . . . , 2N }. Theorem 1.7 Let α < 1. We assume that the initial condition is either flat or wedge. Then, the process mN converges in probability, for the RSkorohod topology x on D([0, ∞), C([0, 1])), to the deterministic process m(t, x) = 0 (2η(t, y) − 1)dy where η is the entropy solution of the following inviscid Burgers equation ( ∂t η = 2σ∂x (η(1 − η)) , (1.11) η(t, 0) = 1 , η(t, 1) = 0 , with initial condition η(0, ·) = 1/2 if we start from flat, while if we start from wedge then η(0, x) = 0 on [0, 1/2] and η(0, x) = 1 on [1/2, 1]. The definition of the entropy solution of (1.11) will be recalled in Section 4.3. The process η should be seen as the limiting density of particles. Actually, if we let ̺N (t, dx) =

2N X(t(2N )1+α , k) + 1 1 X N ηt (k) δ k (dx) , ηtN (k) = , (1.12) 2N 2N 2 k=1

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Nα N 2α

0

λN t γN

N

λ

2N − γ N t N

2N

Figure 3: A plot of (1.13): the bold black line is the initial condition, the dashed line is the solution at some time 0 < t < 1/(2σ), and the dotted line is the solution at the terminal time 1/(2σ). The blue box corresponds to the window where we see KPZ fluctuations. then the main step in the proof of Theorem 1.7 consists in showing that ̺N converges in the Skorohod space of finite measure-valued processes to the process η(t, x)dx. The proof is inspired by the seminal work of Rezakhanlou [Rez91] which establishes, in particular, the convergence of the density of the asymmetric simple exclusion process, on the whole space Z, to the solution of the inviscid Burgers equation on R. Let us also mention the work of Bahadoran [Bah12] that extends the results of Rezakhanlou to domains with open boundaries. Even though our result is in the spirit of these two works, there are two important differences that require some additional care. First, our asymmetry is not fixed but vanishes with N : our space-time scaling is therefore not the so-called Euler scaling considered in [Bah12, Rez91]. Second, their proofs work when the density of particles is initially distributed according to a product measure (this is in particular required in the probabilistic proof of the two-block estimates of [Rez91]). The first point can be handled by splitting the generator into a symmetric and a fully-asymmetric part, and by adapting the arguments of [Rez91] accordingly. The second point is problematic if we start from the flat initial condition, which is not a product measure. We circumvent the difficulty by using the monotonicity (in law) of the height function with respect to its initial profile, and the fact that the flat initial condition can be bounded from above and below by suitably chosen profiles whose densities are distributed according to product measures, we refer to the beginning of Section 4.3 for details. Notice that the wedge initial condition is a product measure so that we do not face this issue in that case. Let us finally provide explicitly the solution of the Burgers equation (1.11) starting from the flat initial condition. At the level of the height function, we have: m(t, x) = x ∧ (1 − x) ∧ (σt) , t > 0 ,

x ∈ [0, 1] ,

(1.13)

see Figure 3 for an illustration. Notice that the stationary state is reached at the finite time T = 1/(2σ). 1.4 KPZ fluctuations Let us recall the famous result of Bertini and Giacomin [BG97] on the Kardar Parisi Zhang (KPZ) equation. Consider a simple exclusion process on the infinite

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√ lattice Z with jump rates 1/2 + ǫ to the left and 1/2 to the right. If one starts from a flat initial profile, then results in [G¨ar88, DMPS89] ensure that the hydro√ dynamic limit grows evenly at speed ǫ. Then, Bertini and Giacomin look at the fluctuations around this hydrodynamic limit and show that the random process √ ǫ (S(tǫ−2 , xǫ−1 ) − ǫ−3/2 t) converges to the solution of the KPZ equation, whose expression is given in (1.16) below (in Bertini and Giacomin’s case, σ = 1/2). Although our setting is similar to the one considered by Bertini and Giacomin, the “zero-flux” boundary condition induces a major difference: our process admits a reversible probability measure, while this is not the case on the infinite lattice Z. However, if one starts the interface “far” from equilibrium, then we are in an irreversible setting up to the time needed by the interface to reach the stationary regime, and one would expect the fluctuations to be described by the KPZ equation. Bertini and Giacomin’s result suggests to rescale the height function by 1/(2N )α , the space variable by (2N )2α and the time variable by (2N )4α . The space scaling immediately forces one to take α ≤ 1/2 since, otherwise, the lattice {0, 1, . . . , 2N } would be mapped onto a singleton in the limit. It happens that the geometry of our model imposes a further constraint: Theorem 1.7 and Equation (1.13) show that the interface reaches the stationary state in finite time in the time scale (2N )α+1 ; therefore, as soon as 4α > α + 1, Bertini and Giacomin’s scaling yields an interface which is already at equilibrium in the limit N → ∞. Consequently, we have to restrict α to (0, 1/3] for this scaling to be meaningful. We set hN (t, x) := γN S(t(2N )4α , N + x(2N )2α ) − λN t ,

(1.14)

where pN (2N )4α 1 log , cN := γ , λN := cN (eγN − 2 + e−γN ) . 2 1 − pN e N + e−γN (1.15) The last result of the present paper is the following. γN :=

Theorem 1.8 Take α ≤ 1/3 and consider the flat initial condition. As N → ∞, the sequence hN converges in distribution to the solution of the KPZ equation: ( ˙ , x∈R, t>0, ∂t h = 12 ∂x2 h − σ(∂x h)2 + W (1.16) h(0, x) = 0 . The convergence holds on D([0, T ), C(R)) where T = 1/(2σ) when α = 1/3, and T = ∞ when α < 1/3. Here D([0, T ), C(R)) is endowed with the topology of uniform convergence on compact subsets of [0, T ). Remark 1.9 Amir, Corwin and Quastel [ACQ11] proved that the convergence result of Bertini and Giacomin [BG97] still holds when one starts from the (very singular) wedge initial condition on Z. To that end, one needs to add another scaling factor in the discrete Hopf-Cole transform. Using this additional scaling factor, it is possible to adapt our proof in order to start from the wedge initial condition on {0, . . . , 2N }: we then get a similar dichotomy between α = 1/3 and α < 1/3.

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Observe that for α = 1/3, T is the time needed by the hydrodynamic limit to reach the stationary state. Indeed, in that case the time-scale of the hydrodynamic limit coincides with the time-scale of the KPZ fluctuations. Although one could have thought that the fluctuations continuously vanish as t ↑ T , our result show that they don’t: the limiting fluctuations are given by the solution of the KPZ equation, restricted to the time interval [0, T ). This means that the fluctuations suddenly vanish at time T ; let us give a simple explanation for this phenomenon. At any time t ∈ [0, T ), the particle system is split into three zones: a high density zone {1, . . . , λγNN t}, a low density zone {2N − λγNN t, . . . , 2N } and, in between, the bulk where the density of particles is approximately 1/2, we refer to Figure 3. The KPZ fluctuations occur in a window of order N 2α around the middle point of the bulk: from the point of view of this window, the boundaries of the bulk are “at infinity” but move “at infinite speed”. Therefore, inside this window the system does not feel the effect of the boundary conditions until the very final time T where the boundaries of the bulk merge. Let us recall that the KPZ equation is a singular SPDE: indeed, the solution of the linearised equation is not differentiable in space so that the non-linear term would involve the square of a distribution. While it was introduced in the physics literature [KPZ86] by Kardar, Parisi and Zhang, a first rigorous definition was given by Bertini and Giacomin [BG97] through the so-called Hopf-Cole transform h 7→ ξ = e−2σh that maps formally the equation (1.16) onto ( ˙ , x∈R, t>0, ∂t ξ = 21 ∂x2 ξ + 2σξ W (1.17) ξ(0, x) = 1 . This SPDE is usually referred to as the multiplicative stochastic heat equation: it admits a notion of solution via Itˆo integration, see for instance [DPZ92, Wal86]. M¨uller [Mue91] showed that the solution is strictly positive at all times, if the initial condition is non-negative and non-zero. This allows to take the logarithm of the solution, and then, one can define the solution of (1.16) to be h := − log ξ/2σ. This is the notion of solution that we consider in Theorem 1.8. There exists a more direct definition of this SPDE (restricted to a bounded domain) due to Hairer [Hai13, Hai14] via his theory of regularity structures. Let us also mention the notion of “energy solution” introduced by Gonc¸alves and Jara [GJ14], for which uniqueness has been proved by Gubinelli and Perkowski [GP15]. It provides a new framework for characterising the solution to the KPZ equation but it requires the equation to be taken under its stationary measure. The literature on the KPZ equation and related discrete models has been growing fast these last years, we refer to the reviews of Corwin [Cor12], Quastel [Qua12] or Spohn [Spo16]. The paper is organised as follows. In Section 2, we study the scaling limit of the invariant measure. Section 3 is devoted to the fluctuations at equilibrium. In Section 4 we prove Theorems 1.5 and 1.7 on the hydrodynamic limit, and in Section 5 we present the proof of the convergence of the fluctuations to the KPZ equation. Some technical bounds are postponed to the Appendix. The sections are essentially independent: at some localised places, we will rely on results obtained on the static of the model in Section 2.

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INVARIANT MEASURE

Acknowledgements. I am indebted to Reda Chhaibi for some deep discussions on this work at an early stage of the project. I would also like to thank Christophe Bahadoran for a fruitful discussion on the notion of entropy solutions for the inviscid Burgers equation, Nikos Zygouras for pointing out the article [DH96] and Julien Reygner for his helpful comments on a preliminary version of the paper. Notations. At some places in the paper, we rely on the one-to-one correspondence between Pk particle systems and discrete interfaces. Namely, to every interface S(k) = i=1 Xi , 1 ≤ k ≤ 2N , we associate the particle system η ∈ {0, 1}2N defined by η(i) = (Xi + 1)/2 for all i ∈ {1, . . . , 2N }. We let τk denote the shift by k ∈ Z, namely τk η := (η(k + 1), η(k + 2), . . . , η(k − 1), η(k)) .

(1.18)

where indices are taken modulo 2N . At several occasions in the paper, we will use microscopic variables in macroscopic functions: for instance with the rescaled interface uN with α ≥ 1, we will write uN (k) to denote uN (k/2N ). This abusive notation will never raise any confusion, but will greatly simplify the notations.

2 The invariant measure Let Be(q) denote the Bernoulli ±1 distribution with parameter q ∈ [0, 1], and let L be the log-Laplace functional associated with Be(1/2), namely L(h) = log cosh h for all h ∈ R. For each given N ≥ 1, we will work on the set {−1, +1}2N endowed with its natural sigma-field. The canonical process will be denoted by P X1 , . . . , X2N , and will be viewed as the steps of the walk S(n) := X k≤n k . Recall that A(S) is the area under the walk S, defined in (1.1). The strategy of the proof consists in introducing an auxiliary measure νN which is the same as µN except that, under νN , the walk is not conditioned on coming back to 0 but satisfies νN [S(2N )] = 0. This makes νN more amenable to limit theorems: we establish a Central Limit Theorem and a Local Limit Theorem for the marginals of the walk under νN . Since µN is equal to νN conditioned on the event S(2N ) = 0, and since the νN -probability of this event can be estimated with the Local Limit Theorem, we are able to get the convergence of the marginals of the walk under µN . Let us now provide the details. Let πN be the law of the simple random walk, that is 2N

πN := ⊗ Be(1/2) , k=1

and let νN be the measure defined by  p  A(S) dνN N 2 N = e̺N S(2N )−LS (h ) , dπN 1 − pN P N where LS (hN ) := 2N k=1 L(hk ) and 2σ  1 1 2σ  N = , h , k ∈ {1, . . . , 2N } . N + N − k + ̺N = − k (2N )α 2 (2N )α 2

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INVARIANT MEASURE

Remark 2.1 Under the measure νN , the total number of particles is not equal to N almost surely, but is equal to N in mean. The measure νN can be seen as a mixture of 2N + 1 measures, each of them being supported by an hyperplane of configurations with ℓ ∈ {0, . . . , 2N } particles. It is easy to check that our dynamics is reversible with respect to each of these measures, and therefore, with respect to νN . A simple calculation yields the identities P N N dνN = e k hk Xk −LS (h ) , dπN

2N

νN = ⊗ Be(qkN ) ,

(2.1)

k=1

where qkN = (L′ (hN k ) + 1)/2. From there, we deduce that νN [S(k)] =

k X

L′ (hN i ) , VarνN [S(k), S(ℓ)] =

k∧ℓ X

L′ (hN i ).

i=1

i=1

Observe that the curve ΣN α defined in (1.2) is nothing but the mean of S under νN . A simple calculation then yields the following asymptotics. For α ≥ 1, we have ( (2N )2−α σx(1 − x) + O(N 4−3α ) α>1, Rx ′ (2.2) ΣN (x) = α 2N 0 L (σ(1 − 2y))dy + O(1) α=1, and

( (2N ) qα (0, x ∧ y) + O(N 3−2α ) VarνN [S(x2N ), S(y2N )] = (2N ) qα (0, x ∧ y) + O(1) for all x, y ∈ [0, 1]. For α < 1, we find Z ∞   N α (L′ (2σy) − 1)dy + O(1) Σα (x) = N + (2N ) x +

α>1, α=1,

α 0. Lemma 2.3 Uniformly over all ~y ∈ Dα~x,N and all N ≥ 1, we have k

(2N ) 2 (α∧1) νN (uN (~xN ) = ~y ) − g˜α~x (~y ) = o(1) , 2k ˜α (~x). where g˜α~x is the density of the random vector B Let xf = 1 if α ≥ 1 and xf = +∞ if α < 1. In the case k = 1, let Eαx,N be the set of values y such that νN (uN (xf ) − uN (xN ) = y) > 0. Lemma 2.4 Uniformly over all y ∈ Eαx,N and all N ≥ 1, we have 1

(2N ) 2 (α∧1) νN (uN (xf ) − uN (xN ) = y ) − 2

Z

(x,xf )

g˜α

(z, z + y)dz = o(1) .

z∈R

Below, we provide the proof of the first lemma. The second lemma follows from exactly the same arguments, one simply has to notice that uN (xf ) − uN (xN ) R (x,x ) ˜α (xf )− B ˜α (x), and that converges in law to B ˜α f (z, z +y)dz is the density z∈R g at y of this limiting r.v. Proof of Lemma 2.3. Let us prove the case α < 1 which is more involved. The main difference in the proof with the case α ≥ 1 lies in the fact that the forthcoming bound (2.6) cannot be applied to all hN i simultaneously when α < 1. Indeed, these coefficients are not bounded uniformly over i and N when α < 1. However, for any given a > 0, they are bounded uniformly over all i ∈ IN,a := [N − a(2N )α , N + a(2N )α ] and all N ≥ 1. Without loss of generality, we can assume that x1 < x2 < . . . < xk so that only xk can take the value +∞. Let ϕh (t) = exp(L(h + it) − L(h)) for t, h ∈ R. This is the characteristic function of the Bernoulli ±1 r.v. with mean L′ (h) so that ϕh (t) = cos(t) + iL′ (h) sin(t) ,

t∈R, h∈R.

(2.5)

In particular, the characteristic function of the r.v. Xi under νN is given by ϕhN . i The function ϕ is 2π-periodic and |ϕh (t)| ≤ 1 for all h, t. From (2.5), one deduces that for any compact set K ⊂ R, there exists r(K) > 0 such that |ϕh (t)| ≤ exp(−rt2 L′′ (h)) ,

h 2π 2π i , , ∀t ∈ − 3 3

∀h ∈ K .

(2.6)

˜α (~x). Classical Let Φα denote the characteristic function of the Gaussian vector B ~ x,N arguments from Fourier analysis entail that for all y ∈ Dα RN := (2N )

αk 2

νN (uN (~xN ) = ~y ) − 2k g˜α~x (~y ) ,

can be rewritten as RN

1 = k π

1 ~ ΦN (~t)e−iht,~yi d~t − k π D

Z

Z

Rk

~ Φα (~t)e−iht,~yi d~t ,

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15

INVARIANT MEASURE

where ΦN is the characteristic function of uN (~xN ) under νN and o n α π D := ~t ∈ Rk : |tℓ | ≤ (2N ) 2 , ℓ = 1, . . . , k . 2 Notice that the factor 1/2 in the definition of D comes from the simple fact that our step distribution charges {−1, 1}, and therefore has a maximal span equal to 2. 1 ) and we bound |RN |π k by the sum of the following Then, we take ̺ ∈ (0, 2(3+k) three terms Z |ΦN (~t) − Φα (~t)|d~t , D1 = [−N ̺α , N ̺α ]k , J1 = Z D1 |Φα (~t)|d~t , D2 = Rk \D1 , J2 = D2 Z |ΦN (~t)|d~t , D3 = D\D1 . J3 = D3

It suffices to show that these three terms vanish as N → ∞. Regarding J1 , the proof of Lemma 2.2 shows that |ΦN (~t) − Φα (~t)| . |Φα (~t)|

|~t |3 α , N2

uniformly over all |~t|3 = o(N α/2 ). Since ̺(3 + k) < 21 , a simple calculation shows that J1 goes to 0 as N → ∞. The convergence of J2 to 0 as N → ∞ is immediate. We turn to J3 . For each ℓ ∈ {1, . . . , k}, we set o n D3,ℓ = D3 ∩ |tℓ | > 3−ℓ N ̺α ; ∀j > ℓ, |tj | ≤ 3−j N ̺α , so that D3 = ∪ℓ D3,ℓ . The important feature of these sets is that for all N large enough |tℓ | |tℓ + . . . + tk | 2π , ≤ α ≤ α 3 2(2N ) 2 (2N ) 2

∀~t ∈ D3,ℓ , ∀ℓ ∈ {1, . . . , k} .

(2.7)

We bound separately each term J3,ℓ arising from the restriction of the integral in J3 to D3,ℓ . Take a > 0 such that −a < x1 < xk−1 < a and recall that xk can be infinite. Let K be a compact set that contains all the values hN i , i ∈ IN,a , and let r be the corresponding constant introduced above (2.6). We also define jp = N + ⌊xp (2N )α ⌋ for all p ∈ {1, . . . , k} and IN,a,ℓ := IN,a ∩ (N + xℓ−1 (2N )α , N + xℓ (2N )α ] .

Using the independence of the Xi ’s under νN and the fact that the modulus of a characteristic function is smaller than 1 at the second line, as well as (2.6) and (2.7) at the third line, we get k 2N h  X X ~ 1{j≤jp } X(j) ΦN (t) = νN exp i j=1

p=1

tp i α (2N ) 2

 t + . . . + t  Y ℓ k ≤ ϕhN α j 2 (2N ) j∈IN,a,ℓ   (tℓ + . . . + tk )2 X ′′ N ≤ exp − r L (h ) . j (2N )α j∈IN,a,ℓ

(2.8)

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INVARIANT MEASURE

For all N large enough we have 1 (2N )α

X

j∈IN,a,ℓ

1 L′′ (hN j ) ≥ q(xℓ−1 , xℓ ∧ a) , 2

so that, using (2.7), we get Z Z (k−1)α r 2 e− 8 tℓ q(xℓ−1 ,xℓ ∧a) d~t . N 2 J3,ℓ ≤

r 2

|tℓ |>3−ℓ N ̺α

D3,ℓ

e− 8 tℓ q(xℓ−1 ,xℓ ∧a) dtℓ ,

which goes to 0 as N → ∞. This concludes the proof. Remark 2.5 It is possible to push the expansion of the local limit theorem one step further, in the spirit of [Pet75, Thm VII.12]. Then, a simple calculation shows the following. Let x ∈ (0, 1) if α ≥ 1, and x ∈ R if α < 1. We have µN [S(k)] − νN [S(k)] = o(N

1∧α 2

),

uniformly over all k ≤ x(2N ) if α ≥ 1, and all k ≤ N + x(2N )α if α < 1. Corollary 2.6 For α ≥ 1, let x ∈ (0, 1), and for α < 1 let x ∈ R. Let GxN be the sigma-field generated by all the Xk , with k ≤ x(2N ) when α ≥ 1, with k ≤ N + x(2N )α when α < 1. Then, the Radon-Nikodym derivative of µN with respect to νN , restricted to GxN , is bounded uniformly over all N ≥ 1. Proof. Let k = ⌊x(2N )⌋ if α ≥ 1, and k = ⌊N + x(2N )α ⌋ if α < 1. Using (2.4) at the first line and the independence of the Xi ’s at the second line, we get k

k

µN ( ∩ {S(i) = yi }) = i=1

νN ( ∩ {S(i) = yi }; S(2N ) = 0) i=1

νN (S(2N ) = 0) k νN (S(2N ) − S(k) = −yk ) = νN ( ∩ {S(i) = yi }) , i=1 νN (S(2N ) = 0)

for all y1 , . . . , yk ∈ R. By Lemmas 2.3 and 2.4, the fraction on the r.h.s. is uniformly bounded over all yk ∈ R and all N ≥ 1, thus yielding the statement of the corollary. Lemma 2.7 Let J = [0, 1] if α ≥ 1, and J = [−A, A] for some arbitrary A > 0 if α < 1. For any p ≥ 1 and β ∈ (0, 1/2), we have i h sup µN kuN kpC β (J,R) < ∞ . N ≥1

As a consequence, the sequence of processes uN under µN is tight in C([0, 1], R) if α ≥ 1, in C(R, R) if α < 1. Proof. Observe that the law of uN under µN is invariant under the reparametrisation x 7→ 1 − x when α ≥ 1, and x 7→ −x when α < 1. Therefore, it suffices to prove the statement of the lemma with the β-H¨older norm restricted to [0, 1/2] in the first case, and to [−A, 0] in the second case. The uniform absolute continuity

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17

INVARIANT MEASURE

of Corollary 2.6 ensures in turn that it suffices to bound, uniformly over all N ≥ 1, the p-th moment of the β-H¨older norm of uN under νN . First, we bound the p-th moment of uN (0). In the case α ≥ 1, the latter is actually equal to 0, while for α < 1, we have for any λ ∈ R log νN [eλu

N (0)

λ2 qα (−∞, 0) + O(N −α/2 ) , 2

]=

uniformly over all N ≥ 1, which ensures that all the moments of uN (0) are uniformly bounded in N ≥ 1. Regarding the H¨older semi-norm, a direct computation shows that for all δ > 0 h  uN (y) − uN (x) i log νN exp ≤ kL′′ k∞ |x − y|1−2δ , |y − x|δ

uniformly over all x, y of the form k/2N with k ∈ {1, . . . , 2N } when α ≥ 1, and of the form (k − N )/(2N )α with k ∈ {N − ⌊A(2N )α ⌋, . . . , N + ⌊A(2N )α ⌋} when α < 1. Taking δ ∈ (0, 1/2), this yields a finite bound uniformly over all N ≥ 1 and all such discrete x, y. Using classical interpolation arguments, we deduce that this bound is still finite for non-discrete x, y lying in [0, 1] or [−A, A]. Henceforth, the Kolmogorov Continuity Theorem ensures that for any β ∈ (0, 1/2) and any p ≥ 1 we have:   |uN (x) − uN (y)|p sup νN sup 1, we use the fact that L(0) = L′ (0) = L(3) (0) = 0, L′′ (0) = 1 and kL(4) k∞ < ∞, to get LS (hN ) = =

2N X

L(hN i )=

i=1 σ2

6

2N  X 2σ 2 1 2 ′′ N − i + + O(N 5−4α ) L (0) (2N )2α 2 i=1

(2N )3−2α + O(N (5−4α)∨(1−2α) ) ,

and the asserted result follows in that case. For α = 1, the result follows from the convergence of Riemann approximations of integrals. Finally, when α < 1, we use the simple facts that L is even and that L(x) − x + log 2 is integrable on [0, ∞) to get N

LS (h ) = 2

N X

L(hN i )

=2

i=1

N X i=1

hN i

− 2N log 2 + 2

σ = (2N )2−α − 2N log 2 + O(N α ) , 2

N X i=1

N (L(hN i ) − hi + log 2)

thus concluding the proof.

3 Equilibrium fluctuations The goal of this section is to establish Theorem 1.4. Our method of proof is classical: first, we show tightness of the sequence of processes uN , then we identify the limit via a martingale problem. Recall that we work under the reversible measure µN . 3.1 Tightness From now on, we set J = [0, 1] when α ≥ 1 and J = [−A, +A] for an arbitrary value A > 0 when α < 1, along with (√ 2 sin(nπx) for α ≥ 1 ,   en (x) = √1 sin nπ (x + A) for α < 1 . 2A A

E QUILIBRIUM

19

FLUCTUATIONS

This is an orthonormal basis of L2 (J). For all β > 0, we define the associated Sobolev spaces n o X H −β (J) := f ∈ S ′ (J) : kf k2H −β := n−2β hf, en i2 < ∞ . n≥1

Recall that for α < 1, the value A > 0 is arbitrary. In order to prove tightness of the sequence uN in the Skorohod space D([0, ∞), C([0, 1]) for α > 1 and in D([0, ∞), C(R)) for α < 1, it suffices to show that the sequence of laws of uN (t = 0, ·) is tight in C(J), and that for any T > 0 there exists p > 0 such that   p N N N sup ku (t) − u (s)kC(J) = 0 , (3.1) lim lim EµN h↓0 N →∞

s,t≤T |t−s|≤h

see for instance [Bil99, Thm 13.2]. Since we start from the stationary measure, the first condition is ensured by Lemma 2.7. To check the second condition, we proceed as follows. We introduce a piecewise linear interpolation in time u ¯N of our original process by setting   tN , · u ¯N (t, ·) := (tN + 1 − t(2N )2α∧2 )uN (2N )2α∧2  t +1  N + (t(2N )2α∧2 − tN )uN ,· , (2N )2α∧2

where tN := ⌊t(2N )2α∧2 ⌋.

Lemma 3.1 For all β > 1/2 and all p ≥ 1, we have i1 h √ p p N N . t−s, k¯ u (t) − u ¯ (s)k EN −β µN H (J)

uniformly over all 0 ≤ s ≤ t ≤ T and all N ≥ 1. Proof. Assume that we have the bound EN µN

h

N

N

ku (t) − u

(s)kpH −β (J)

i1

p

.

√

3

t − s + N − 2 (1∧α) ,

(3.2)

uniformly over all 0 ≤ s ≤ t and all N ≥ 1. Let 0 ≤ s ≤ t ≤ T . We distinguish two cases. If tN = sN or t = (sN + 1)/(2N )2α∧2 , then t − s ≤ 1/(2N )2α∧2 and    s  sN + 1  N N N N 2α∧2 N ,· −u ,· u ¯ (t, ·) − u ¯ (s, ·) = (t − s)(2N ) , u (2N )2α∧2 (2N )2α∧2 so that the asserted bound follows from (3.2). If tN ≥ sN + 1, then we write  s +1    tN N N , · − u ,· u ¯N (t, ·) − u ¯N (s, ·) = uN (2N )2α∧2 (2N )2α∧2   tN +u ¯N (t, ·) − u ¯N , · (2N )2α∧2  s +1  N ,· − u ¯N (s, ·) . +u ¯N (2N )2α∧2

E QUILIBRIUM

20

FLUCTUATIONS

The second and third increments √ on the r.h.s. can be bounded using the first case above, yielding a term of order t − s. Regarding the first increment, either tN = sN +1 and it vanishes, or tN ≥ sN +1 + 1 and (3.2) yields a bound of order r √ √ 3 3 tN sN + 1 − + N − 2 (1∧α) . t − s + (t − s) 4 . t − s , 2α∧2 2α∧2 (2N ) (2N ) as required. To complete the proof R of the lemma, it suffices to show (3.2). For all n ≥ 1, we let u ˆ(t, n) := J u(t, x)en (x)dx. Since β > 1/2, (3.2) is proved as soon as we show that for all p ≥ 1 i1 √ h 3 p u(t, n) − u ˆ(s, n)|p . t − s + N − 2 (1∧α) , EN µN |ˆ

(3.3)

uniformly over all N ≥ 1, all n ≥ 1 and all 0 ≤ s ≤ t. Let LN be the generator of uN . Using the reversibility of the process, we have the following identities Z t N LN u ˆ(r, n)dr + Ms,t (n) , u ˆ(t, n) − u ˆ(s, n) = s Z t ˜ N (n) , LN u ˆ(r, n)dr + M u ˆ(T − (T − t), n) − u ˆ(T − (T − s), n) = − t,s s

N (n), t ≥ s is a martingale adapted to the natural filtration of uN , and where Ms,t N , s ≤ t is a martingale in the reversed filtration. Summing up these two iden˜ t,s M tities, we deduce that it suffices to control the p-th moment of the martingales N (n) and M ˜ N (n). Using the Burkh¨older-Davis-Gundy inequality (6.3), we get Ms,t t,s

i1 i1 h h i1 h p/2 p N N N p p N N p p N hM (n)i sup |M (n)−M (n)| . E +E , (n)| |M EN s,· s,r s,r− µN µN s,t µN t r∈(s,t]

uniformly over all N ≥ 1, all n ≥ 1 and all 0 ≤ s ≤ t. It is then a simple calculation to check that (3.3) is satisfied. The same bound holds for the reversed martingale by symmetry, thus concluding the proof. We need an interpolation inequality to conclude the proof of the tightness. Lemma 3.2 Let η = 1/2 − ǫ and β = 1/2 + ǫ. For ǫ > 0 small enough, there exist c > 0 and γ, κ ∈ (0, 1) such that 1−κ kf kC γ (J) ≤ c kf kκC η (J) kf kH −β (J) ,

∀f ∈ C η (J) ∩ H −β (J) .

(3.4)

Proof. We rely on two classical interpolation results, we refer to the book of Triebel [Tri78] for the proofs. For q ≥ 1 and δ ∈ (0, 1), let W δ,q (J) be the space of functions f : J → R such that  Z Z |f (t) − f (s)|q 1 q kf kW δ,q := kf kLq + ds dt 0 and κ ∈ (0, 1), we set δ := κη − (1 − κ)β as well as q := 2/(1 − κ). Then, there exists c′ > 0 such that , kf kW δ,q ≤ c′ kf kκC η kf k1−κ H −β

∀f ∈ C η ∩ H −β .

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FLUCTUATIONS

Furthermore, for any γ > 0 such that (δ − γ)q > 1 there exists c′′ > 0 such that kf kC γ ≤ c′′ kf kW δ,q ,

∀f ∈ W δ,q .

Therefore, taking κ ∈ (2/3, 1), η = 1/2 − ǫ and β = 1/2 + ǫ with ǫ small enough, we deduce the statement of the lemma. Using Lemma 2.7, the stationarity of the process uN and the definition of u ¯N , we deduce that for all p ≥ 1 and all η ∈ (0, 1/2) i h p N N k¯ u (t) − u ¯ (s)k sup sup EN µN C η (J) < ∞ . N ≥1 0≤s≤t

Using Lemmas 3.1 and 3.2 together with H¨older’s inequality, we deduce that there exist γ, κ ∈ (0, 1) such that for all p ≥ 1 i h p(1−κ) uN (t) − u ¯N (s)kpC γ (J) . (t − s) 2 , EN µN k¯

uniformly over all N ≥ 1 and all 0 ≤ s ≤ t ≤ T . Applying Kolmogorov’s Continuity Theorem, we deduce that for all ν ∈ (0, (1 − κ)/2) and all p ≥ 1, we have   k¯ uN (t) − u ¯N (s)kpC γ (J) N 0} + (1 − pN )1{∆S(s,·) 0 and let ιN be a measure on {0, 1}2N . For α < 1, we suppose that Assumption 4.1 is fulfilled. For every δ > 0, we have lim lim

ǫ↓0 N →∞

PN ιN

Z

2N  1 X VǫN (τk ηs )ds ≥ δ = 0 . N

t 0

(4.1)

k=1

The proof of this theorem relies on the classical one-block and two-blocks estimates. First, let us introduce the Dirichlet form associated to our dynamics: Xp p f (η)LN f (η) νN (η) , DN (f ) = − η

where f : {0, 1}2N → R+ and LN is the generator of our sped up process, that is LN g(η) = (2N )(1+α)∧2

2N −1 X k=1

(g(η k,k+1 ) − g(η))(pN η(k + 1)(1 − η(k)) + (1 − pN ) η(k)(1 − η(k + 1))) ,

where η k,k+1 is obtained from η by permuting the values at sites k and k + 1. The reference measure in the Dirichlet form is taken to be νN , as defined in Section 2, which is reversible for our dynamics. This is because we do not work only on the hyperplane of configurations with N particles but on the whole set {0, 1}2N . In the statements of the lemmas below, the function f will always be non-negative and such that νN [f ] = 1. Lemma 4.3 (One-block estimate) For any α > 0 and any C > 0, we have lim lim

sup

ℓ→∞ N →∞ f :D (f )≤CN (2−α)∨1 N

2N i 1 X h νN Vℓ (τk η)f (η) = 0 . N k=1

H YDRODYNAMIC

28

LIMIT

The proof of this result is standard, we refer to [KOV89, Lemma 2.2] for instance, or to [EL15, Lemma 23]. Actually, the case α ≥ 1 can be derived directly from these references, while the case α < 1 is slightly different due the time-scaling of our process. However, the arguments in the proof work verbatim up to simple modifications. Lemma 4.4 (Two-blocks estimate) For any α ≥ 1 and any C > 0, we have 2N −1 1 X 1 N →∞ f :DN (f )≤CN N (2ǫN + 1)2 k=1   X X ′ × νN MTℓ (j ) (η) − MTℓ (j) (η) f (η) = 0 .

lim lim lim

ℓ→∞ ǫ↓0

sup

j:|j−k|≤ǫN j ′ :|j ′ −k|≤ǫN

For α < 1, if ιN satisfies Assumption 4.1, we have for all t, δ > 0

2N −1 X X 1 1 X N →∞ N (2ǫN + 1)2 k=1 j:|j−k|≤ǫN j ′ :|j ′ −k|≤ǫN Z t   ′ ) (ηs ) − MT (j) (ηs )| ≥ δ ds = 0 . |M PN × T (j ιN ℓ ℓ

lim lim lim

ℓ→∞ ǫ↓0

0

In the case α ≥ 1, this two-blocks estimate is a classical result of the theory of hydrodynamic limits. We refer to Lemma 2.3 in Kipnis, Olla and Varadhan [KOV89] for a very similar statement but for the WASEP with periodic boundary conditions, or to Lemma 24 in [EL15]. In the case α < 1, the result is due to Rezakhanlou [Rez91, Sect. 6] when the lattice does not have boundaries: to adapt his proof to our setting, it suffices to use the coupling introduced in the forthcoming Subsection 4.3 and to apply the arguments presented in [Rez91, Lemma 6.6]. Proof of Theorem 4.2. Denote by PtN the semigroup associated to our discrete dynamics and by ftN the Radon-Nikodym derivative of Z 1 t ιN PsN ds , t 0 with respect to νN . Let G : {0, 1}2N → R+ . Then, for any measure ιN we have  Z t Z t  N −1 N P ιN G(ηs )ds ≥ δ ≤ δ EιN G(ηs )ds (4.2) 0

0

Classical arguments (see for instance Section 5.2 in [KL99]) ensure that DN (ftN ) is bounded by HN (ιN |νN )/2t where HN is the relative entropy defined by   dιN dιN log . HN (ιN |νN ) := νN dνN dνN A simple calculation shows that, for any measure ιN , this relative entropy is bounded by a term of order N 1∨(2−α) . Consequently, we get h i Z t  −1 G(η)f (η) , (4.3) ν G(η )ds ≥ δ ≤ δ t sup PN N s ιN 0

f :DN (f )≤CN 1∨(2−α)

H YDRODYNAMIC

29

LIMIT

where the supremum is taken over all f : {0, 1}2N → R+ such that νN [f ] = 1. The calculation performed on p.120 of [KOV89] yields VǫN (τk η) ≤

˜ ′ k∞ kΦ (2ǫN + 1)2

1 + 2ǫN + 1

X

X

j:|j−k|≤ǫN j ′ :|j ′ −k|≤ǫN

X

j:|j−k|≤ǫN

MTℓ (j ′ ) (η) − MTℓ (j) (η)

ℓ Vℓ (τj η) + O , N

(4.4)

where the O( Nℓ ) is uniform in k and η, so that it has a negligible contribution in (4.1). Using (4.3), we bound the contribution of the second term as follows: PN ιN

Z

≤δ

−1

t 0

2N 1 1 X N 2ǫN + 1 k=1

t

X

j:|j−k|≤ǫN

νN

sup f :DN (f )≤CN 1∨(2−α)



Vℓ (τj ηs )ds ≥ δ

2N 1 X 1 N 2ǫN + 1 k=1



X

j:|j−k|≤ǫN

 Vℓ (τj η)f (η) ,

so that Lemma 4.3 ensures that this term has a vanishing contribution as N → ∞, ǫ ↓ 0 and then ℓ → ∞. Similarly, for α ≥ 1 the contribution of the first term of (4.4) is handled by Lemma 4.4 combined with (4.3) for α ≥ 1. For α < 1, the contribution of the first term of (4.4) is dealt with as follows. Using (4.2), we get PN ιN

Z

0

t

2N ˜ ′ k∞ 1 X kΦ N (2ǫN + 1)2 k=1

2N ˜ ′ k∞ 1 X kΦ ≤ δ−1 N (2ǫN + 1)2 k=1

X

j:|j−k|≤ǫN j ′ :|j ′ −k|≤ǫN

X

j:|j−k|≤ǫN j ′ :|j ′ −k|≤ǫN

Z

 MTℓ (j ′ ) (ηs ) − MTℓ (j) (ηs ) ds ≥ δ t

0

EN ιN

h

(4.5) i |MTℓ (j ′ ) (ηs ) − MTℓ (j) (ηs )| ds .

Then, for any κ > 0 we write   h i N ′ ′ |M (η ) − M (η )| ≥ δκ + δκ , |M (η ) − M (η )| ≤ P EN Tℓ (j) s Tℓ (j) s Tℓ (j ) s Tℓ (j ) s ιN ιN

where we have used the fact that MTℓ (j) (η) belongs to [0, 1] for all ℓ, j, η. By Lemma 4.4, we deduce that (4.5) goes to 0 as N → ∞, ǫ ↓ 0 and ℓ → ∞. This concludes the proof. 4.2 Hydrodynamic limit: the parabolic case The goal of this subsection is to prove Theorem 1.5: we will write PN for the law of the process starting from an initial condition satisfying the hypothesis of that theorem. Recall that we write mN (t, k) instead of mN (t, k/2N ) for simplicity. To prove tightness of the sequence mN in the Skorohod space D([0, ∞), C([0, 1])), it suffices to show that the sequence mN (t = 0, ·) is tight in C([0, 1]), and that we have for any T > 0 i h (4.6) sup kmN (t, ·) − mN (s, ·)k∞ = 0 . lim lim EN h↓0 N →∞

t,s≤T,|t−s|≤h

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The former is actually an hypothesis of our theorem. To prove the latter, we introduce a piecewise linear time interpolation of mN , namely we set tN := ⌊t(2N )2 ⌋ and   t N , · m ¯ N (t, ·) :=(tN + 1 − t(2N )2 )mN (2N )2 t + 1  N + (t(2N )2 − tN )mN ,· . (2N )2 First, we control the distance between mN and m ¯ N.

Lemma 4.5 For all T > 0, we have i h lim EN sup kmN (t, ·) − m ¯ N (t, ·)k∞ = 0 . N →∞

t∈[0,T ]

The proof of this lemma is almost the same as the proof of Lemma 3.3, so we omit it. This result ensures that it is actually sufficient to show (4.6) with mN replaced by m ¯ N in order to get tightness. The following proposition ensures that m ¯ N satisfies (4.6). Proposition 4.6 For any T > 0, there exists δ > 0 such that   km ¯ N (t, ·) − m ¯ N (s, ·)k∞ N t. Given the expression (4.8), the increment mN (t′ , ℓ) − mN (t, ℓ) can be written as the sum of three terms: the contribution of the initial condition, of the asymmetry and of the martingale terms. We bound separately the p-th moments of these three terms. First, we let p¯N be the fundamental solution of the discrete heat equation on the whole line Z: contrary to pN , p¯N is translation invariant. Let us also extend mN into a function on the whole line Z: we simply consider the 4N -periodic, odd function that coincides with mN on [0, 2N ]. By (6.11), we get 2N −1 X k=1

=

N N (pN t′ (k, ℓ) − pt (k, ℓ))m (0, k)

X

k∈Z

=

X

N (¯ pN ¯N t (ℓ − k))m (0, k) t′ (ℓ − k) − p

p¯N t (k)

X j∈Z

k∈Z

N N p¯N t′ −t (j)(m (0, ℓ − k − j) − m (0, ℓ − k)) ,

so that, using the H¨older regularity of the initial condition and Lemma 6.2, we deduce that −1 X 2N  |j| 2δ X N N N N . (t′ − t)δ , . p ¯ (j) (p (k, ℓ) − p (k, ℓ))m (0, k) t t′ −t t′ 2N j∈Z

k=1

uniformly over all ℓ ∈ {1, . . . , 2N − 1}, all t ≤ t′ ∈ [0, T ] and all N ≥ 1. We turn to the contribution of the asymmetry. Using the estimates on pN recalled in Appendix 6.2, we get the following almost sure bound Z tX Z t′ X N pt′ −s (k, ℓ)1{∆S(s(2N )2 ,k)6=0} ds − pN t−s (k, ℓ)1{∆S(s(2N )2 ,k)6=0} ds 0

≤

0

k

Z tX 0

k

′

N |pN t′ −s (k, ℓ) − pt−s (k, ℓ)|ds +

δ

. (t − t) ,

Z

t

t′

k

X k

pN t′ −s (k, ℓ)ds

uniformly over all ℓ ∈ {1, . . . , 2N − 1}, all t, t′ ∈ [0, T ] and all N ≥ 1. Finally, we treat the martingale term. We introduce the following martingales Z t+u X t,t′ N ′ Au (ℓ) := pN t′ −s (k, ℓ)dM (s, k) , u ∈ [0, t − t] , t

t,t′

Bu (ℓ) :=

Z

0

k uX k

N N (pN t′ −s (k, ℓ) − pt−s (k, ℓ))dM (s, k) , u ∈ [0, t] ,

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′

′

t,t and we observe that Ntt′ (ℓ) − Ntt (ℓ) = At,t t′ −t (ℓ) + Bt (ℓ). We bound separately the p-th moments of these two terms. In both cases, we will apply (6.3). First, we observe that the jumps of these two martingales are almost surely bounded by a term of order 2/(2N )2−α . Second, by Lemma 6.1 we have the following almost sure bounds Z t′ X 4(2N )2 (t′ − t)1/2 t,t′ 2 N , hA· (ℓ)it′ −t ≤ (k, ℓ) ds . p ′ t −s (2N )4−2α t (2N )3−2α ′

k

and ′

hB·t,t (ℓ)it ≤

4(2N )2 (2N )4−2α

Z tX 0

k

2 N (pN t′ −s (k, ℓ) − pt−s (k, ℓ)) ds .

(t′ − t)2δ , (2N )3−2α

uniformly over all t < t′ ∈ [0, T ], all ℓ ∈ {1, . . . , 2N − 1} and all N ≥ 1. Applying (6.3), we get the desired bound, thus concluding the proof. Lemma 4.8 For all δ ∈ (0, δi ∧ 21 ), all T > 0 and all p ≥ 1, we have E

N

h

N

N

p

|m (t, x) − m (t, y)|

i1

p

. |x − y|δ ,

uniformly over all t ∈ [0, T ], all x, y ∈ [0, 1] and all N ≥ 1. This result is immediate when α = 1 since, in that case, the scaling makes all the discrete interfaces uniformly 1-Lipschitz. Proof. It suffices to establish the bound for x, y of the form ℓ/2N, ℓ′ /2N since the remaining cases follow by interpolation. Given the expression (4.8), the increment mN (t, ℓ) − mN (t, ℓ′ ) can be written as the sum of three terms: the contribution of the initial condition, of the asymmetry and of the martingale terms. We bound separately the p-th moments of these three terms. As in the proof of the previous lemma, we use the fact that mN (0, ·) is δ-H¨older uniformly in N ≥ 1 to deduce that −1 2N  |ℓ − ℓ′ | δ X N ′ N , (pt (k, ℓ) − pN (k, ℓ ))m (0, k) . t 2N k=1

uniformly over all N ≥ 1, all ℓ, ℓ′ and all t ≥ 0, as required. We turn to the contribution of the asymmetry. Using classical estimates on the heat kernel, recalled in Appendix 6.2 we have almost surely Z tX N ′ (pN (k, ℓ) − p (k, ℓ ))1 ds 2 {∆S(t(2N ) ,k)6=0} t−s t−s 0

≤

k

Z tX 0

k

N ′ |pN t−s (k, ℓ) − pt−s (k, ℓ )|ds .

 |ℓ − ℓ′ | δ 2N

,

uniformly over all N ≥ 1, all t ∈ [0, T ] and all ℓ, ℓ′ ∈ {1, . . . , 2N − 1}, thus yielding the desired bound. Finally, we treat the martingale term. We aim at applying (6.3) to the martingale

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s 7→ Nst (ℓ) − Nst (ℓ′ ) at time t. First, observe that the absolute value of the jumps of this martingale are bounded by 2/(2N )2−α . Second, we have the following almost sure bound Z t 2N −1 X 4 t t ′ N ′ 2 hN· (ℓ) − N· (ℓ )it ≤ |pN t−s (k, ℓ) − pt−s (k, ℓ )| ds (2N )2−2α 0 k=1 Z t  |ℓ − ℓ′ | 2δ 1 1 √ ds . (2N )2−2α 0 2N t − s 2N  |ℓ − ℓ′ | 2δ 1 . , (2N )3−2α 2N

uniformly over all N ≥ 1, all ℓ, ℓ′ ∈ {1, . . . , 2N − 1} and all t ∈ [0, T ]. Applying (6.3), we get a bound of order  |ℓ − ℓ′ | δ 1 1 + , 3−2α 2N (2N )2−α (2N ) 2

for the p-th moment of Ntt (ℓ) − Ntt (ℓ′ ). As soon as ℓ 6= ℓ′ , we find (2N )−(2−α) . (|ℓ − ℓ′ |/(2N ))δ thus concluding the proof. Proof of Proposition 4.6. Fix T > 0. Recall the definition of m ¯ N . Arguing differently according to the relative values of |t′ − t| and (2N )−2 , one can deduce from Lemma 4.7 that there exists δ ∈ (0, δ2i ∧ 14 ) such that for any p ≥ 1 i1 h p ¯ N (t′ , x) − m ¯ N (t, x)|p . |t′ − t|δ , EN |m

uniformly over all t′ , t ∈ [0, T ], all x ∈ [0, 1] and all N ≥ 1. Using Lemma 4.8 and the definition of m ¯ N , we also get i1 h p ¯ N (t, x) − m ¯ N (t, y)|p EN |m ≤

1 X j=0

t + j  h   p i 1 p N N tN + j EN mN , x − m , y (2N )2 (2N )2

. |x − y|δ ,

uniformly over all x, y ∈ [0, 1], all t ∈ [0, T ] and all N ≥ 1. Combining these two bounds, we obtain for all p ≥ 1, 1

EN [|m ¯ N (t′ , x) − m ¯ N (t, y)|p ] p . (|t′ − t| + |x − y|)δ , uniformly over the same set of parameters. Kolmogorov’s Continuity Theorem then ensures that m ¯ N admits a modification satisfying the bound stated in Proposition 4.6 uniformly in N ≥ 1 for some δ > 0. Since m ¯ N is already continuous, it coincides with its modification PN -a.s., thus concluding the proof. We now proceed to the proof of Theorem 1.5: we argue differently in the cases α ∈ (1, 3/2) and α = 1. In both cases, we set hf, giN

2N 1 X  k   k  g . = f 2N 2N 2N k=1

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Proof of Theorem 1.5, α ∈ (1, 3/2). We already know that the sequence mN , N ≥ 1 is tight. Let m be the limit of a converging subsequence. To conclude the proof, we only need to show that for any ϕ ∈ C 2 ([0, 1]) such that ϕ(0) = ϕ(1) = 0, we have Z 1 t hm(t), ϕi = hm(0), ϕi + hm(s), ϕ′′ ids + σth1, ϕi . (4.9) 2 0 This suffices to identify the unique weak solution of the PDE (1.8). The definition of our dynamics implies that for all ϕ ∈ C 2 ([0, 1]) such that ϕ(0) = ϕ(1) = 0, we have Z 1 t N N N hm (t), ϕiN =hm (0), ϕiN + hm (s), (2N )2 ∆ϕiN ds 2 0 (4.10) Z tD E N α 1{∆S(s(2N )2 ,·)6=0} , ϕ ds + Mt (ϕ) , + (2pN − 1)(2N ) N

0

where M N (ϕ) is a martingale with bracket Z t 4 N hϕ2 , pN 1{∆S(s(2N )2 ,·)>0} + (1 − pN )1{∆S(s(2N )2 ,·) 0 , (4.12) 2 ξ(t, 0) = ξ(t, 1) = e2σ t , ξ(0, ·) = e−2σm(0,·) . This equation admits a unique weak solution in the space of continuous space-time functions, and it is well-known that the unique weak solution of (1.9) coincides with the latter solution upon reverse Hopf-Cole transform. A famous result due to G¨artner [G¨ar88] shows that a similar transform, performed at the level of the exclusion process, linearises the drift of the stochastic differential equations solved by our discrete process. Namely, if one sets γN = and

2σ , 2N

cN =

(2N )2 , λN = cN (eγN − 2 + e−γN ) , eγN + e−γN 2 ,2N x)+λ t N

ξ N (t, x) := e−γN S(t(2N )

,

x ∈ [0, 1] ,

t≥0,

then, using the abusive notation ξ N (t, k) for ξ N (t, x) when x = k/2N , we have ( ˜ N (t, k) , dξ N (t, k) = cN ∆ξ N (t, k)dt + dM ξ N (t, 0) = ξ N (t, 1) = eλN t , where ∆ is the discrete Laplacian and M N (t, k) is a martingale with quadratic variation given by Z t  ˜ N (·, k)it = (2N )2 hM ξ N (s, k)2 (e−2γN − 1)2 1{∆S(s(2N )2 ,k)>0} pN 0 (4.13)  2 2γN +(e − 1) 1{∆S(s(2N )2 ,k) 0 such that |ξ N (t, k)| ≤ C for all t in a compact set of R+ , all k ∈ {1, . . . , 2N − 1} and all N ≥ 1. Consequently ˜ N (·, k)it ≤ C ′ t uniformly over the same set there exists C ′ > 0 such that hM of parameters. Moreover, the jumps of this martingale are uniformly bounded by some constant on the same set of parameters. Then, a simple calculation based on (6.3) shows that for any given t ≥ 0, the moments of RtN (ϕ) vanish as N → ∞. Hence, any limit ξ of a converging subsequence of ξ N satisfies Z  1 t 2 hξ(t), ϕi = hξ(0), ϕi + hξ(s), ϕ′′ i + e2σ s (ϕ′ (0) − ϕ′ (1)) ds , 2 0 for all ϕ as above, and therefore coincides with the unique weak solution of (4.12), thus concluding the proof. 4.3 Hydrodynamic limit: the hyperbolic case Definition 4.9 We say that η ∈ L∞ ((0, ∞)×(0, 1)) is an entropy solution of (1.11) if for all c ∈ [0, 1] and all ϕ ∈ Cc∞ ([0, ∞) × [0, 1], R+ ) we have Z ∞Z 1 Z 1 ± ± (η0 − c)± ϕ(0, x)dx ((η − c) ∂t ϕ + h (η, c)∂x ϕ)dx dt + 0 0 0 (4.14) Z ∞  ± ± (1 − c) ϕ(t, 0) + (0 − c) ϕ(t, 1) dt ≥ 0 , + 2σ 0

where (x)± denotes the positive/negative part of x ∈ R, sgn± (x) = ±1(0,∞) (±x) and h± (η, c) := −2σ sgn± (η − c)(η(1 − η) − c(1 − c)).

There is existence and uniqueness of the entropy solution of (1.11), we refer for instance to Vovelle [Vov02]. Notice that the meaning given to the boundary conditions is not immediate in general. Actually, if η admits a trace on the boundary, then the trace satisfies the so-called BLN conditions, see [BlRN79]. We let M([0, 1]) be the space of finite measures on [0, 1] endowed with the topology of weak convergence. Recall that ̺N is the rescaled process of the density of particles, defined in (1.12). Proposition 4.10 Let ιN be any measure on {0, 1}2N . The sequence of processes (̺N t , t ≥ 0), starting from ιN , is tight in the space D([0, ∞), M([0, 1])). Furthermore, the associated sequence of processes (mN (t, x), t ≥ 0, x ∈ [0, 1]) is tight in D([0, ∞), C([0, 1])). For a generic measure ιN on {0, 1}2N , mN (t, 1) is not necessarily equal to 0. Proof. Let ϕ ∈ C 2 ([0, 1]). It suffices to show that h̺N 0 , ϕi is tight in R, and that for all T > 0 h i N N sup |h̺ − ̺ , ϕi| =0. lim lim EN t s ιN h↓0 N →∞

s,t≤T,|t−s|≤h

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N The former is immediate since |h̺N 0 , ϕi| ≤ kϕk∞ . Regarding the latter, we let L be the generator of our process and we write

h̺N t

− ̺N s , ϕi

1 = 2N

Z tX 2N s k=1

N ϕ(k)LN ηr (k)dr + Ms,t (ϕ) ,

N (ϕ) is a martingale. Its bracket can be bounded almost surely as follows where Ms,t

N hMs,· (ϕ)it

≤

Z

t s

2N −1 X 1 t−s . (∇ϕ(k))2 (2N )1+α dr . 2 (2N ) (2N )2−α k=1

Since the jumps of this martingale are bounded by a term of order kϕ′ k∞ /(2N )2 , N (ϕ)| vanthe BDG inequality (6.3) ensures that the expectation of supt∈[s,s+h] |Ms,t ishes as N → ∞. Consequently, h i 1 N sup |M (ϕ)| =0. sup EN s,t ιN h↓0 N →∞ h s∈[0,T ] t≤T,|t−s|≤h

lim lim

Let us bound the term involving the generator. Decomposing the jump rates into the symmetric part (of intensity 1−pN ) and the totally asymmetric part (of intensity 2pN − 1), we find 2N −1 2N X 1 X ∇η(k)∇ϕ(k) ϕ(k)LN η(k) = −(2N )α (1 − pN ) 2N k=1

− (2N )α (2pN − 1)

k=1 2N −1 X k=1

η(k + 1)(1 − η(k))∇ϕ(k) .

A simple integration by parts shows that the first term on the right is bounded by a term of order N α−1 while the second term is of order 1. Consequently EN ιN

h

2N 1 Z tX i N ϕ(k)L η (k)dr .h, r 2N s s,t≤T,|t−s|≤h

sup

k=1

uniformly over all N ≥ 1 and all h > 0. Taking the limit as N → ∞ and h ↓ 0, this vanishes. We turn to the tightness of the interface mN . First, the profile mN (t, ·) is 1Lipschitz for all t ≥ 0 and all N ≥ 1. Second, we claim that for some β ∈ (α, 1) 1

N N p p EN ιN [|m (t, k) − m (s, k)| ] . |t − s| +

1 N 1−β

,

(4.15)

uniformly over all 0 ≤ s ≤ t ≤ T , all k ∈ {1, . . . , 2N } and all N ≥ 1. This being given, the proof proceeds as in Subsection 4.2: one introduces a piecewise linear time-interpolation m ¯ N of mN and shows tightness for this process, and then one shows that the difference between m ¯ N and mN is uniformly small. We are left with the proof of (4.15). Let ψ : R → R+ be a non-increasing, smooth function such that ψ(x) = 1 for all x ≤ 0 and ψ(x) = 0 for all x ≥ 1. Fix β ∈ (α, 1).

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For any given k ∈ {1, . . . , 2N }, we define ϕN k : {0, . . . , 2N } → R by setting β ). Then, we observe that (ℓ) = ψ((ℓ − k)/(2N ) ϕN k 2N 1 X N β−1 ), (2ηt (ℓ) − 1)ϕN k (ℓ) = m (t, k) + O(N 2N ℓ=1

uniformly over all k ∈ {1, . . . , 2N } and all t ≥ 0. Then, similar computations to those made in the first part of the proof show that EN ιN

r 2N p i 1 h 1 X 1 t−s p N + 1+β , . (t − s) + (ηt (ℓ) − ηs (ℓ))ϕk (ℓ) 1+β−α 2N N N ℓ=1

uniformly over all k, all 0 ≤ s ≤ t ≤ T and all N ≥ 1. This yields (4.15).

The main step in the proof of Theorem 1.7 is to prove the convergence of the density of particles, starting from a product measure satisfying Assumption 4.1. This is the content of the following result. Theorem 4.11 Let ιN be a measure satisfying Assumption 4.1, so that ̺N 0 under ιN converges to some deterministic limit ̺0 (dx) = η0 (x)dx. Then, under PN ιN , the N process ̺ converges in distribution in the Skorohod space D([0, ∞), M([0, 1])) to the deterministic process (η(t, x)dx, t ≥ 0), where η is the entropy solution of Definition 4.9 starting from η0 . Given this result, the proof of the hydrodynamic limit is simple. Proof of Theorem 1.7. Let ιN be as in Theorem 4.11. We know that ̺N converges to some limit ̺ and that mN is tight. Let m be some limit point and let mNi be an associated converging subsequence. By Skorohod’s representation theorem, we can assume that (̺Ni , mNi ) converges almost surely to (̺, m). Recall R xthat ̺ is of the form ̺(t, x) = η(t, x)dx. Our first goal is to show that m(t, x) = 0 (2η(t, y) − 1)dy for all t, x. Fix x0 ∈ (0, 1). WeRintroduce an approximation of the indicator of (−∞, x0 ] by · setting ϕp (·) = 1 − −∞ P1/p (y − x0 )dy, p ≥ 1 where Pt is the heat kernel on R at time t. This is a collection of smooth functions on R such that for any δ > 0 kϕp − 1[0,x0 ] kL1 (0,1) → 0 ,

|hf, δx0 + ∂x ϕp i| →0, kf kC δ f ∈C δ ([0,1]) sup

as p → ∞. If we set I(t, x0 ) = m(t, x0 )− and some t > 0, then we have

R x0 0

(4.16)

(2η(t, y)− 1)dy for some x0 ∈ (0, 1)

|I(t, x0 )| ≤ km(t) − mN (t)k∞ + |hmN (t), δx0 + ∂x ϕp i|

N + |hmN (t), ∂x ϕp i + h2̺N t − 1, ϕp i| + 2|h̺t − ̺t , ϕp i|

+ |h2̺(t) − 1, ϕp − 1[0,x0 ] i| .

Recall that mN is 1-Lipschitz in space, so that the second term on the right vanishes as p → ∞ by (4.16). A discrete integration by parts shows that the third term vanishes as N go to ∞. The first and fourth terms vanish as N → ∞ by the

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convergence of mN and ̺N , and the last term is dealt with using (4.16). Choosing p and then N large enough, we deduce that |I(t, x0 )| is almost surely as small as desired. This identifies completely the limit m of any converging subsequence, under PN ιN . Since the wedge initial condition is defined as the product of Bernoulli measures with mean 0 on the N first sites, and 1 on the N last sites, we are in the scope of Theorem 4.11, and consequently Theorem 1.7 is proved in this case. On the other hand, when we start from the flat initial condition, we need to argue differently. Let m◦ be the integrated solution of the Burgers equation (1.11) when the initial condition is taken to be η(t = 0, x) = 1/2: its expression is given in (1.13). We know that mN is tight by Proposition 4.10 when starting from the flat initial condition, so we only need to show that any limit m coincides with m◦ . Let η + (resp. η − ) be the solution of (1.11) when the initial condition is taken to be η + (t = 0, x) = 1 (resp. η − (t = 0, x) = 0) for all x ∈ [0, ǫ] and η ± (t = 0, x) = 1/2 for all x ∈ [ǫ, 1]. Let m± be the corresponding integrated solutions. We check in Appendix 6.3 that m− and m+ converge pointwise to m◦ as ǫ ↓ 0. Now, let ι+ N (resp. ι− N ) be the product measure with mean +1 (resp. 0) on the ǫ2N first sites, and then with mean 1/2 on the remaining sites. The initial conditions ι± N fall into the scope of Theorem 4.11 and the corresponding (integrated) scaling limits are given by the deterministic processes m± . The key point is that the height function − associated with ι+ N (resp. ιN ) is, with a probability that goes to 1 when N → ∞, above (resp. below) the height function associated with the flat initial condition. It is easy to check that the dynamics preserves (stochastically) the ordering at any time. Therefore, any limit m belongs to [m− , m+ ]. This ensures that m = m◦ , thus concluding the proof. To prove Theorem 4.11, we need to show that the limit of any converging subsequence of ̺N is of the form ̺(t, dx) = η(t, x)dx and that η satisfies the entropy inequalities of Definition 4.9. To make appear the constant c in these inequalities, the usual trick is to define a coupling of the particle system η N with another particle system ζ N which is stationary with density c so that, at larger scales, one can replace the averages of ζ N by c. Such a coupling has been defined by Rezakhanlou [Rez91] in the case of the infinite lattice Z. The specificity of the present setting comes from the boundary conditions of our system: one needs to choose carefully the flux of particles at 1 and 2N for ζ N . The precise definition of our coupling goes as follows. We set p(1) = 1 − pN ,

p(−1) = pN , and

p(k) = 0 ∀k 6= {−1, 1} ,

as well as b(a, a′ ) = a(1 − a′ ). Then, we define L˜bulk f (η, ζ) = (2N )1+α h

2N X

k,ℓ=1

p(ℓ − k) ×

(b(η(k), η(ℓ)) ∧ b(ζ(k), ζ(ℓ)))(f (η k,ℓ , ζ k,ℓ ) − f (η, ζ))

+ (b(η(k), η(ℓ)) − b(η(k), η(ℓ)) ∧ b(ζ(k), ζ(ℓ)))(f (η k,ℓ , ζ) − f (η, ζ)) i + (b(ζ(k), ζ(ℓ)) − b(η(k), η(ℓ)) ∧ b(ζ(k), ζ(ℓ)))(f (η, ζ k,ℓ ) − f (η, ζ)) ,

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and, using the notation ζ ± δk to denote the particle configuration which coincides with ζ everywhere except at site k where the occupation is ζ(k) ± 1, L˜bdry f (η, ζ) = (2N )1+α (2pN − 1)(1 − c)ζ(1)(f (η, ζ − δ1 ) − f (η, ζ))

+ (2N )1+α (2pN − 1)c(1 − ζ(2N ))(f (η, ζ + δ2N ) − f (η, ζ)) .

We consider the stochastic process (ηtN , ζtN ), t ≥ 0 associated to the generator L˜ = L˜bulk + L˜bdry . From now on, we will always assume that η0N has law ιN , where ιN satisfies Assumption 4.1, and that ζ0N is distributed as a product of Bernoulli measures with parameter c. Furthermore, we will always assume that the coupling at time 0 is such that sgn(η0N (k) − ζ0N (k)) = sgn(f (k/2N ) − c) ,

∀k ∈ {1, . . . , 2N } ,

where f is the macroscopic density profile arising in Assumption 4.1. Notice that ˜ N be the law of the this is always possible to construct such a coupling. We let P ιN ,c process (η N , ζ N ). Remark 4.12 The bulk part of the generator prescribes the dynamics of the simple exclusion process for both particle systems. The coupling is such that the order of ζ N and η N is preserved: this is the reason why the expression of the generator is non-trivial. The boundary part of the generator prescribes the minimal jump rates for ζ N to be stationary with density c: there is no flux from 0 to 1 nor from 2N to 2N + 1. It appears that this choice is convenient for establishing the entropy inequalities. It will actually be important to track the sign changes in the pair (η N , ζ N ). To that end, we let Fk,ℓ (η, ζ) = 1 if η(k) ≥ ζ(k) and η(ℓ) ≥ ζ(ℓ); and Fk,ℓ (η, ζ) = 0 otherwise. We say that a subset C of consecutive integers in {1, . . . , 2N } is a cluster with constant sign if for all k, ℓ ∈ C we have Fk,ℓ (η, ζ) = 1, or for all k, ℓ ∈ C we have Fk,ℓ (ζ, η) = 1. For a given configuration (η, ζ), we let n be the minimal number of clusters needed to cover {1, . . . , 2N }: we will call n the number of sign changes. There is not necessarily a unique choice of covering into n clusters. Let C(i), i ≤ n be any such covering and let 1 = k1 < k2 < . . . kn < kn+1 = 2N + 1 be the integers such that C(i) = {ki , ki+1 − 1}. ˜ N , the process η N has law PN while the process ζ N is Lemma 4.13 Under P ιN ιN ,c Be(c). Furthermore, the number of sign changes n(t) is stationary with law ⊗2N k=1 smaller than n(0) + 3 at all time t ≥ 0. Proof. It is simple to check the assertion on the laws of the marginals η N and ζ N . Regarding the number of sign changes, the key observation is the following. In the bulk {2, . . . , 2N −1}, to create a new sign change we need to have two consecutive sites k, ℓ such that η N (k) = ζ N (k) = 1, η N (ℓ) = ζ N (ℓ) = 0 and we need to let one particle jump from k to ℓ, but not both. However, our coupling does never allow such a jump. Therefore, the number of sign changes can only increase at the boundaries due to the interaction of ζ N with the reservoirs: this can create at most 2 new sign changes, thus concluding the proof.

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Assumption 4.1 ensures the existence of a constant C > 0 such that n(0) < C almost surely for all N ≥ 1. We now derive the entropy inequalities at the microscopic level. Recall that τk stands for P the shift operator with periodic boundary conditions, and let hu, viN = (2N )−1 2N k=1 u(k/2N )v(k/2N ) denote the discrete 2 L product. Lemma 4.14 (Microscopic inequalities) Let ιN be a measure on {0, 1}2N satisfying Assumption 4.1. For all ϕ ∈ Cc∞ ([0, ∞) × [0, 1], R+ ), all δ > 0 and all c ∈ [0, 1], we have  Z ∞D E D E N ˜ ∂s ϕ(s, ·), (ηsN (·) − ζsN (·))± + ∂x ϕ(s, ·), H ± (τ· ηsN , τ· ζsN ) lim PιN ,c N N N →∞ 0   ± ± + 2σ (1 − c) ϕ(s, 0) + (0 − c) ϕ(s, 1) ds  E D ± N N + ϕ(0, ·), (η0 (·) − ζ0 (·)) ≥ −δ = 1 , N

where

H + (η, ζ) = −2σ(b(η(1), η(0))−b(ζ(1), ζ(0)))F1,0 (η, ζ) ,

H − (η, ζ) = H + (ζ, η) .

This is an adaptation of Theorem 3.1 in [Rez91]. Proof. We define Z t  ˜ h∂s ϕ(s, ·), (ηs (·) − ζs (·))± iN + Lhϕ(s, ·), (ηs (·) − ζs (·))± iN ds Bt = 0 E D . + ϕ(0, ·), (η0N (·) − ζ0N (·))± N

We have the identity D

ϕ(t, ·), (ηtN (·) − ζtN (·))±

E

N

= Bt + Mt ,

(4.17)

where M is a mean zero martingale. Since ϕ has compact support, the l.h.s. vanishes for t large enough. Below, we work at an arbitrary time s so we drop the subscript s in the calculations. Moreover, we write ϕ(k) instead of ϕ(k/2N ) to simplify notations. We treat separately the boundary part and the bulk part of the generator. Regarding the former, we have L˜bdry hϕ(·), (η(·) − ζ(·))+ iN   = (2N )α (2pN − 1) ϕ(1)η(1)ζ(1)(1 − c) − ϕ(2N )η(2N )(1 − ζ(2N ))c ≤ 2σϕ(0)(1 − c) + O(N −α ) ,

since ϕ is non-negative and 2pN − 1 ∼ 2σ(2N )−α . Similarly, we find

L˜bdry hϕ(·), (η(·) − ζ(·))− iN ≤ 2σϕ(2N )(0 − c)− + O(N −α ) .

We turn to the bulk part of the generator. Recall the map Fk,ℓ (η, ζ), and set Gk,ℓ (η, ζ) = 1 − Fk,ℓ (η, ζ)Fk,ℓ (ζ, η). By checking all the possible cases, one easily gets the following identity X  + bulk 1+α ˜ p(ℓ − k)(b(ζ(k), ζ(ℓ)) − b(η(k), η(ℓ))) L (η(k) − ζ(k)) = (2N ) ℓ

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 − p(k − ℓ)(b(ζ(ℓ), ζ(k)) − b(η(ℓ), η(k))) Fk,ℓ (η, ζ)    − p(ℓ − k)b(η(k), η(ℓ)) + p(k − ℓ)b(ζ(ℓ), ζ(k)) Gk,ℓ (η, ζ) . Since η and ζ play symmetric rˆoles in L˜bulk , we can find a similar identity for L˜bulk (η(k) − ζ(k))− . Notice that the term on the third line is non-positive, so we will drop it in the inequalities below. We thus get L˜bulk hϕ(·), (η(·) − ζ(·))± iN ≤ (2N )α

2N X

k,ℓ=1 ℓ=k±1

± p(ℓ − k)(ϕ(k) − ϕ(ℓ))Ik,ℓ (η, ζ) ,

where + Ik,ℓ (η, ζ) = (b(ζ(k), ζ(ℓ)) − b(η(k), η(ℓ)))Fk,ℓ (η, ζ) ,

− + Ik,ℓ (η, ζ) = Ik,ℓ (ζ, η) .

Up to now, we essentially followed the calculations made in the first step of the proof of [Rez91, Thm 3.1]. At this point, we argue differently: we decompose p(±1) into the symmetric part 1 − pN , which is of order 1/2, and the asymmetric part which is either 0 or 2pN − 1 ∼ 2σ(2N )−α . We start with the contribution of the symmetric part. Recall the definition of the number of sign changes n and of the integers k1 < . . . < kn+1 . Using a discrete integration by parts, one easily deduces that for all i ≤ n ki+1 −1

X

k,ℓ=ki ℓ=k±1

(ϕ(k) −

± ϕ(ℓ))Ik,ℓ (η, ζ)

=

ki+1 −2

X

k=ki

(η(k) − ζ(k))± ∆ϕ(k)

− (η(ki+1 − 1) − ζ(ki+1 − 1))± ∇ϕ(ki+1 − 2)

+ (η(ki ) − ζ(ki ))± ∇ϕ(ki − 1) .

Since n(s) is bounded uniformly over all N ≥ 1 and all s ≥ 0, we deduce that the boundary terms arising at the second and third lines yield a negligible contribution. Thus we find (2N )α

2N X

± (η, ζ) = O (1 − pN )(ϕ(k) − ϕ(ℓ))Ik,ℓ

k,ℓ=1 ℓ=k±1



1 N 1−α



.

Regarding the asymmetric part p(±1) − (1 − pN ), a simple calculation yields the identity (2N )α

2N X

k,ℓ=1 ℓ=k±1

± (p(ℓ − k) − 1 + pN )(ϕ(k) − ϕ(ℓ))Ik,ℓ (η, ζ)

2N −1 1 X = ∂x ϕ(k)τk H ± (η, ζ) + O(N −α ) , 2N k=1

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uniformly over all N ≥ 1. Therefore 2N −1   1 1 X ± bulk ˜ . ∂x ϕ(k)τk H ± (η, ζ) + O L hϕ(·), (η(·) − ζ(·)) iN ≤ 2N N α∧(1−α) k=1

Putting together the two contributions of the generator, we get Z t h∂s ϕ(s, ·), (ηsN (·) − ζsN (·))± iN + h∂x ϕ(s, ·), τ· H ± (ηsN , ζsN )iN Bt ≤ 0  + 2σ((1 − c)± ϕ(s, 0) + (0 − c)± ϕ(s, 1)) ds E D   1 + ϕ(0, ·), (η0N (·) − ζ0N (·))± +O . N α∧(1−α) N

1 ˜ N hM it . 1−α Recall the equation (4.17). A simple calculation shows that E ιN ,c N uniformly over all N ≥ 1 and all t ≥ 0. Moreover, the jumps of M are almost surely bounded by a term of order N −1 . Applying the BDG inequality (6.3), we deduce that i1 h 1 2 2 ˜N . 1−α , E ιN ,c sup Ms s≤t N 2

uniformly over all N ≥ 1 and all t ≥ 0. Since ϕ has compact support, Bt = −Mt for t large enough. The assertion of the lemma then easily follows. Recall that MTℓ (u) η is the average of η on the box Tℓ (u) for any u ∈ {1, . . . , 2N }.

Lemma 4.15 (Macroscopic inequalities) Let ιN be a measure on {0, 1}2N satisfying Assumption 4.1. For all ϕ ∈ Cc∞ ([0, ∞) × [0, 1], R+ ), all δ > 0 and all c ∈ [0, 1], we have Z ∞D  ± E N lim lim PιN ∂s ϕ(s, ·), MTǫN (·) (ηsN ) − c ǫ↓0 N →∞ N 0 D  E + ∂x ϕ(s, ·), h± MTǫN (·) (ηsN ), c N (4.18)   ± ± + 2σ (1 − c) ϕ(s, 0) + (0 − c) ϕ(s, 1) ds  D  ± E N + ϕ(0, ·), MTǫN (·) (η0 ) − c ≥ −δ = 1 . N

Proof. Since at any time s ≥ 0, ζ N (s, ·) is distributed according to a product of Bernoulli measures with parameter c, we deduce that ˜N lim lim E ιN ,c ǫ↓0 N →∞

2N i h 1 X MTǫN (u) (ζsN ) − c = 0 . 2N u=1

and consequently, by the Dominated Convergence Theorem, we have ˜N lim lim E ιN ,c ǫ↓0 N →∞

hZ

t 0

2N i 1 X MTǫN (u) (ζsN ) − c ds = 0 . 2N u=1

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˜ N almost surely Now we observe that for all ǫ > 0, we have P ιN ,c E E D D + O(ǫ) . = ϕ(0, ·), MTǫN (·) (η0N − ζ0N )± ϕ(0, ·), (η0N (·) − ζ0N (·))± N

N

˜ N almost surely the Recall the coupling we chose for (η0N (·), ζ0N (·)). Since P ιN ,c number of sign changes n(0) is bounded by some constant C > 0 uniformly over all N ≥ 1, we deduce using the previous identity that E E D D ϕ(0, ·), (η0N (·) − ζ0N (·))± = ϕ(0, ·), (MTǫN (·) η0N − MTǫN (·) ζ0N )± + O(ǫ) . N

N

Therefore, by Lemma 4.14, we deduce that the statement of the lemma follows if we can show that for all δ > 0 Z t 2N 1 X ± N ˜ lim lim PιN ,c MTǫN (u) (ηsN − ζsN ) ǫ↓0 N →∞ 2N 0 u=1  N N ± − (MTǫN (u) (ηs − ζs )) ds > δ = 0 , (4.19) Z t 2N X 1 ˜N lim lim P MTǫN (u) H ± (ηsN , ζsN ) ι ,c ǫ↓0 N →∞ N 0 2N u=1    N N ± − h MTǫN (u) (ηs ), MTǫN (u) (ζs ) ds > δ = 0 .

We restrict ourselves to proving the second identity, since the first is simpler. Let Ns+ , resp. Ns− , be the set of u ∈ {1, . . . , 2N } such that ηs ≥ ζs , resp. ζs ≥ ηs , on the whole box TǫN (u). By Lemma 4.13, 2N − #Ns+ − #Ns− is of order ǫN uniformly over all s, all N ≥ 1 and all ǫ. Therefore, we can neglect the contribution ˜ of all u ∈ / Ns+ ∪ Ns− . If we define Φ(η) = −2ση(1)(1 − η(0)) and if we let Φ(a) be the expectation of Φ under a product of Bernoulli measures with parameter a, then for all u ∈ Ns+ we have   MTǫN (u) H − (ηsN , ζsN ) − h− MTǫN (u) (ηsN ), MTǫN (u) (ζsN ) = 0 , as well as

  MTǫN (u) H + (ηsN , ζsN ) − h+ MTǫN (u) (ηsN ), MTǫN (u) (ζsN )     ˜ MT (u) ζ N . ˜ MT (u) η N − MT (u) Φ(ζ N ) + Φ = MTǫN (u) Φ(ηsN ) − Φ s s s ǫN ǫN ǫN

Similar identities hold for every u ∈ Ns− . We deduce that (4.19) follows if we can show that for all δ > 0 Z t  2N   1 X N N ˜ MT (u) ηs ds > δ = 0 , lim lim PιN MTǫN (u) Φ(ηs ) − Φ ǫN ǫ↓0 N →∞ 0 2N u=1 Z t 2N    1 X N N N ˜ ˜ lim lim EιN ,c MTǫN (u) Φ(ζs ) − Φ MTǫN (u) ζs ds = 0 . ǫ↓0 N →∞ 0 2N u=1

The first convergence is ensured by Theorem 4.2, while the second follows from the stationarity of ζ N and the Central Limit Theorem for strongly mixing sequences of r.v. This completes the proof of the lemma.

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Proof of Theorem 4.11. For any given ǫ > 0, we have  h k i k 1 − ǫ, +ǫ MT2ǫN (k) (ηs ) = ̺N s, 2ǫ 2N 2N (4.20)  1 N −1 = ̺ s, [x − ǫ, x + ǫ] + O(N ) , 2ǫ h i k k+1 uniformly over all k ∈ {1, . . . , 2N − 1}, all x ∈ 2N , 2N and all N ≥ 1. Notice that the O(N −1 ) depends on ǫ. For all ̺ ∈ D([0, ∞), M([0, 1])), we set Z ∞D i ± E 1  h ̺ s, · −ǫ, · + ǫ − c ∂s ϕ(s, ·), Vc (ǫ, ̺) := 2ǫ 0 1  h D i E + ∂x ϕ(s, ·), h± ̺ s, · −ǫ, · + ǫ , c 2ǫ   ± + 2σ (1 − c) ϕ(s, 0) + (0 − c)± ϕ(s, 1) ds i ± E D 1  h ̺ 0, · −ǫ, · + ǫ − c . + ϕ(0, ·), 2ǫ

Combining (4.20), (4.18) and the continuity of the maps h± (·, c) and (·)± , we deduce that for any δ > 0, we have N lim lim PN ιN (Vc (ǫ, ̺ ) ≥ −δ) = 1 . ǫ↓0 N →∞

At this point, we observe that for all ϕ ∈ C([0, 1], R+ ) we have h̺N (t), ϕi ≤

2N 1 X ϕ(k/2N ) , 2N k=1

so that a simple argument ensures that for every limit point ̺ of ̺N and for all t ≥ 0, the measure ̺(t, dx) is absolutely continuous with respect to the Lebesgue measure, and its density is bounded by 1. Therefore, any limit point is of the form ̺(t, dx) = η(t, x)dx with η ∈ L∞ ([0, ∞) × (0, 1)). Let P be the law of the limit of a converging subsequence ̺Ni . Since ̺ 7→ Vc (ǫ, ̺) is a P-a.s. continuous map on D([0, ∞), M([0, 1])), we have for all ǫ > 0 Ni i lim PN ιN (Vc (ǫ, ̺ ) ≥ −δ) ≤ P(Vc (ǫ, ̺) ≥ −δ) .

i→∞

i

For any ̺ of the form ̺(t, dx) = η(t, x)dx, we set Z ∞D  E  ± E D ∂s ϕ(s, ·), η(s, ·) − c + ∂x ϕ(s, ·), h± η(s, ·), c Vc (̺) := 0 E   D + 2σ (1 − c)± ϕ(s, 0) + (0 − c)± ϕ(s, 1) ds + ϕ(0, ·), (η0 − c)± ,

and we observe that by Lebesgue Differentiation Theorem, we have P-a.s. Vc (̺) = limǫ↓0 Vc (ǫ, ̺). Therefore, P(Vc (̺) ≥ −δ) = P( lim Vc (ǫ, ̺) ≥ −δ) ≥ E[ lim 1{Vc (ǫ,̺)≥−δ/2} ] ǫ↓0

ǫ↓0

Ni i ≥ lim E[1{Vc (ǫ,̺)≥−δ/2} ] ≥ lim lim PN ιN (Vc (ǫ, ̺ ) ≥ −δ/2) = 1 , ǫ↓0

ǫ↓0 i→∞

i

so the process (η(t, x), t ≥ 0, x ∈ (0, 1)) under P coincides with the unique entropy solution of (1.11), thus concluding the proof.

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5 KPZ fluctuations To prove Theorem 1.8, we follow the method of Bertini and Giacomin [BG97]. Due to our boundary conditions, there are two important steps that need some specific arguments: first the bound on the moments of the discrete process, see Proposition 5.3, second the bound on the error terms arising in the identification of the limit, see Proposition 5.7. In order to simplify the notations, we will regularly use the microscopic variables k, ℓ ∈ {1, . . . , 2N − 1} in rescaled quantities: for instance, hN (t, ℓ) stands for hN (t, x) with x = (ℓ − N )/(2N )2α . The proof relies on the discrete Hopf-Cole transform, which was introduced by G¨artner [G¨ar88]. The important feature of this transform is that it linearises the drift terms of the stochastic differential equations solved by the discrete process. Indeed, if one sets ξ N (t, x) := exp(−hN (t, x)), where hN was introduced in (1.14), then the stochastic differential equations solved by ξ N are given by  N N N  dξ (t, ℓ) = cN ∆ξ (t, ℓ)dt + dM (t, ℓ) , (5.1) ξ N (t, 0) = ξ N (t, 2N ) = eλN t ,  N (0,ℓ)  N −h ξ (0, ℓ) = e , for all ℓ ∈ {1, . . . , 2N − 1}, where M N is a martingale with bracket given by hM N (·, k), M N (·, ℓ)it = 0 whenever k 6= ℓ, and   dhM N (·, k)it = λN ξ N (t, k)∆ξ N (t, k) + 2ξ N (t, k)2 dt (5.2) − (2N )4α ∇+ ξ N (t, k)∇− ξ N (t, k)dt ,

where we rely on the notation

∇+ f (ℓ) := f (ℓ + 1) − f (ℓ) , ∇− f (ℓ) := f (ℓ) − f (ℓ − 1) . Observe that |dhM N (·, k)it | . ξ N (t, k)2 (2N )2α ,

uniformly over all t ≥ 0, all k and all N ≥ 1. As usual, we let Ft , t ≥ 0 be the natural filtration associated with the process (ξ N (t), t ≥ 0). In order to analyse this random process, we need to define a few objects first. N t, 2N − λγNN t] ⊂ [0, 2N ]. The hydrodynamic limit We define B0N (t) := [ λγN obtained in Theorem 1.7 shows that this is the window where the density of particles is approximately 1/2 at time t (in the time scale (2N )4α ). On the left of this window, the density is approximately 1, and on the right it is approximately 0. For technical reasons, it is convenient to introduce an ǫ-approximation of this window by setting: i hλ λN N t + ǫN, 2N − t − ǫN , t ∈ [0, T ) . BǫN (t) := γN γN We let pN t (k, ℓ) be the discrete heat kernel on {0, . . . , 2N } sped up by 2cN , we refer to Appendix 6.2 for a definition and some properties. We set  N,◦ N,◦  ∂t ξ (t, ℓ) = cN ∆ξ (t, ℓ) , ξ N,◦ (t, 0) = ξ N,◦ (t, 2N ) = eλN t ,   N,◦ ξ (0, ℓ) = 1 .

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For all t ≥ 0, we define the martingale Nrt (ℓ)

=

Z

0

−1 r 2N X

N pN t−s (k, ℓ)dM (s, k) ,

k=1

r ∈ [0, t] ,

(5.3)

Classical arguments ensure that the unique solution of (5.1) is given by N

ξ (t, ℓ) = ξ

N,◦

(t, ℓ) +

2N −1 X k=1

N t pN t (k, ℓ)(ξ (0, k) − 1) + Nt (ℓ) .

(5.4)

In this expression, the first term gives the hydrodynamic behaviour of ξ N , while the second term is a negligible contribution coming from the lattice approximation of the flat initial condition. Indeed we have the bound −1 2N X N N p (k, ℓ)(ξ (0, k) − 1) . γN , t

(5.5)

k=1

uniformly over all t ≥ 0, all ℓ ∈ {1, . . . , 2N − 1} and all N ≥ 1. The third term in (5.4) is the stochastic term that provides the fluctuations of our particle system. Lemma 5.1 Fix a compact set K ⊂ [0, T ) and ǫ > 0. There exists δ > 0 such that |ξ N,◦ (t, ℓ) − 1| . exp(−δN 2α ) uniformly over all t ∈ K, all ℓ ∈ BǫN (t) and all N ≥ 1. Furthermore, if we set   bN (t, ℓ) := 2 + exp λN t − γN (ℓ ∧ (2N − ℓ)) ,

then for N large enough, we have ξ N,◦ (t, ℓ) ≤ bN (t, ℓ) for all t ∈ K and all ℓ ∈ {1, . . . , 2N − 1}.

The value 2 in the expression of bN could be replaced by any arbitrary value strictly larger than 1. Remark 5.2 The hydrodynamic limit of Theorem   1.5, upon Hopf-Cole transform, is given by 1 ∨ exp λN t − γN (ℓ ∧ (2N − ℓ)) .

Proof. Given the equation solved by ξ N,◦ (t, ℓ) − eλN t , it is simple to check that we have the following identity ξ N,◦ (t, ℓ) = 1 + λN

Z t 2N −1  X pN (k, ℓ) eλN s ds . 1− t−s 0

k=1

The first part of the lemma will be ensured if we are able to show that there exists δ > 0 such that 2N −1   X 2α λN s (5.6) pN . e−δN , 1− t−s (k, ℓ) e k=1

uniformly over all s ∈ [0, t], all t ∈ K and all ℓ ∈ BǫN (t). The proof of this estimate on the heat kernel is provided in Appendix 6.2.

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We turn to the second part of the statement. Using the estimate on ξ N,◦ (t, N ) obtained in the first part, we deduce that for N large enough, bN solves  N N  ∂t b (t, ℓ) = cN ∆b (t, ℓ) , ℓ ∈ {1, . . . , N − 1} , bN (t, 0) ≥ ξ N,◦ (t, 0) , bN (t, N ) ≥ ξ N,◦ (t, N ) ,   N b (0, k) ≥ ξ N,◦ (0, k) .

By the maximum principle, one deduces that bN (t, ℓ) ≥ ξ N,◦ (t, ℓ) for all t ∈ K and all ℓ ∈ {0, . . . , N }. By symmetry, this inequality also holds for ℓ ∈ {N, . . . , 2N }.

To alleviate the notation, we define N N qs,t (k, ℓ) = pN t−s (k, ℓ)b (s, k) .

(5.7)

We now have all the ingredients at hand to bound the moments of ξ N . Proposition 5.3 For all n ≥ 1 and all compact set K ⊂ [0, T ), we have sup

sup

sup E

N ≥1 ℓ∈{1,...,2N −1} t∈K

h ξ N (t, ℓ) n i bN (t, ℓ)

0 such that for all A > 0, we have h i 1 √ E |E[∇+ ξ N (t, ℓ)∇− ξ N (t, ℓ) | Fs ]| . , 2α+κ (2N ) t−s uniformly over all ℓ ∈ [N − A(2N )2α , N + A(2N )2α ], all t in a compact set of [N −α , T ), all s ∈ [0, t] and all N ≥ 1. Remark 5.8 Quastel suggested in [Qua12] a proof of such a bound via a replacement lemma: this is the approach that Dembo and Tsai [DT16] followed in the context of the weakly asymmetric non-simple exclusion process on Z. It is not clear whether one can follow this approach in our setting. To prove this proposition, we need to collect some preliminary results. First, we set 2N −1 X N pN ξ˜N (t, ℓ) := ξ N,◦ (t, ℓ) + t (k, ℓ)(ξ (0, k) − 1) , k=1

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and we observe that ∇± ξ N (t, ℓ) = ∇± ξ˜N (t, ℓ) + ∇± Ntt (ℓ) , where the process N·t was introduced in (5.3). If we set N − N Kt−r (k, ℓ) := ∇+ pN t−r (k, ℓ)∇ pt−r (k, ℓ) ,

(here the gradients act on the variable ℓ), then using the martingale property of N·t (ℓ) we obtain for all s ≤ t i h E ∇+ ξ N (t, ℓ)∇− ξ N (t, ℓ) | Fs = (∇+ ξ˜N (t, ℓ) + ∇+ Nst (ℓ))(∇− ξ˜N (t, ℓ) + ∇− Nst (ℓ))  Z t 2N −1  X N +E Kt−r (k, ℓ) dhM N (·, k)ir Fs . s

Let

fsN (t, ℓ)

k=1

 h i  + N − N := E E ∇ ξ (t, ℓ)∇ ξ (t, ℓ) | Fs .

Using the expression of the bracket (5.2) of M N , we get Z t X N N N fs (t, ℓ) ≤ Ds (t, ℓ) + (2N )4α |Kt−r (k, ℓ)|fsN (r, k)dr , s

(5.13)

N (r) k∈Bǫ/2

where DsN (t, ℓ) = DsN,1 (t, ℓ) + DsN,2 (t, ℓ) + DsN,3 (t, ℓ) with h i DsN,1 (t, ℓ) := E |(∇+ ξ˜N (t, ℓ) + ∇+ Nst (ℓ))(∇− ξ˜N (t, ℓ) + ∇− Nst (ℓ))| , Z t X N,2 N Ds (t, ℓ) := (2N )4α |Kt−r (k, ℓ)|fsN (r, k)dr , s

DsN,3 (t, ℓ)

N (r) k ∈B / ǫ/2

 h Z := λN E E

s

−1 t 2N X

N Kt−r (k, ℓ)(ξ N (r, k)∆ξ N (r, k)

k=1

i  + 2ξ (r, k) )dr | Fs . N

2

From now on, we fix a compact set K ⊂ [0, T ). We start with a technical lemma on the discrete heat kernel, whose proof is postponed to Appendix 6.2. Lemma 5.9 Fix ǫ > 0. There exist β ∈ (0, 1) such that Z t X N |Kt−r (k, ℓ)|(2N )4α dr < β , s

N (r) k∈Bǫ/2

uniformly over all s ≤ t ∈ K, all ℓ ∈ BǫN (t) and all N large enough. We now establish a bound on DsN .

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Lemma 5.10 Fix ǫ > 0. There exists κ > 0 such that DsN (t, ℓ) . 1 ∧

1 (2N )2α+κ

√

t−s

,

(5.14)

uniformly over all ℓ ∈ BǫN (t), all N −α ≤ s < t ∈ K and all N ≥ 1. Proof. Let us observe that we have the simple bound 2 fsN (r, k) . bN (r, k)2 γN ,

(5.15)

uniformly over all the parameters. Recall also that bN (t, ℓ) is of order 1 whenever ℓ ∈ BǫN (t). Let p¯N be the discrete heat kernel on the whole line Z sped up by 2cN , see Ap− ¯N (ℓ − k). ¯ tN (k, ℓ) = ∇+ p¯N pendix 6.2, and set K t (ℓ − k)∇ p t N,1 Bound of Ds . We write i h DsN,1 (t, ℓ) . E (∇+ ξ˜N (t, ℓ))2 + (∇+ Nst (ℓ))2 + (∇− ξ˜N (t, ℓ))2 + (∇− Nst (ℓ))2 , and we bound separately each of the four terms on the right. Notice that the proofs will be the same for ∇+ and ∇− so we only provide the details for the former. By Lemma 5.1, there exists δ > 0 such that (∇+ ξ N,◦ (t, ℓ))2 . e−δN

2α

,

uniformly over all ℓ ∈ BǫN (t), all t ∈ K and all N ≥ 1. Furthermore, using Lemmas 6.4 and 6.5, we get 2N −1 X k=1

N ∇+ pN t (k, ℓ)(ξ (0, k) − 1) =

X

N ∇+ p¯N t (ℓ − k)(ξ (0, k) − 1)

N (0) k∈Bǫ/2

2α

+ O(N 1−α e−δN ) , uniformly over all ℓ ∈ BǫN (t), all t ∈ K and all N ≥ 1. Then, we write X |∇+ p¯N pN pN ¯N t (ℓ − k)| = −¯ t (ℓ − i− − 1) + 2¯ t (0) − p t (ℓ − i+ ) , N (0) k∈Bǫ/2

N (0). Using Lemma 6.2 and the fact where i± are the first and last integers in Bǫ/2 that |ξ N (0, k) − 1| . γN , we deduce that X   1 N √ , 1 ∧ . γ ∇+ p¯N (ℓ − k)(ξ (0, k) − 1) N t t(2N )2α N k∈Bǫ/2 (0)

uniformly over the same set of parameters. Putting everything together, we deduce that (∇+ ξ˜N (t, ℓ))2 . N −5α uniformly over all N −α ≤ t ∈ K, all ℓ ∈ BǫN (t) and all N ≥ 1. We now treat ∇+ Nst (ℓ). Using again Lemmas 6.4 and 6.5, we have −1 Z i h 2N h X 2 + t E (∇ Ns (ℓ)) . E k=1

0

s

2 (∇+ pN t−r (k, ℓ)) dhM (·, k)ir

i

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2α

. (2N )

X Z

N (r) k∈Bǫ/2

s

0

2α

2 1+2α −δN ), (∇+ p¯N e t−r (k, ℓ)) dr + O(N

uniformly over all ℓ ∈ BǫN (t), all t ∈ K and all N ≥ 1. Using Lemma √6.2, we easily deduce that the last expression is bounded by a term of order 1∧1/( t − s(2N )4α ) as required. Bound of D N,2 . Using the exponential decay of Lemma 6.4 and (5.15), we deduce that there exists δ > 0 such that Z t X Z t X 2α 4α N N (2N ) |Kt−r (k, ℓ)|fs (r, k)dr . (2N )2α e−2δN dr , s

s

N (r) k ∈B / ǫ/2

N (r) k ∈B / ǫ/2

uniformly over all ℓ ∈ BǫN (t), all s ≤ t ∈ K and all N ≥ 1. This trivially yields a bound of order N −3α as required. Bound of D N,3 . By Lemmas 6.5 and 6.4, there exists δ > 0 such that DsN,3 (t, ℓ) can be rewritten as  h Z t X i  ¯ N (k, ℓ)(ξ N (r, k)∆ξ N (r, k) + 2ξ N (r, k)2 )dr | Fs , λN E E K t−r s

N (r) k∈Bǫ/2

(5.16) 2α up to an error of order N 2α+1 e−δN , uniformly over all ℓ ∈ BǫN (t), all t ∈ K and all N ≥ 1. The error term satisfies the bound of the statement. We bound separately the two contributions arising in (5.16). First, using the almost sure bound |∆ξ N (r, k)| . γN ξ N (r, k), we get  h Z t X i  N N N ¯ t−r (k, ℓ)ξ (r, k)∆ξ (r, k)dr | Fs K λN E E s

α

. (2N )

Z

N (r) k∈Bǫ/2

t

s

X

N (r) k∈Bǫ/2

¯ N (k, ℓ)|dr , |K t−r

uniformly over all ℓ ∈ BǫN (t), all t ∈ K and all N ≥ 1. Using Lemma 6.2, this easily yields a bound of order 1/(2N )2α+κ with κ > 0, as required. Second, we have Z t X i h N ¯ t−r K (k, ℓ)E ξ N (r, k)2 | Fs dr s

=

+

N (r) k∈Bǫ/2

Z Z

t s

i h ¯ N (k, ℓ)E ξ N (r, k)2 − ξ N (t, ℓ)2 | Fs dr K t−r

X

i h N ¯ t−r K (k, ℓ)dr E ξ N (t, ℓ)2 | Fs .

N (r) k∈Bǫ/2

t s

X

N (r) k∈Bǫ/2

Using a simple integration by parts, we get Z t X Z ∞X Z ¯ N (k, ℓ)dr = ¯ N (k, ℓ)dr − K K s

t−r

N (r) k∈Bǫ/2

t−s k∈Z

r

t s

X

N (r) k ∈B / ǫ/2

¯ N (k, ℓ)dr . K t−r

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The second term on the right can be bounded using Lemma 6.4: it has a negligible contribution. Using Lemma 6.2 on the first term, we easily deduce that  Z t X i  h 1 N N 2 ¯ Kt−r (k, ℓ)dr E ξ (t, ℓ) | Fs . 1 ∧ √ λN E , t − s(2N )4α s N k∈Bǫ/2 (r)

uniformly over all ℓ ∈ BǫN (t), all t ∈ K and all N ≥ 1. On the other hand, for any given β ∈ (0, 1/4), the Cauchy-Schwarz inequality together with Lemmas 5.4 and 5.5 yields

h i h i1 h i1 2 2 E |ξ N (r, k)2 − ξ N (t, ℓ)2 | . E (ξ N (r, k) + ξ N (r, ℓ))2 E (ξ N (r, k) − ξ N (r, ℓ))2 i1 h h i1 2 2 + E (ξ N (r, ℓ) + ξ N (t, ℓ))2 E (ξ N (r, ℓ) − ξ N (t, ℓ))2  ℓ − k 2β 1  β + |t − r| + , .1∧ (2N )2α (2N )α N (r), all r ≤ t ∈ K and all N ≥ 1. uniformly over all ℓ ∈ BǫN (t), all k ∈ Bǫ/2 Using Lemma 6.2, it is simple to deduce the existence of κ ∈ (0, 1) such that  Z t X i  h 1 N N 2 N 2 ¯ , Kt−r (k, ℓ)E ξ (r, k) − ξ (t, ℓ) | Fs dr . λN E 2α+κ (2N ) s N k∈Bǫ/2 (r)

uniformly over all s < t ∈ K, all ℓ ∈ BǫN (t) and all N ≥ 1.

We have all the elements at hand to prove the main result of this section. Proof of Proposition 5.7. Iterating (5.13) and using Lemma 5.9 and the bound (5.15), we deduce that X Is (t, ℓ, n) , fsN (t, ℓ) ≤ DsN (t, ℓ) + n≥1

where for all n ≥ 1, we set tn+1 = t, kn+1 = ℓ and Z n X Y Is (t, ℓ, n) := DsN (t1 , k1 ) (2N )4α |KtNi+1 −ti (ki , ki+1 )|dti . i=1

N s≤t1 ≤...≤tn ≤t ki ∈Bǫ/2 (ti )

By Lemma 5.10, we already know that DsN (t, ℓ) satisfies the bound of the statement of Proposition 5.7. To conclude the proof of the proposition, we only need to show that this is also the case for the sum over n ≥ 1 of Is (t, ℓ, n). Fix A > 0. Let n0 = c log N , for an arbitrary c > −3α/ log β, where β < 1 is taken from Lemma 5.9. Using Lemmas 5.10 and 5.9, we easily deduce that Is (t, ℓ, n) . β n uniformly over all n ≥ 1, all ℓ ∈ {N − A(2N )2α , N + A(2N )2α } P and all N −α ≤ s ≤ t ∈ K. Given the definition of n0 , we deduce that n≥n0 Is (t, ℓ, n) . (2N )−3α uniformly P over the same set of parameters, as required. Let us now treat n 0 such that As (t, ℓ, n) .

β n−1 √ , (2N )2α+κ′ t − s

(5.17)

uniformly over all ℓ ∈ [N − A(2N )2α , N + A(2N )2α ], all t ∈ K, all s ∈ [0, t], all n < n0 and all N ≥ 1. Finally, we set Bs (t, ℓ, n) := Is (t, ℓ, n) − As (t, ℓ, n). As for the previous term, we can replace each occurrence of pN by p¯N up to a negligible term, using Lemma 6.5. Among the parameters k1 , . . . , kn involved in the definition of Bs (t, ℓ, n), N (t )\B N (t ). Then, using the bound at least one them, say ki0 , belongs to Bǫ/2 i0 i0 ǫ N N ¯ |Kt (k, ℓ)| ≤ p¯t (k, ℓ) together with the semigroup property of the discrete heat kernel at the second line and the exponential decay of Lemma 6.4, we get X

n Y

ki0 +1 ,...,kn i=i0

¯ tN −t (ki , ki+1 )| |K i+1 i

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≤

X

n Y

ki0 +1 ,...,kn i=i0

−δN p¯N ¯N ti+1 −ti (ki , ki+1 ) = p t−ti (ki0 , ℓ) . e

2α

0

,

uniformly over all the parameters. Using Lemma 5.10, one easily gets 2α

Bs (t, ℓ, n) . (2N )n4α e−δ(2N )

,

uniformly over all n ≥ 1, all s < t ∈ K and all ℓ ∈ [N − A(2N )2α , N + A(2N )2α ]. Given the definition of n0 , we deduce that the sum over all n < n0 of the latter is negligible w.r.t. (2N )−3α , uniformly over the same set of parameters. This concludes the proof. 5.3 Identification of the limit We use the notation hf, gi to denote the inner product of f and g in L2 (R). Similarly, for all maps f, g : [0, 2N ] → R, we set hf, giN

2N −1  X 1 k−N  k−N  := f g . (2N )2α (2N )2α (2N )2α k=1

To conclude the proof of Theorem 1.8, it suffices to show that any limit point ξ of a converging subsequence of ξ N satisfies the following martingale problem (see Proposition 4.11 in [BG97]). Definition 5.11 (Martingale problem) Let (ξ(t, x), t ∈ [0, T ), x ∈ R) be a continuous process satisfying the following two conditions. Let t0 ∈ [0, T ). First, there exists a > 0 such that sup sup e−a|x| E[ξ(0, x)2 ] < ∞ .

t≤t0 x∈R

Second, for all ϕ ∈ Cc∞ (R), the processes Z 1 t hξ(s), ϕ′′ ids , M (t, ϕ) := hξ(t), ϕi − hξ(0), ϕi − 2 0 Z t 2 2 L(t, ϕ) := M (t, ϕ) − 4σ hξ(s)2 , ϕ2 ids , 0

are local martingales on [0, t0 ]. Then, ξ is a solution of (1.5) on [0, T ). The first condition is a simple consequence of Proposition 5.3. To prove that the second condition is satisfied, we introduce the discrete analogues of the above processes. For all ϕ ∈ Cc∞ (R), the processes Z 1 t hξ(s), (2N )4α ∆ϕiN ds , M N (t, ϕ) = hξ N (t), ϕiN − hξ(0), ϕiN − 2 0 Z t 2λN N N 2 L (t, ϕ) = M (t, ϕ) − hξ N (s)2 , ϕ2 iN ds + R1N (t, ϕ) + R2N (t, ϕ) , (2N )2α 0 are martingales, where R1N (t, ϕ)

λN := − (2N )2α

Z

t 0

hξ N (s)∆ξ N (s), ϕ2 iN ds ,

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R2N (t, ϕ) := (2N )2α

Z

0

t

h∇+ ξ N (s)∇− ξ N (s), ϕ2 iN ds .

If we show that R1N (t, ϕ) and R2N (t, ϕ) vanish in probability when N → ∞, then passing to the limit on a converging subsequence, we easily deduce that the martingale problem above is satisfied. Below, we will be working on [N − A(2N )2α , N + A(2N )2α ] where A is a large enough value such that [−A, A] contains the support of ϕ. The moments of ξ N on this interval are of order 1 thanks to Proposition 5.3. Since |∆ξ N | . γN ξ N , we have Z t X k−N  1 N ds . γN , E[|R1 (t, ϕ)|] . γN ϕ2 2α (2N )2α 0 (2N ) k

so that R1N (t, ϕ) converges to 0 in probability as N → ∞. To control R2N , we write Z tZ sX  k − N  2  k′ − N  N 2 ϕ2 E[R2 (t, ϕ) ] . ϕ (2N )2α (2N )2α 0 0 k,k ′ × E[∇+ ξ N (s, k)∇− ξ N (s, k)∇+ ξ N (s′ , k′ )∇− ξ N (s′ , k′ )] ds ds′ . By the Cauchy-Schwarz inequality, for all C > 0 we have

|E[1{ξ N (s′ ,k′ )>C} ∇+ ξ N (s, k)∇− ξ N (s, k)∇+ ξ N (s′ , k′ )∇− ξ N (s′ , k′ )]| 1 1 4 1 4 √ , . γN P(ξ N (s′ , k′ ) > C) 2 E[ξ N (s, k)4 ξ N (s′ , k′ )4 ] 2 . γN C uniformly over all k, k ′ ∈ [N − A(2N )2α , N + A(2N )2α ] and all s, s′ ∈ [0, t]. On the other hand, by Proposition 5.7 there exists κ > 0 such that for all C > 0 we have |E[1{ξ N (s′ ,k′ )≤C} ∇+ ξ N (s, k)∇− ξ N (s, k)∇+ ξ N (s′ , k′ )∇− ξ N (s′ , k′ )]| i h 2 . γN E 1{ξ N (s′ ,k′ )≤C} ξ N (s′ , k′ )2 E[∇+ ξ N (s, k)∇− ξ N (s, k) | Fs′ ] 2 . C 2 γN

1

√ . (2N )2α+κ s − s′

uniformly over all k, k′ , all s ∈ [0, t], all s′ ∈ [N −α , s] and all N ≥ 1. This being given, we easily deduce that E[R2N (t, ϕ)2 ] .

t2 t C 2 t3/2 √ + , + Nα (2N )κ C

uniformly over all C > 0 and all N ≥ 1. Taking the limit as N → ∞ and then as C → ∞, we deduce that R2N (t, ϕ) converges to 0 in probability, thus concluding the proof.

6 Appendix 6.1 Martingale inequalities Let X(t), t ≥ 0 be a c`adl`ag, mean zero, square-integrable martingale. Let hXit , t ≥ 0 denote the bracket of X, that is, the unique predictable process such that X 2 −

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62

hXi is a martingale. Let [X]t denote its quadratic variation: in the case where the martingale is of finite variation, we have X [X]t = (Xτ − Xτ − )2 . τ ∈(0,t]

The Burkholder-Davis-Gundy inequality ensures that for every p ≥ 1, there exists c(p) > 0 such that h p i1 1 p E[|Xt |p ] p ≤ c(p)E [X]t2 . (6.1)

It happens that the process Dt = [X]t − hXit is also a martingale. Thus, using twice the Burkholder-Davis-Gundy inequality, one gets that for every p ≥ 2 there exists c′ (p) > 0 such that h  h pi1  p i1 1 p p . (6.2) E[|Xt |p ] p ≤ c′ (p) E hXit2 + E [D]t4

We will also rely on the following inequality i1 h i1   h h p i1 p p ′′ p p 2 p E sup |Xs | , ≤ c (p) E hXit + E sup |Xs − Xs− | s≤t

(6.3)

s≤t

which can be found in [LLP80] for instance. 6.2 Discrete heat kernel estimates

We introduce the fundamental solution pN t (k, ℓ) of the discrete heat equation  N N  ∂t pt (k, ℓ) = cN ∆pt (k, ℓ) , (6.4) pN 0 (k, ℓ) = δk (ℓ) ,   N N pt (k, 0) = pt (k, 2N ) = 0 , for all k, ℓ ∈ {1, . . . , 2N − 1}, as well as its analogue p¯N t (ℓ) on Z: ( ∂t p¯N pN t (ℓ) = cN ∆¯ t (ℓ) , N p¯0 (ℓ) = δ0 (ℓ) ,

(6.5)

for all ℓ ∈ Z. The latter is more tractable than the former since it is translation invariant. Using a coupling between a simple random walk on Z and a simple ¯N random walk killed at 0 and 2N , we get the elementary bound pN t (ℓ−k) t (k, ℓ) ≤ p for all k, ℓ ∈ {1, . . . , 2N − 1} and all t ≥ 0. The following estimates are classical, see for instance Lemma A.1 in [DT16] or Lemma 26 in [EL15]. Lemma 6.1 For all β ∈ [0, 1], we have

1 , tcN 1 ℓ − ℓ′ β N ′ √ |pN (k, ℓ) − p (k, ℓ )| . 1 ∧ √ , t t cN tcN 1 t − t′ β N N |pt (k, ℓ) − pt′ (k, ℓ)| . 1 ∧ √ , t tcN pN t (k, ℓ) . 1 ∧ √

uniformly over all 0 ≤ t < t′ , all k, ℓ, ℓ′ ∈ {1, . . . , 2N − 1} and all N ≥ 1. The same bounds hold for p¯N .

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Let us also state the following simple bound. Lemma 6.2 We have X k∈Z

P

k

p¯N t (k)|k| .

|∇¯ pN pN t (k)| ≤ 2¯ t (0) ,

X k∈Z

√

cN t as well as

|∇¯ pN pN t (k)||k| . 1 , |∇¯ t (ℓ)| . 1 ∧

1 , tcN

uniformly over all ℓ ∈ Z, all t ≥ 0 and all N ≥ 1. P Proof. Notice that k p¯N t (k)|k| is smaller than the square root of the variance of a simple random walk on Z at time 2cN t. This easily yields the first bound. We turn to the bounds involving the gradient of p¯N . First, ∇¯ pN t (k) is positive if k < 0, and negative otherwise. Then, we have the simple identity X X ∇¯ pN ∇¯ pN ¯N t (k) = − t (k) = p t (0) , k 3, then a simple calculation ensures that ( e−λN (t−s)+γN (ℓ−k) if k ≤ N , bN (s, k) . bN (t, ℓ) e−λN (t−s)+γN (k−ℓ) if k ≥ N , uniformly over all s < t, all k ∈ {1, . . . , 2N − 1}, all ℓ ∈ {1, . . . , N } and all N ≥ 1. Therefore, it suffices to show that 2N −1 X k=1

pN t−s (k, ℓ) . 1 ,

1 , pN t−s (k, ℓ) . 1 ∧ √ t − s(2N )2α

(6.7)

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as well as 2N −1 X

−λN (t−s)+γN |ℓ−k| pN .1, t−s (k, ℓ)e

(6.8)

k=1

−λN (t−s)+γN |ℓ−k| pN t−s (k, ℓ)e

1 . .1∧ √ t − s(2N )2α

Regarding (6.7), the first bound is immediate since pN t (·, ℓ) is a sub-probability measure, while the second bound is proved in Lemma 6.1. We turn to (6.8). Since k 7→ eak is an eigenvector of the discrete Laplacian on Z with eigenvalue 2(cosh a − 1), we deduce that 2N −1 X k=1

γN |ℓ−k| pN ≤ t−s (k, ℓ)e

X

γN |ℓ−k| p¯N t−s (ℓ − k)e

k∈Z 2cN (t−s)(cosh γN −1)

≤ 2e

= 2eλN (t−s) ,

thus yielding the first bound. To get the second bound, it suffices to use the Fourier γN · . decomposition of p¯N t−s (·)e We also recall a simple bound on the heat kernel, in the flavour of large deviations techniques. For all a > 0, t > 0 and N ≥ 1, we have X

k≥a

2tcN g ( 2ca

p¯N t (k) ≤ e

Nt

)

,

(6.9)

where g(x) = cosh(argsh x)− x argsh x− 1 for all x ∈ R. By studying the function g(x)/x, one easily deduces that the term on the r.h.s. is increasing with t. N (k, ℓ) := p N We let q¯s,t ¯N t−s (k, ℓ)b (s, k). Lemma 6.4 Fix a compact set K ⊂ [0, T ) and ǫ > 0. There exists δ > 0 such that N N qs,t (k, ℓ) ≤ q¯s,t (k, ℓ) . e−δN

2α

,

N (s), all ℓ ∈ B N (t) and all 0 ≤ s ≤ t ∈ K. uniformly over all k ∈ / Bǫ/2 ǫ

Proof. Let us consider the case where bN (s, k) ≥ 3; by symmetry we can assume that k ∈ {1, . . . , N }. Then, we apply (6.9) to get 2(t−s)cN g ( 2c ℓ−k )+λN s−γN k (t−s)

λN s−γN k p¯N ≤e t−s (ℓ − k)e

N

.

(6.10)

We argue differently according to the value of α. If 4α ≤ 1, then (ℓ − k)/cN is bounded away from 0 uniformly over all N ≥ 1, all k ∈ / Bǫ/2 (s) and all ℓ ∈ Bǫ (t). Using the concavity of g, we deduce that there exists d > 0 such that the logarithm of the r.h.s. of (6.10) is bounded by −d(ℓ − k) + λN s − γN k . −dǫ

N , 2

thus concluding the proof in that case. We now treat the case 4α > 1. Let η > 0. First, we assume that s ∈ [0, t − η]. For

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65

any c > 1/4!, we have g(x) ≤ −x2 /2 + cx4 for all x in a neighbourhood of the origin. Then, for N large enough we bound the logarithm of the r.h.s. of (6.10) by f (s) = −

1 (ℓ − k)2 (ℓ − k)4 +c + λN s − γN k . 2 2cN (t − s) (2cN (t − s))3

A tedious but simple calculation shows the following. There exists δ′ > 0, only depending on ǫ, such that sups∈[0,t−η] f (s) ≤ −δ′ N 2α for all N large enough. This ensures the bound of the statement in the case where s ∈ [0, t − η]. Using the monotonicity in t of (6.9), we easily deduce that for all s ∈ [t − η, t], we have λN s−γN k p¯N ≤ ef (t−η)+λN (s−t+η) . t−s (ℓ − k)e

Recall that λN ∼ N 2α . Choosing η < δ′ small enough and applying the bound obtained above, we deduce that the statement of the lemma holds true. The case where bN (s, k) is smaller than 3 is simpler, one can adapt the above arguments to get the required bound. P2N −1 N Proof of (5.6). The quantity 1 − k=1 pt−s (k, ℓ) is equal to the probability that a simple random walk, sped up by 2cN and started from ℓ, has hit 0 or 2N by time t − s. By the reflexion principle, this is smaller than twice X X p¯N p¯N t−s (k) + t−s (k) . k≥ℓ

k≥2N −ℓ

We restrict ourselves to bounding the first term, since one can proceed similarly for the second term. Using (6.9), we deduce that it suffices to bound exp(2(t − s)cN g(ℓ/(2(t − s)cN )) + λN s). This is equal to the l.h.s. of (6.10) when k = 0, so that the required bound follows from the arguments presented in the last proof. Finally, we rely on the following representation of pN : X pN p¯N ¯N t (k, ℓ) = t (k + j4N − ℓ) − p t (−k + j4N − ℓ) . j∈Z

The next lemma shows that pN ¯N t (k, ℓ) can be replaced by p t (ℓ − k) up to some negligible term, whenever ℓ is in the ǫ-bulk at time t. Lemma 6.5 Fix ǫ > 0 and a compact set K ⊂ [0, T ). There exists δ > 0 such that uniformly over all s ≤ t ∈ K, all k ∈ {1, . . . , 2N − 1}, all ℓ ∈ BǫN (t) and all N ≥ 1, we have N −δN |pN ¯N t−s (k, ℓ) − p t−s (k, ℓ)|b (s, k) . e

2α

.

Proof. We only consider the case where bN (s, k) > 3 since the other case is simpler. Observe that there exists C > 0 such that log bN (s, k) ≤ CN 2α for all s ∈ K and all k ∈ {1, . . . , 2N − 1}. Arguing differently according to the relative values of 4α and 1, and using the bound (6.9), we deduce that there exists j0 ≥ 1 such that X N N −δN 2α , p¯N ¯N t−s (k + j4N − ℓ)b (s, k) + p t−s (−k + j4N − ℓ)b (s, k) . e j∈Z:|j|≥j0

(6.11)

A PPENDIX

66

uniformly over all s ≤ t ∈ K, all k ∈ {1, . . . , 2N − 1} and all ℓ ∈ Bǫ (t). On the other hand, the arguments in the proof of Lemma 6.4 yield that X N −δN 2α , p¯N t−s (−k + j4N − ℓ)b (s, k) . e j∈Z:|j|<j0

X

j∈Z:0