The hydrodynamic limit getdoc884c. 330KB Jun 04 2011 12:04:38 AM

or by a probability measure µ stochastically dominated by ¯ ν ϕ . In this last case, as proven in [A], by the monotonicity of g one obtains that the zero range process η t starting from µ is stochastically dominated by the zero range process ζ t starting from ¯ ν ϕ , i.e. one can construct on an enlarged probability space both processes η t and ζ t s.t. η t x 6 ζ t x almost surely. Finally, we recall that all measures ¯ ν ϕ are reversible for the zero range process and that ¯ ν ϕ e θ ηx ∞ for some θ 0, thus implying that ¯ ν ϕ ηx k ∞ for all k 0 cf. Section 2.3 of [KL]. As usually done, we assume that lim ϕ↑ϕ c Z ϕ = ∞. Then, cf. Section 2.3 in [KL], the function R ϕ := ¯ ν ϕ η0 is strictly increasing and gives a bijection from [0, ϕ c to [0, ∞. Given ρ ∈ [0, ∞ we will write ϕρ for the unique value such that Rϕ = ρ. Then we set ν ρ := ¯ ν ϕρ , φρ := ν ρ gη x x ∈ C ω . 2.4 As proven in Section 2.3 of [KL], φ is Lipschitz with constant g ∗ .

2.3 The hydrodynamic limit

Given an integer N 1 and a probability measure µ N on N C ω , we denote by P ω,µ N the law of the zero range process with generator N 2 L see 2.3 and with initial distribution µ N assuming this dynamics to be admissible. We denote by E ω,µ N the associated expectation. In order to state the hydrodynamic limit, we define BΩ as the family of bounded Borel functions on Ω and let D be the d × d symmetric matrix characterized by the variational formula a, Da = 1 m inf ψ∈BΩ    X e ∈B ∗ Z Ω ω0, ea e + ψτ e ω − ψω 2 I 0,e ∈C ω Q dω    , ∀a ∈ R d , 2.5 where B ∗ denotes the canonical basis of Z d , m := Q 0 ∈ C ω 2.6 and the translated environment τ e ω is defined as τ e ωx, y = ωx + e, y + e for all bonds {x, y} in E d . In general, I A denotes the characteristic function of A. The above matrix D is the diffusion matrix of the random walk among random conductances on the supercritical percolation cluster and it equals the identity matrix multiplied by a positive constant see the discussion in [F] and references therein. Theorem 2.1. For Q–almost all environments ω the following holds. Let ρ : R d → [0, ∞ be a bounded Borel function and let {µ N } N 1 be a sequence of probability measures on N C ω such that for all δ 0 and all continuous functions G on R d with compact support shortly G ∈ C c R d , it holds lim N ↑∞ µ N N −d X x ∈C ω Gx N ηx − Z R d Gx ρ xd x δ = 0 . 2.7 Moreover, suppose that there exist ρ , ρ ∗ , C 0 such that µ N is stochastically dominated by ν ρ and the entropy H µ N |ν ρ ∗ is bounded by C N d . 263 Then, for all t 0, G ∈ C c R d and δ 0, it holds lim N ↑∞ P ω,µ N N −d X x ∈C ω Gx N η t x − Z R d Gx ρx, td x δ = 0 , 2.8 where ρ : R d × [0, ∞ → R is assumed to be the unique weak solution of the heat equation ∂ t ρ = m∇ · D∇φρm 2.9 with boundary condition ρ at t = 0. We define the empirical measure π N η associated to the particle configuration η as π N η := 1 N d X x ∈C ω ηxδ x N ∈ M R d , where M R d denotes the Polish space of non–negative Radon measures on R d endowed with the vague topology namely, ν n → ν in M if and only if ν n f → ν f for each f ∈ C c R d . We refer to the Appendix of [S] for a detailed discussion about the space M endowed of the vague topology. We write π N t for the empirical measure π N η t , η t being the zero range process with generator N 2 L . Then condition 2.7 simply means that under µ N the random measure π N converges in probability to ρ xd x, while under P ω,µ N the random measure π N t converges in probability to ρx, td x, for each fixed t 0. In order to prove the conclusion of Theorem 2.1 one only needs to show that the law of the random path π N · ∈ D[0, T ], M weakly converges to the delta distribution concentrated on the path [0, T ] ∋ t → ρx, td x ∈ M see [KL][Chapter 5]. It is this stronger result that we prove here. Let us give some comments on our assumptions. We have restricted to increasing functions g in order to assure attractiveness and therefore that the dynamics is well defined whenever the initial distributions are stochastically dominated by some invariant measure ν ρ . This simplifies also some technical estimates. One could remove the monotone assumption on g and choose other conditions assuring a well defined dynamics and some basic technical estimates involved in the proof, which would be similar to the ones appearing in [KL][Chapter 5]. The entropy bound H µ N |ν ρ ∗ 6 C N d is rather restrictive. Indeed, given a locally Riemann in- tegrable bounded profile ρ : R d → [0, ∞, let µ N be the product measure on N C ω with slowly varying parameter associated to the profile ρ m at scale N . Namely, µ N is the product measure on N C ω such that µ N ηx = k = ν ρ xN m ηx = k . Due to the ergodicity of Q condition 2.7 is fulfilled and, setting ρ ′ := sup x ρ x, µ N is stochasti- cally dominated by ν ρ ′ m . On the other hand, the entropy H µ N |ν ρ ∗ is given by 1 N d H µ N |ν ρ ∗ = 1 N d X x ∈C ω ρ xN m h log ϕ ρ xN m − log ϕ ρ ∗ io + 1 N d X x ∈C ω n log Z ϕ ρ ∗ − log Z ϕ ρ xN m 2.10 264 Hence, H µ N |µ ρ ∗ 6 C N d only if ρ approaches sufficiently fast the constant m ρ ∗ at infinity. All these technical problems are due to the infinite space. In order to weaken the entropic assumption one should proceed as in [LM]. Since here we want to concentrate on the Moving Particle Lemma, which is the real new problem, we keep our assumptions. Finally, we have assumed uniqueness of the solution of differential equation 2.9 with initial con- dition ρ . Results on uniqueness can be found in [BC], [KL][Chapter 5] and [Va]. Proceeding as in [KL][Section 5.7] and using the ideas developed below, one can prove that the limit points of the sequence {π N t } t ∈[0,T ] are concentrated on paths t → ρx, td x satisfying an energy estimate. 3 Tightness of {π N t } t ∈[0,T ] As already mentioned, in order to reduce the proof of Theorem 2.1 to the Replacement Lemma one has to adapt the method of the corrected empirical measure developed in [GJ1] and after that invoke some homogenization properties proved in [F]. The discussion in [GJ1] refers to the zero range process on a finite toroidal grid and has to be modified in order to solve technical problems due to the presence of infinite particles. Given N ∈ N + , we define L N as the generator of the random walk among random conductances ω on the supercritical percolation cluster, after diffusive rescaling. More precisely, we define C N ω = {xN : x ∈ C ω} and set L N f x N = N 2 X e ∈B: x+e ∈C ω ωx, x + e n f x + e N − f xN o 3.1 for all x ∈ C ω and f : C N ω → R. We denote by ν N ω the uniform measure on C N ω given by ν N ω = N −d P x ∈C N ω δ x . Below we will think of the operator L N as acting on L 2 ν N ω . We write ·, · ν N ω and k · k L 2 ν N ω for the scalar product and the norm in L 2 ν N ω , respectively. Note that L N is a symmetric operator, such that f , −L N f ν N ω 0 for each nonzero function f ∈ L 2 ν N ω . In particular, λI − L N is invertible for each λ 0. Moreover, it holds f , −L N g ν N ω = 1 2 X x ∈C ω X e ∈B: x+e ∈C ω ωx, x + e f x + e − f x · gx + e − gx for all functions f , g ∈ L 2 ν N ω . Given λ 0, G ∈ C ∞ c R d and N ∈ N + we define G λ N as the unique element of L 2 ν N ω such that λG λ N − L N G λ N = G λ , 3.2 where G λ is defined as the restriction to C N ω of the function λG − ∇ · D∇G ∈ C ∞ c R d . Let us collect some useful facts on the function G λ N : Lemma 3.1. Fix λ 0. Then, for each G ∈ C ∞ c R d it holds G λ N , −L N G λ N ν N ω 6 cG, λ, 3.3 kG λ N k L 1 ν N ω , kG λ N k L 2 ν N ω 6 c λ, G , 3.4 kL N G λ N k L 1 ν N ω , kL N G λ N k L 2 ν N ω 6 c λ, G , 3.5 265 for a suitable positive constant c λ, G depending on λ and G, but not from N . Moreover, for Q–a.s. conductance fields ω it holds lim N ↑∞ kG λ N − Gk L 2 ν N ω = 0 , ∀G ∈ C ∞ c R d , 3.6 lim N ↑∞ kG λ N − Gk L 1 ν N ω = 0 , ∀G ∈ C ∞ c R d . 3.7 Proof. By taking the scalar product with G λ N in 3.2 one obtains that λkG λ N k 2 L 2 ν N ω + G λ N , −L N G λ N ν N ω = G λ N , G λ ν N ω 6 kG λ N k L 2 ν N ω kG λ k L 2 ν N ω . Using that sup N kG λ k L 2 ν N ω ∞, from the above expression one easily obtains the uniform up- per bounds on G λ N , −L N G λ N ν N ω and kG λ N k L 2 ν N ω . Since sup N 1 kG λ k L 2 ν N ω ∞, by difference one obtains the uniform upper bound on kL N G λ N k L 2 ν N ω . In order to estimate kG λ N k L 1 ν N ω let us write p N t x, y for the probability that the random walk on C N ω with generator L N and starting point x is at site y at time t. Then, since the jump rates depend on the unoriented bonds, p N t x, y = p N t y, x. Since G λ N x = X y ∈C N ω Z ∞ e −λt p N t x, yG λ y , 3.8 for all x ∈ C N ω, the above symmetry allows to conclude that kG λ N k L 1 ν N ω 6 1 N d X x, y ∈C N ω Z ∞ e −λt p N t x, y|G λ x| = 1 λN d X x ∈C N ω |G λ x| → 1 λ Z R d |G λ u|du ∞ . 3.9 Again, since sup N 1 kG λ k L 1 ν N ω ∞, by difference one obtains the uniform upper bound on kL N G λ N k L 1 ν N ω . The homogenization result 3.6 follows from Theorem 2.4 iii in [F]. Finally, let us consider 3.7. Given ℓ 0, using Schwarz inequality, one can bound kG λ N − Gk L 1 ν N ω 6 kG λ N uI|u| ℓk L 1 ν N ω + kGuI|u| ℓk L 1 ν N ω + cℓ d 2 kG λ N − Gk 1 2 L 2 ν N ω . Since G ∈ C ∞ c R d the second term in the r.h.s. is zero for ℓ large enough. The last term in the r.h.s. goes to zero due to 3.6. In order to conclude we need to show that lim ℓ↑∞ lim N ↑∞ kG λ N uI|u| ℓk L 1 ν N ω = 0 . 3.10 Since G λ ∈ C ∞ c R d we can fix nonnegative functions F, f ∈ C ∞ c R d such that − f 6 G λ 6 F . We call F λ N , f λ N the solutions in L 2 ν N ω of the equations λF λ N − L N F λ N = F , λ f λ N − L N f λ N = f , 266 respectively. From the integral representation 3.8 we derive that F λ N , f λ N are nonnegative, and that − f λ N 6 G λ N 6 F λ N on C N ω. In particular, in order to prove 3.10 it is enough to prove the same claim with F λ N , f λ N instead of G λ N . We give the proof for F λ N , the other case is completely similar. Let us define H ∈ C ∞ R d as the unique solution in L 2 d x of the equation λH − ∇ · D∇H = F . 3.11 Again, by a suitable integral representation, we get that H is nonnegative. Applying Schwarz in- equality, we can estimate kF λ N uI|u| ℓk L 1 ν N ω = kF λ N k L 1 ν N ω − kF λ N uI|u| 6 ℓk L 1 ν N ω 6 kF λ N k L 1 ν N ω − kHuI|u| 6 ℓk L 1 ν N ω + kHu − F λ N I|u| 6 ℓk L 1 ν N ω 6 kF λ N k L 1 ν N ω − kHuI|u| 6 ℓk L 1 ν N ω + cℓ d 2 kF λ N − Hk 1 2 L 2 ν N ω . 3.12 Since F λ N and F are nonnegative functions, when repeating the steps in 3.9 with F λ N , F instead of G λ N , G λ respectively, we get the the inequality is an equality and therefore kF λ N k L 1 ν N ω → λ −1 kF k L 1 d x = kHk L 1 d x . This observation, the above bound 3.12 and Theorem 2.4 iii in [F] imply that lim N ↑∞ kF λ N uI|u| ℓk L 1 ν N ω 6 kHuI|u| ℓk L 1 ν N ω At this point it is trivial to derive 3.10 for F λ N . In the rest of this section, we will assume that ω is a good conductance field, i.e. the infinite cluster C ω is well–defined and ω satisfies Lemma 3.1. We recall that these properties hold Q–a.s. The first step in proving the hydrodynamic limit consists in showing that the sequence of processes {π N t } t ∈[0,T ] is tight in the Skohorod space D[0, T ], M . By adapting the proof of Proposition IV.1.7 in [KL] to the vague convergence, one obtains that it is enough to show that the sequence of pro- cesses {π N t [G]} t ∈[0,T ] is tight in the Skohorod space D[0, T ], R for all G ∈ C ∞ c R. A key relation between the zero range process and the random walk among random conductances is given by N 2 L € π N η[G] Š = 1 N d X x ∈C ω g ηx L N G xN . 3.13 The check of 3.13 is trivial and based on integration by parts. At this point, due to the disorder given by the conductance field ω, a second integration by parts as usually done for gradient systems cf. [KL][Chapter 5] would be useless since the resulting object would remain wild. A way to over- come this technical problem is given by the method of the corrected empirical measure: as explained below, the sequence of processes {π N t [G]} t ∈[0,T ] behaves asymptotically as {π N t [G λ N ]} t ∈[0,T ] , thus the tightness of the former follows from the tightness of the latter. We need some care since the total number of particles can be infinite, hence it is not trivial that the process {π N t [G λ N ]} t ∈[0,T ] is well defined. We start with a technical lemma which will be frequently used: 267 Lemma 3.2. Let H be a nonnegative function on C N ω belonging to L 1 ν N ω ∩ L 2 ν N ω and let k 0. Then P ω,µ N sup 0 6 t 6 T 1 N d X x ∈C ω η t x k Hx N A 6 A −1 ck, ρ q kHk 2 L 1 ν N ω + kHk 2 L 2 ν N ω 3.14 for all A 0. Proof. We use a maximal inequality for reversible Markov processes due to Kipnis and Varadhan [KV] cf. Theorem 11.1 in Appendix 1 of [KL]. Let us set F η = 1 N d X x ∈C ω ηx k Hx N , 3.15 supposing first that H has bounded support. Note that F η 6 F η ′ if ηx 6 η ′ x for all x ∈ C ω. Hence by the stochastic domination assumption, it is enough to prove 3.14 with P ω,ν ρ0 always referred to the diffusively accelerated process instead of P ω,µ N . We recall that ν ρ is reversible w.r.t. the the zero range process. Moreover ν ρ F 2 = 1 N 2d X x, y ∈C ω Hx N H yN ν ρ ηx k η y k 6 ck, ρ kHk 2 L 1 ν N ω , 3.16 while ν ρ F, −N 2 L F = N 2 N 2d X x, y ∈C ω Hx N H yN ν ρ ηx k , −L η y k 6 ck, ρ N 2 N 2d X x ∈C ω X y ∈C ω: |x− y|=1 Hx N H yN . Using the bound Hx N H yN 6 HxN 2 + H yN 2 and the fact that d 2, we conclude that ν ρ F, −N 2 L F 6 ck, ρ kHk 2 L 2 ν N ω . 3.17 By the result of Kipnis and Varadhan it holds P ω,ν ρ0 sup 0 6 t 6 T 1 N d X x ∈C ω η t x k Hx N A 6 e A Æ ν ρ F 2 + T ν ρ F, −N 2 L F . 3.18 At this point the thesis follows from the above bounds 3.16 and 3.17. In order to remove the assumption that H is local, it is enough to apply the result to the sequence of functions H n x := Hx χ|x| 6 n and then apply the Monotone Convergence Theorem as n ↑ ∞. Remark 1. We observe that the arguments used in the proof of Lemma 4.3 in [CLO] imply that, given a function H of bounded support and defining F as in 3.15, it holds E ω,ν ρ0 sup 0 6 t 6 T F η t − F η 2 6 c T ν ρ F, −N 2 L F . 268 In particular, it holds E ω,ν ρ0 sup 0 6 t 6 T F η t 2 6 c ν ρ F 2 + c T ν ρ F, −N 2 L F . Using the bounds 3.16 and 3.17, the domination assumption and the Monotone Convergence The- orem, under the same assumption of Lemma 3.2 one obtains E ω,µ N sup 0 6 t 6 T h 1 N d X x ∈C ω η t x k Hx N i 2 6 ck, ρ h kHk 2 L 1 ν N ω + kHk 2 L 2 ν N ω i . 3.19 Using afterwards the Markov inequality, one concludes that P ω,µ N sup 0 6 t 6 T 1 N d X x ∈C ω η t x k Hx N A 6 ck, ρ A −2 h kHk 2 L 1 ν N ω + kHk 2 L 2 ν N ω i 3.20 for all A 0. The use of 3.14 or 3.20 in the rest of the discussion is completely equivalent. Due to Lemma 3.2 and Lemma 3.1 the process {π N t [G λ N ]} t ∈[0,T ] is well defined w.r.t E ω,µ N . Let us explain why this process is a good approximation of the process {π N t [G]} t ∈[0,T ] : Lemma 3.3. Let G ∈ C ∞ c R d . Then, given δ 0, it holds lim N ↑∞ P ω,µ N sup 0 6 t 6 T π N t [G λ N ] − π N t [G] δ = 0 . 3.21 Proof. By Lemma 3.2 we can bound the above probability by c ρ δ −1 q kG λ N − Gk 2 L 1 ν N ω + kG λ N − Gk 2 L 2 ν N ω . At this point the thesis follows from Lemma 3.1. Due to the above Lemma, in order to prove the tightness of {π N t [G]} t ∈[0,T ] it is enough to prove the tightness of {π N t [G λ N ]} t ∈[0,T ] . Now we can go on with the standard method based on martingales and Aldous criterion for tightness cf. [KL][Chapter 5], but again we need to handle with care our objects due to the risk of explosion. We fix a good realization ω of the conductance field. Due to Lemma 3.1, Lemma 3.2 and the bound gk 6 g ∗ k, we conclude that the process {M N t } 0 6 t 6 T where M N t G := π N t G λ N − π N G λ N − Z t 1 N d X x ∈C ω g η s xL N G λ N xN ds , 3.22 is well defined P ω,µ N –a.s. Lemma 3.4. Given δ 0, lim N ↑∞ P ω,µ N sup 0 6 t 6 T |M N t G| δ = 0 . 3.23 269 Proof. Given n 1, we define the cut–off function G λ N ,n : C N ω → R as G λ N ,n x = G λ N xI|x| 6 n. Then G λ N ,n is a local function and by the results of [A] together with the stochastic domination assumption we know that M N ,n t G := π N t G λ N ,n − π N G λ N ,n − Z t 1 N d X x ∈C ω g η s xL N G λ N ,n xN ds is an L 2 –martingale of quadratic variation M N ,n t G = Z t N 2 N 2d X x ∈C ω X y ∈C ω: |x− y|=1 g η s xωx, y G λ N ,n yN − G λ N ,n xN 2 ds . Note that, by the stochastic domination assumption and the bound gk 6 g ∗ k, E ω,µ N M N ,n t G 6 g ∗ Z t N 2 N 2d X x ∈C ω X y ∈C ω: |x− y|=1 E ω,ν ρ0 η s x ωx, yG λ N ,n yN − G λ N ,n xN 2 ds = g ∗ ρ t N −d G λ N ,n , −L N G λ N ,n ν N ω . 3.24 By Doob’s inequality and 3.24, we conclude that P ω,µ N sup 0 6 t 6 T |M N ,n t G| δ 6 c δ 2 E ω,µ N |M N ,n T G| 2 6 c g ∗ T ρ δ 2 N d G λ N ,n , −L N G λ N ,n ν N ω 6 c ′ g ∗ T ρ δ 2 N d . 3.25 Above we have used that lim n ↑∞ G λ N ,n , −L N G λ N ,n ν N ω = G λ N , −L N G λ N ν N ω 6 c λ see Lemma 3.1. The above process {M N ,n t G} t ∈[0,T ] is a good approximation of {M N t G} t ∈[0,T ] as n ↑ ∞. Indeed, it holds lim n ↑∞ P ω,µ N sup 0 6 t 6 T |M N ,n t G − M N t G| δ = 0 , δ 0 . 3.26 Indeed, since kG λ N ,n − G λ N k L 1 ν N ω ,L 2 ν N ω and kL N G λ N ,n − L N G λ N k L 1 ν N ω ,L 2 ν N ω converge to zero as N ↑ ∞ and since gk 6 g ∗ k, the above claim follows from Lemma 3.2. At this point, 3.26 and 3.25 imply 3.23. Let us prove the tightness of {π N t [G λ N ]} t ∈[0,T ] using Aldous criterion cf. Proposition 1.2 and Propo- sition 1.6 in Section 4 of [KL]: Lemma 3.5. Given G ∈ C ∞ c R d , the sequence of processes {π N t [G λ N ]} t ∈[0,T ] is tight in D[0, T ], R. 270 Proof. Fix θ 0 and uppose that τ is a stopping time w.r.t. the canonical filtration bounded by T . With some abuse of notation we write τ + θ for the quantity min{τ + θ , T }. Then, given ǫ 0, by Lemmata 3.1 and 3.2, P ω,µ N Z τ+θ τ 1 N d X x ∈C ω g η s xL N G λ N xN ds ǫ 6 P ω,µ N θ g ∗ sup s ∈[0,T ] 1 N d X x ∈C ω η s x|L N G λ N xN |ds ǫ 6 C g ∗ θ ǫ . 3.27 In particular, lim γ↓0 lim sup N ↑∞ sup τ,θ ∈[0,γ] P ω,µ N Z τ+θ τ 1 N d X x ∈C ω g η s xL N G λ N xN ds ǫ = 0 . 3.28 An estimate similar to 3.27 implies that lim ǫ↑∞ sup N 1 P ω,µ N Z t 1 N d X x ∈C ω g η s xL N G λ N xN ds ǫ = 0 . 3.29 Let us now come back to Lemma 3.4. Let τ, θ as above. Then lim sup N ↑∞ P ω,µ N |M N τ+θ − M N τ | ǫ 6 lim sup N ↑∞ P ω,µ N sup 0 6 s 6 T |M N s | ǫ2 = 0 3.30 Collecting 3.28, 3.29 and 3.30, together with Lemma 3.4, we conclude that lim γ↓0 lim sup N ↑∞ sup τ,θ ∈[0,γ] P ω,µ N π N τ+θ [G λ N ] − π N τ [G λ N ] ǫ = 0 , 3.31 lim ǫ↑∞ sup N 1 P ω,µ N |π N t [G λ N ] ǫ = 0 . 3.32 Aldous criterion for tightness allows to derive the thesis from 3.31 and 3.32. Let us come back to 3.22 and investigate the integral term there. The following holds: Lemma 3.6. Let It := R t N −d P x ∈C ω g η s x G λ N xN − GxN ds. Then, for all δ 0, lim N ↑∞ P ω,µ N sup 0 6 t 6 T |It| δ = 0 . 3.33 Proof. Since gk 6 g ∗ k and by Schwarz inequality we can bound I t 6 J := T g ∗ kG λ N − Gk 1 2 L 1 ν N ω sup 0 6 s 6 T {N −d X x ∈C ω η s x 2 G λ N xN − GxN } 1 2 . Using the stochastic domination assumption and applying Lemma 3.2 we obtain P ω,µ N sup 0 6 t 6 T |It| δ 6 P ω,ν ρ0 J δ 6 1δ 2 T 2 g ∗ 2 kG λ N − Gk L 1 ν N ω q kG λ N − Gk 2 L 1 ν N ω + kG λ N − Gk 2 L 2 ν N ω . 3.34 The thesis now follows by applying Lemma 3.1. 271 We are finally arrived at the conclusion. Indeed, due to Lemma 3.3 and Lemma 3.5 we know that the sequence of processes {π N t } t ∈[0,T ] is tight in the Skohorod space D[0, T ], M . Moreover, starting from the identity 3.22, applying Lemma 3.4, using the identity 3.2 which equivalent to L N G λ N = λG λ N − G λ = λG λ N − G + ∇ · D∇G , and finally invoking Lemma 3.6 we conclude that, fixed a good conductance field ω, for any G ∈ C ∞ c R d and for any δ 0 lim N ↑∞ P ω,µ N sup 0 6 t 6 T π N t G − π N G − Z t 1 N d X x ∈C ω g η s x∇ · D∇GxN ds δ = 0 . 3.35 Using the stochastic domination assumption it is trivial to prove that any limit point of the sequence {π N t } t ∈[0,T ] is concentrate on trajectories {π t } t ∈[0,T ] such that π t is absolutely continuous w.r.t. to the Lebesgue measure. Moreover, in order to characterize the limit points as solution of the differential equation 2.9 one would need non only 3.35. Indeed, it is necessary to prove that, given ω good, for each function G ∈ C ∞ c [0, T ] × R d it holds lim N ↑∞ P ω,µ N sup 0 6 t 6 T π N t G t − π N G − Z t 1 N d X x ∈C ω g η s x∇ · D∇G s xN ds − Z t π N s ∂ s G s ds δ = 0 , 3.36 where G s x := Gs, x. One can easily recover 3.36 from the same estimates used to get 3.35 and suitable approximations of G which are piecewise linear in t as in the final part of Section 3 in [GJ1]. In order to avoid heavy notation will continue the investigation of 3.35 only. 4 The Replacement Lemma As consequence of the discussion in the previous section, in order to prove the hydrodynamical limit stated in Theorem 2.1 we only need to control the term Z t 1 N d X x ∈C ω g η s x∇ · D∇GxN ds . 4.1 To this aim we first introduce some notation. Given a family of parameters α 1 , α 2 , . . . , α n , we will write lim sup α 1 →a 1 , α 2 →a 2 ,..., α n →a n instead of lim sup α n →a n lim sup α n −1 →a n −1 · · · lim sup α 1 →a 1 . Below, given x ∈ Z d and k ∈ N, we write Λ x,k for the box Λ x,k := x + [ −k, k] d ∩ Z d , 272 and we write η k x for the density η k x := 1 2k + 1 d X y ∈Λ x,k ∩C ω η y . If x = 0 we simply write Λ k instead of Λ 0,k . Then, we claim that for Q–a.a. ω, given G ∈ C c R d , δ 0 and a sequence µ N of probability measures on N C ω stochastically dominated by some ν ρ and such that H µ N |ν ρ ∗ 6 C N d , it holds lim sup N ↑∞,ǫ↓0 P ω,µ N Z t 1 N d X x ∈C ω g η s x GxN ds− Z t m N d X x ∈Z d φ η ǫN s xm GxN ds δ = 0 . 4.2 Let us first assume the above claim and explain how to conclude, supposing for simplicity of notation that ǫN ∈ N. Given u ∈ R d and ǫ 0, define ι u, ǫ := 2 ǫ −d I {u ∈ [−ǫ, ǫ] d }. Then the integral π N ι x N ,ǫ , x ∈ Z d , can be written as π N ι x N ,ǫ = 2ǫN + 1 d 2ǫN d η ǫN x . 4.3 Let us define Z t m N d X x ∈Z d φπ N s [ι x N ,ǫ m]∇ · D∇GxN ds . 4.4 Then, due to 4.3 and since φ is Lipschitz with constant g ∗ , we can estimate from above the difference between 4.4 and the second integral term in 4.2 with G substituted by ∇ · D∇G as 2ǫN + 1 d − 2ǫN d 2ǫN d Z t 1 N d X x η ǫN s x ∇ · D∇GxN ds 4.5 Since the integral term in 4.5 has finite expectation w.r.t P ω,ν ρ0 and therefore also w.r.t. P ω,µ N , we conclude that the above difference goes to zero in probability w.r.t. P ω,µ N . At this point the conclusion of the proof of Theorem 2.1 can be obtained by the same arguments used in [KL][pages 78,79]. Let us come back to our claim. Since X x ∈C ω g η s xGxN = X x ∈Z d g η s xIx ∈ C ωGxN , by a standard integration by parts argument and using that G ∈ C ∞ c R d one can replace the first integral in 4.2 by Z t 1 N d X x ∈Z d h 1 2ℓ + 1 d X y: y ∈Λ x, ℓ ∩C ω g η s y i Gx N . Then the claim 4.2 follows from 273 Lemma 4.1. Replacement Lemma For Q–a.a. ω, given δ 0, M ∈ N and given a sequence of prob- ability measures µ N on N C ω stochastically dominated by some ν ρ and such that H µ N |ν ρ ∗ 6 C N d , it holds lim sup N ↑∞,ǫ↓0 P ω,µ N h Z t 1 N d X x ∈Λ M N V ǫN τ x η s , τ x ωds δ i = 0 , 4.6 where V ℓ η, ω = 1 2ℓ + 1 d X y: y ∈Λ ℓ ∩C ω g η y − mφ η ℓ 0m . Let us define Υ C ,N as the set of measurable functions f : N C ω → [0, ∞ such that i ν ρ ∗ f = 1, ii D f := ν ρ ∗ p f , −L p f 6 C N d −2 and iii f d ν ρ ∗ is stochastically dominated by ν ρ shortly, f d ν ρ ∗ ≺ dν ρ . Using the assumption H µ N |ν ρ ∗ 6 C N d and entropy production arguments as in [KL][Chapter 5], in order to prove the Replacement Lemma it is enough to show that for Q–a.a. ω, given ρ , ρ ∗ , C , M 0, it holds lim sup N ↑∞,ǫ↓0 sup f ∈Υ C0,N Z 1 N d X x ∈Λ M N V ǫN τ x η, τ x ω f ην ρ ∗ dη = 0 . 4.7 Trivially, since ν ρ 1 stochastically dominates ν ρ 2 if ρ 1 ρ 2 , it is enough to prove that, given ρ , ρ ∗ , C , M 0, for Q–a.a. ω 4.7 is verified. We claim that the above result follows from the the One Block and the Two Blocks estimates: Lemma 4.2. One block estimate Fix ρ , ρ ∗ , C , M 0. Then, for Q–a.a. ω it holds lim sup N ↑∞,ℓ↑∞ sup f ∈Υ C0,N Z 1 N d X x ∈Λ M N V ℓ τ x η, τ x ω f ην ρ ∗ dη = 0 . 4.8 Lemma 4.3. Two blocks estimate Fix ρ , ρ ∗ , C , M 0. Then, for Q–a.a. ω it holds lim sup N ↑∞,ǫ↓0,ℓ↑∞ sup f ∈Υ C0,N Z 1 N d X x ∈Λ M N h 1 2ǫN + 1 d X y ∈Λ x, ǫN η ℓ y − η ǫN x i f ην ρ ∗ dη = 0 . 4.9 We point out that the form of the Two Blocks Estimate is slightly weaker from the one in [KL][Chapter 5], nevertheless it is strong enough to imply, together with the One Block Estimate, equation 4.7. Indeed, let us define a y := I y ∈ C ω and I 1 η := 1 N d X x ∈Λ M N Av y ∈Λ x, ǫN g η ya y − Av y ∈Λ x, ǫN Av z ∈Λ y, ℓ g ηzaz , 4.10 I 2 η := 1 N d X x ∈Λ M N Av y ∈Λ x, ǫN Av z ∈Λ y, ℓ g ηzaz − mφη ℓ ym , 4.11 I 3 η := 1 N d X x ∈Λ M N Av y ∈Λ x, ǫN m φη ℓ ym − mφη ǫN xm , 4.12 274 where Av denotes the standard average. Then 1 N d X x ∈Λ M N V ǫN τ x η, τ x ω = 1 N d X x ∈Λ M N Av y ∈Λ x, ǫN g η ya y − mφη ǫN xm 6 I 1 + I 2 + I 3 η . 4.13 Let us explain a simple bound that will be frequently used below, often without any mention. Con- sider a family of numbers bx, x ∈ Z d . Then, taking L, ℓ 0 we can write Av x ∈Λ L bx − Av x ∈Λ L Av u ∈Λ x, ℓ bu = 1 |Λ L | X x ∈Λ L+ ℓ bx Ix ∈ Λ L − 1 |Λ ℓ | ♯{u ∈ Λ L : |x − u| ∞ 6 ℓ} . In particular, it holds Av x ∈Λ L bx − Av x ∈Λ L Av u ∈Λ x, ℓ bu 6 1 |Λ L | X x ∈Λ L+ ℓ \Λ L −ℓ |bx| . 4.14 Due to the above bound we conclude that I 1 η 6 g ∗ N d 2ǫN + 1 d X x ∈Λ M N X u ∈Λ x, ǫN +ℓ \Λ x, ǫN −ℓ ηuau . In particular, using that f d ν ρ ∗ ≺ dν ρ , we conclude that R I 1 η f ην ρ ∗ dη 6 cℓǫN . The sec- ond term I 2 η can be estimated for ǫ 6 1 as I 2 η 6 1 N d X x ∈Λ M N Av y ∈Λ x, ǫN V ℓ τ z η, τ z ω 6 1 N d X x ∈Λ M +1N V ℓ τ x η, τ x ω . Due to the One block estimate one gets that lim sup N ↑∞,ǫ↓0,ℓ↑∞ sup f ∈Υ C0,N Z I 2 η f ην ρ ∗ dη = 0 . The same result holds also for I 3 η due to the Lipschitz property of φ and the Two Blocks estimate. The above observations together with 4.13 imply 4.7. 5 Proof of the Two Blocks Estimate For simplicity of notation we set ℓ ∗ = 2ℓ + 1 d and we take M = 1 the general case can be treated similarly. Moreover, given ∆ ⊂ Z d , we write N ∆ for the number of particles in the region ∆, namely N ∆ := P x ∈∆∩C ω ηx. 275 Let us set A η := 1 N d X x ∈Λ N Av y ∈Λ x, ǫN η ℓ y − η ǫN x , 5.1 B η := 1 N d X x ∈Λ N Av y ∈Λ x, ǫN Av z ∈Λ x, ǫN η ℓ y − η ℓ z , 5.2 C η := 1 N d |Λ ǫN | X x ∈Λ N Av y ∈Λ x, ǫN X z ∈Λ x, ǫN : |z− y| ∞ 2 ℓ η ℓ y − η ℓ z . 5.3 Since due to 4.14 η ǫN x = Av z ∈Λ x, ǫN η ℓ z + E , |E | 6 c|Λ ǫN | −1 X u ∈Λ x, ǫN +ℓ \Λ x, ǫN −ℓ ηu , using that f d ν ρ ∗ ≺ dν ρ , in order to prove the Two Blocks estimate we only need to show that lim sup N ↑∞,ǫ↓0,ℓ↑∞ sup f ∈Υ C0,N Z B η f ην ρ ∗ dη = 0 . Since B η 6 Cη + 1 N d |Λ ǫN | X x ∈Λ N Av y ∈Λ x, ǫN X z ∈Λ x, ǫN : |z− y| ∞ 6 2 ℓ η ℓ y + η ℓ z , using again that f d ν ρ ∗ ≺ dν ρ , in order to prove the Two Blocks Estimate we only need to show that lim sup N ↑∞,ǫ↓0,ℓ↑∞ sup f ∈Υ C0,N Z 1 N d ǫN 2d ℓ d X x ∈Λ N X y ∈Λ x, ǫN X z ∈Λ x, ǫN : |z− y| ∞ 2 ℓ N Λ y, ℓ − N Λ z, ℓ f ην ρ ∗ dη = 0 . 5.4 Let us now make an observation that will be frequently used below. Let X ⊂ Z d be a subset possibly depending on ω and on some parameters for simplicity, we consider a real–value parameter L ∈ N. Suppose that for Q–a.a. ω it holds lim sup N ↑∞,L↑∞ |X ∩ Λ N | N d = 0 . Then, in order to prove 5.4 we only need to show that lim sup N ↑∞,ǫ↓0,ℓ↑∞,L↑∞ sup f ∈Υ C0,N Z 1 N d ǫN 2d ℓ d X x ∈Λ N X y ∈Λ x, ǫN X z ∈Λ x, ǫN : |z− y| ∞ 2 ℓ N Λ y, ℓ \ X − N Λ z, ℓ \ X f ην ρ ∗ dη = 0 . 5.5 276 We know that there exists α