KINETIC LIMIT FOR A HARMONIC CHAIN WITH A CONSERVATIVE ORNSTEIN-UHLENBECK STOCHASTIC PERTURBATION

. We consider a one dimensional inﬁnite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserv- ing energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corre- sponding to the chain. This result generalizes the results of [7] obtained for a random momentum exchange that is of a white noise type. In contrast with [7] the scattering term in the limiting Boltzmann equation obtained in the present situation depends also on the dispersion relation.


1.
Introduction. In the present paper we investigate the kinetic limit of the energy density for a one dimensional harmonic chain with a random mechanism of momentum exchange that ensures its conservation. The system is described by a stochastically perturbed discrete linear wave equation in a one dimensional integer lattice      dq y dt = ∂ py H(p, q), dp y dt = −∂ qy H(p, q) + √ ζ y (t), y ∈ Z.
Here (p, q) = ((p y , q y )) y∈Z , where the component labelled by y corresponds to the one dimensional momentum p y and position q y . The Hamiltonian is (formally) given by H(p, q) := 1 2 y∈Z p 2 y + 1 2 y,y ∈Z α y−y q y q y .
The assumptions made about the coupling constants (α y ) y∈Z (see (a1)-(a3) made in Section 2.2) ensure that the potential energy q → y,y ∈Z α y−y q y q y is nonnegative and the interaction between the oscillators is local in space.
The stochastic perturbation √ ζ y (t) describes the momentum exchange mechanism that we superimpose on the Hamiltonian system. It is assumed here to be weak, as we shall later take 1. Its particular form is determined by the stipulation that it should conserve the momentum and be local, i.e. depend only on neighbouring sites. We shall asume therefore that it is the stochastic differential of an Ornstein-Uhlenbeck process ζ := (ζ(t)) t≥0 of the form ζ y (t) := √ z=−1,0,1 (Y y+z p y )ξ y+z (t), y ∈ Z.
Note that they are tangent to the circle described by the system of equation and p y−1 + p y + p y+1 ≡ const (5) guaranteeing that the exchange of momenta, taking place in the chain, conserves locally the kinetic energy and the momentum. The process (ξ(t)) t∈R , appearing in (3), is a zero mean Gaussian with the covariance given by E[ξ y (t)ξ y (s)] = T e −γ(k)|t−s|−2πik(y−y ) σ(k)dk, ∀ t, s ∈ R, y, y ∈ Z.
Here T is the one dimensional unit torus, defined throughout this paper as the interval [−1/2, 1/2] with identified endpoints. We assume that the time mixing rate γ(k) and spatial spectral density σ(k) are non-negative (detailed assumptions are given in Section 2.3) and γ(k) ≥ γ * , k ∈ T for some γ * > 0.
The main result of the present paper, see Theorem 2.5 below, concerns the asymptotics of the Wigner function W (t, x, k), that describes the resolution of the energy in spatial and frequency mode coordinates (x, k) ∈ R × T, see (34) below. We prove that at time t ∼ 1/ the expectation of W (t, x, k) tends, as → 0, to the solution U (t, x, k) of a linear Boltzmann equation where the transport coefficient is determined from the dispersion relation ω(k) = α(k) (see (18)) and α(k) is the Fourier transform of the sequence α := (α y ) y∈Z . It is real and non-negative thanks to the assumption on positive definiteness of the quadratic potential energy. In addition, since we assume that α is a rapidly decaying, real valued sequence (see assumptions (a1)-(a3) below) the functionα(·) is even and C ∞ smooth.
The scattering operator L is of the form with the scattering kernel given by R(k, k ) := 2σ(k + k )γ(k + k )R + (k, k ) The shape of functions R ± (k, k ) is determined by the type of the momentum exchange considered. In our case they equal R ± (k, k ) := 16 sin 2 (πk) sin 2 (πk ) sin 2 (π(k ∓ k )), k, k ∈ T.
It is clear from the formula (9) that R(k, k ) = R(k , k) for all k, k ∈ T. The kinetic limit in the case of a harmonic chain with a random momentum exchange based on a white noise in time has been considered in [7], see also [19], where the compensated wave rather than Wigner function has been studied and [18], where the long time, large scale asymptotics have been obtained using probabilistic representation of the solution of the kinetic equation (7). It has been assumed in [7] that (ξ y (t)) is the cylindrical Wiener process. The limit of the Wigner functions is described then by a linear kinetic equation of the form (8) with the scattering kernel given by R(k, k ) = R + (k, k ) + R − (k, k ).
Note that we can think of the white noise model of [7] as a limiting case of the situation considered here, when both the mixing rate and spectral density are constants γ(k) = γ, σ(k) = 2γ and γ → +∞. Then, the limit of the respective kernels (9) is given by (10). A novel feature of our model, when compared with the one considered in [7], is the fact that the dipersion relation ω(k) (that determines the transport term in (7)) appears also in the formula for the scattering kernel, see (9). This fact may better reflect the coupling between scattering and transport, due to nonlinearity in non-linear chains, e.g. as in the case of a Fermi-Pasta-Ulam chain (see e.g. [21] for necessary definitions). Such a situation does not occur for white noise in time exchange models where the transport and scattering are fully decoupled.
In addition, we believe that the kinetic limit might be valid for a broader family of random perturbations that need not be uniformly mixing for all wavelengths (i.e. the assumption γ(k) ≥ γ * > 0, k ∈ T need not hold). The proof of such a result may require a different approach though, e.g. one could use the Duhamel series expansion for the the Wigner function obtained from the equation system (62) and (65) below and a subsequent analysis of the Feynman diagrams that arise after taking the expectation of the resulting multiple products of Gaussians. As a result, upon a suitable choice of the mixing rate function e.g. of the form γ(k) = | sin(πk)| (implying long time correlations for the proces ξ), one could, in principle, observe a different type of asymptotics of the solutions to the linear Boltzmann equation (7) from the one seen in [14], even in the case when σ(k) ≡ 1, i.e. when the sequence (ξ y (t)) y∈Z is i.i.d. for a fixed t. In particular it seems possible to have a situation when the total scattering kernel satisfies R(k) = T R(k, k )dk = +∞, for all k = 0. It would correspond to the situation when the solution of the kinetic equation has a smoothing property. A somewhat similar phenomenon takes place in the case of the kinetic limit for the Wigner function of the solution of the Schrödinger equation with a random potential, see [10].
Another interesting point is to understand how the particular form of the scattering kernel, as in (9), might influence the long time, large space scale asymptotics of the phonon and also the hydrodynamic limit of the energy distribution in the case when no weak coupling assumption is made for the system (1). Such limits have been considered for harmonic chains with the white noise exchange of momenta and some other closely related models in [9,13,14,15]. It follows from the results of [1,14,23] that in the case of an acoustic chain, i.e. when ω(0) = 0, the 242 TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ limit of the solution of the linear Boltzmann equation (7), under the macroscopic scaling t = N t , x = N 2/3 x , k = k , as N → +∞, is described by the solution of the fractional heat equation ∂ tŪ (t, x) = −|∆ x | 3/4Ū (t, x) (see also [5] for a two dimensional result). The mode coupling argument of [26] (see Appendix 1) seems to suggest that a similar limit should occur in the case of the present model.
Comparing our method of proof with the argument used in [7] we note first that the equations governing the dynamics of the Wigner functions considered here (see (69) and (74) below) differ substantially from their analogues in [7], see (42) and (44) ibid. In particular, both (69) and (74) contain scattering terms that are of apparent order of magnitude −1/2 , as 1. The corresponding terms in (42) and (44) of [7] are of order 1. It is due to the fact that in our model the dynamics of the wave function, see (29) below, is given by an ordinary differential equation containing a random perturbation of the Hamiltonian part (describing the scattering of waves) that is also of order −1/2 , as 1, while the respective term in the corresponding equation in [7], see (14), rewritten in the kinetic time scale t/ , is of order 1. This explains why the equations (42) and (44) for the Wigner functions in [7] do not contain large scattering terms.
For this reason we are prevented to adopt the approach of [7] to the present model. To deal with the large scattering term we resort instead to a technique used in the averaging theory called the perturbed test function method, see e.g [11]. It has been applied to derive the kinetic limit for the Wigner transform of the solution of the Schrödinger equation with a random, time dependent, Markovian potential by G. Bal et al. in [4]. In a nutshell (and with some degree of oversimplification) the method can be described as follows. To obtain the asymptotics of E W (t), G , as 1, for a given (smooth) test function G : R × T → C, where ·, · is the appropriate duality pairing, we replace the test function by a random field of the formḠ (t, x) := G(t, x) + √ G 1, (t, x) + G 2, (t, x), see (95) below (in fact we need three such fields corresponding to three Wigner function considered here). Then, using the temporal dynamics of the random perturbation process, we can find such fields G j, , j = 1, 2 (called correctors) that are both of order 1 (to prove this fact we use the spectral gap assumption γ(k) ≥ γ * ) and the time derivative ∂ t E W (t), G does not contain terms of large magnitude. In the process we also identify the limiting equation satisfied by E W (t), G . Finally, we mention here that an analogous model can be formulated in an arbitrary spatial dimension. One can obtain then a similar result concerning the kinetic limit of the Wigner functions. The only complication occurs on the level of notation (which is already a bit heavy for a one dimensional case).
The organization of the paper is as follows. In Section 2 we introduce the preliminaries and formulate the main result of our paper. The dynamics of the Wigner function, and the accompanying it set of functions needed to close the equations are presented in Section 3. The proof of the main result is contained in Section 5. Some additional results that are needed to make our presentation self-contained are presented in the Appendix.
2. Preliminaries and formulation of the main result.
2.1. Some basic notation. Let p (Z) be the complex Banach space of all complex sequences f = (f y ) y∈Z equipped with the norm f p p := y∈Z |f y | p . Define the Fourier transformf of the sequence f ∈ 1 by letting We denote by · L p (T) the respective L p norm. By the Plancherel theorem we have f L2(T) = f 2 . For a given m ∈ R we define the the Hilbert space h m , made of those sequences, for which where y := (1+y 2 ) 1/2 . Its Fourier transform image shall be denoted by H m (T). In what follows we shall also use the Sobolev spaces W m,p (T) made of those functions that possess m generalized derivatives that are L p integrable, for p ∈ [1, +∞), or have finite essential supremum, if p = +∞, equipped with the respective standard norm.
Denote by S(R), S (R) the spaces of all Schwartz functions on R and the corresponding space of tempered distributions. By S(R × T) we denote the space of complex valued functions on R × T that are C ∞ smooth and such that for any non-negative integers l, m, n we have Let G(p, k) and G(x, y) be the Fourier and the inverse Fourier transforms of G, respectively in the first and second variable, i.e.
Let A be the completion of the space S(R × T) in the norm We let (A , · A ) be the dual of A. For any two sequences J (j) = (J (j) y,y ) y,y ∈Z , j = 1, 2 of elements of L 2 (T) we let By S we denote the space made of those sequences J = (J y,y (k)) y,y ∈Z , for which J y,y (·) ∈ L 1 (T) and for any non-negative integer m we have sup y,y ( y y ) m |J y,y (y − y)| + |J y,y (y − y )| < +∞.
For given p ≥ 1, m ∈ R we consider L p and H m -the completions of S in the respective norms defined by  (2), we assume, that it is real valued and satisfies the following: (a1) there exists C > 0 such that |α y | ≤ Ce −|y|/C , y ∈ Z, (a2) it is even, i.e. α y = α −y , y ∈ Z, (a3) its Fourier transformα(·) is nonnegative on T, andα(k) > 0 for k = 0.

2.3.
Ornstein-Uhlenbeck perturbation. Let ξ = (ξ y (t)) (t,y)∈R×Z be a stationary Gaussian random field defined over some probability space (Ω, F, P). The covariance function of the field is given by (6). The functions γ(·) and σ(·) appearing in (6) are nonnegative and belong to C 2 (T). We suppose furthermore that they are both even, so that the field is real valued, and that for some γ * ∈ (0, 1). Let us fix m > 1/2. From an elementary regularity theory of Gaussian processes, see e.g. Theorem 3.4.1, p. 60 of [2], the above assumptions guarantee that (ξ(t)) t∈R has continuous trajectories in the Hilbert space E := h −m . Let ξ y (t) := ξ y (t/ ). The field ξ := ξ y (t) (t,y)∈R×Z is also Gaussian and homogeneous in (t, y). Therefore, see e.g. Theorem 5.2 of [3], for any , N > 0 we haveC where E is the expectation with respect to measure P and Due to stationarity of the process, the left hand side of (20) does not depend on t.
2.4. Initial data. We assume that the initial data is random and distributed according to a probability law µ supported in 2 (Z). Moreover we require that macroscopically the amount energy per unit length stays finite, i.e. there exists finite K > 0 such that where H(p, q) is given by (2).

KINETIC LIMIT FOR STOCHASTIC WAVE EQUATION 245
2.5. Wave function and its dynamics. Suppose that (q, p) belongs to ( 2 (Z)) 2 . The wave function (see [7]) is defined as Hereω y are the Fourier coefficients of the dispersion relation (18). Since ω ∈ L ∞ (T) we conclude that (ω y ) ∈ 1 (Z). The convolution (ω * q) y is given by y ω y−y q y and belongs to 2 (Z). In fact, when yω y = 0 it suffices only to assume that the sequence (q y − q y−1 ) y∈Z belongs to 2 (Z) in order for (ω * q) to be defined as an element of 2 (Z). Taking into account that the size of the stochastic perturbation is of order √ we expect that its effects shall be significant on the time scale t/ (as it is the scale on which its fluctuations are macroscopically of order O(1)). Adjusting the time variable to the macroscopic scale we define rescaled wave function, that corresponds to (q(·), p(·)) -the solution of (1) -as From (1) we obtain the following equation where θ y,0 (t) ≡ 0 and θ y,∓2 (t) := ±ξ y∓1 (t), θ y,±1 (t) := ± ξ y (t) + ξ y±1 (t) , and ξ y (t) := ξ y (t/ ). (26) Using Theorem A.2 below we conclude that the above Cauchy problem has a unique solution in the sense explained in Appendix A (see (149)), and the 2 -norm of the solution is preserved in time, i.e. y∈Z |ψ y (t)| 2 = y∈Z |ψ y | 2 , t ≥ 0, for a.s. realization of ξ .
By calculating the Fourier transform of the both sides in (25) we obtain the equation for the evolution ofψ (t), which reads ψ (0, k) =ψ(k).
HereV (t, dk) -the spectral measure of the field ξ -is an H −m (T)-valued procesŝ
Remark 2.1. To understand the integral appearing in the right hand side of (29) note that j (k) and The stochastic measureV ( ) (t, ·) belongs to H −m (T). Therefore, its convolutions with r (1) j ψ ( ) ∈ L 2 (T) also belong to H −m (T) and, as a result, the right hand side of (33) is in that space as well. In particular, the time derivative of ψ ( ) (t) appearing on the right hand side of (29) exists in the strong sense in H −m (T). Likewise the derivative appearing on the left hand side of (25) exists strongly in h −m .

Definition of the Wigner functions.
To describe the energy transport on the lattice in large space-time scales we use the lattice Wigner functions corresponding to ψ (t), see [22]. We define them as the distributions: and Here W (t, p, k) and Y (t, p, k) -the respective Fourier-Wigner functions -are given by for (p, k) ∈ R × T and t ≥ 0. The averaging · µ is performed with respect to the initial distribution µ so the expressions in (34) and (35) are random, since they depend on the realizations of ξ . We can write and where To close the equation for the dynamics of W (t) and Y (t) we shall also need Y ,− y,y (k) := Y y,y (−k) * . It is clear that W and Y ,± belong to L 1 , cf (16). Here Y ,+ := Y y,y .
Using Cauchy-Schwarz inequality we conclude that sup t,p, where K is as in (28). Directly from (42) we infer that sup t, where K is the constant defined in (22).

2.7.
The statement of the main result. We formulate the following Cauchy problem for the linear kinetic equation with an appropriate initial condition U 0 . The scattering operator L : A → A has the form with the scattering kernel R(k, k ) given by (9).
We have the following result concerning the existence and uniqueness of solutions of (44).
then the solution is unique. We also have The proof of the above theorem is presented in Section C of the Appendix. The goal of the present paper is to show that the average of the Wigner function converges, as → 0+, to the solution of (44). We consider both the case of an acoustic and pinned chain and i.e. when, respectively In the case (a4) we shall need an additional condition, namely: Remark 2.4. The above condition appeared in [7], see condition (b4) on p. 177. It can be interpreted as the absence of energy at the longest wavelengths in the acoustic chain at time t = 0. This component of energy corresponds to the macroscopic profiles of the elongation r y := q y − q y−1 and momentum p y , see [16].
Our main result can be stated as follows.
Theorem 2.5. Suppose that the sequence of coupling constants α satisfies (a1)-(a3), and either (a4) or (a4'). Assume that the initial distributions (µ ) satisfy (22), and in case (a4) holds, we suppose also that (50) is in force. Furthermore, assume that there exists Then, for each t ≥ 0 fixed, EW (t) converges as → 0+ in the * -weak topology of A to U (t), defined by the solution of the Cauchy problem (44).

Evolution of the Wigner functions.
In the present section we formulate the equations governing the evolution of the Wigner functions W (t) and Y (t), see (69) and (74) below. Note that the right hand side of equation (69) below contains terms that are of order of magnitude −1/2 . This stands in stark contrast with (42) of [7] where we find no large terms (in ) in its right hand side. We start with the following.
Proposition 3.1. Suppose that J ∈ H m for some m > 0 (see (17)). Then for any and Proof. We only prove equality (51), as the argument for (52) is analogous. The equality in question follows, provided we can prove that: Indeed, by the Lebesgue dominated convergence theorem we conclude then that and formula (51) follows from the differentiation of both sides of (54).
Performing the integral over k we can rewrite the series in (53) in the form From (25) we get Substituting into (55) we obtain an expression that can be written as the sum of the series corresponding to each term appearing in the right hand side of (56). We shall prove that and y,y ,|z|≤2 as the remaining terms can be dealt with in a similar fashion. The expression in (57) can be estimated by the Cauchy-Schwarz inequality and we infer that it can be bounded from above by Here E is the expectation with respect to the product measure P⊗µ . The assumptions about regularity of the dispersion relation imply that ω ∈ H 1 (T), therefore, in particular (ω x ) x∈Z ∈ 1 and by Young's inequality thus the series (57) is finite thanks to (28).

TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ
Concerning (58) observe that for a fixed > 0 the expression appearing there can be estimated, using stationarity of ξ , by (cf the definition (20)) where C m (t; ) is defined in (21). Using Cauchy-Schwarz inequality we can estimate the expression by In the last equality we have used the time invariance of ψ (t) 2 2 (cf (28)) In what follows we shall adopt the shorthand notation r o y,y (k, k ) := r + y,y (k, k ) + r − y,y (k, k ). Using Proposition 3.1 and formula (56) we conclude that for any J ∈ S we have (62) for a.e. t ≥ 0 and a.s. realization of ξ . Here, for any J ∈ S we have defined: (64) A stationary Gaussian random field r ι y,y y,y ∈Z is given by For G ∈ S(R × T) we denote, cf (11), G y,y (k) := G( (y + y )/2, k) andG y,y :=G( (y + y )/2, y − y ), y, y ∈ Z. (67) Note that G ∈ S. Thanks to (38) for any > 0 we have for a.s realization of ξ (·). From (68) and (62) we obtain, after a straightforward calculation, that A simple calculation shows that where lim →0+ sup t∈[0,T ] |O(t, )| = 0 for any T > 0 and As a result we conclude that Analogously, we obtain where G e,y,y (k) := G( (y + y )/2, k) + G( (y + y )/2, −k) and 4. Properties of the stochastic perturbation.

4.1.
Markov property of the process ξ. The process ξ, described in Section 2.3, is Markovian and reversible in the following sense. Denote by π the law of ξ(0) on (E, B(E)), with B(E) the Borel σ-algebra on E. Let (F t ) t≥0 be the natural filtration associated with the process. Using a fairly standard argument, relying on the second quantization (see e.g Chapter 4 of [12]), one can show that there exists a strongly continuous semigroup (P t ) t≥0 of non-negative, symmetric contractions on L 2 (π), the transition probability operators, such that P t 1 = 1 and 252 TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ (i) (Markov property) for any F ∈ L 2 (π) with γ * as in (19).
Suppose that y ∈ Z and V y : E → R is the coordinate mapping on E given by Then, V = (V y ) y∈Z is a Gaussian field whose co-variance equals with ·, · L 2 (π) the scalar product in L 2 (π). Its spectral measure V (dk), cf (30), is the L 2 (π)-valued Borel measure on T satisfying For any ψ(·) that belongs to L 2 (T; σ) -the space of complex valued functions on T, square integrable with respect to σ(k)dk -we can define the stochastic integral with respect to the spectral measure as The series on the right hand side of (75) is convergent in L 2 (π). The measure V satisfies V * (dk) = V (−dk) and its structure measure equals Denote by h 1 the closed subspace of L 2 (π) spanned by the stochastic integrals (75) (the first degree Hermite polynomials in the chaos expansion over Gaussian measure π). From the construction of the semigroup, via the second quantization (see p. 45 of [12]), each P t leaves h 1 invariant for any t ≥ 0. For any ψ 1 , ψ 2 ∈ L 2 (T; σ) we conclude from (6) that where ·, · L 2 (T;σ) the scalar product in L 2 (T; σ) and S t ψ 1 (k) := e −γ(k)t ψ 1 (k). Thanks to the invariance of h 1 under P t we conclude from the above that P t I(ψ 1 ) = I(S t ψ 1 ) for all t ≥ 0 and, as a result, where Q is the L 2 (π) generator of the semigroup (P t ) t≥0 . In addition, using formula (4.3), p. 45 of [12], we may conclude that Q (I(ψ 1 )I(ψ 2 )) = − I(γψ 1 )I(ψ 2 ) + I(ψ 1 )I(γψ 2 ) Remark 4.1. Suppose that σ(k) ≡ γ/2, and γ(k) ≡ γ, where γ > 0 is some constant. The respective processes (ξ(t)) t≥0 converge in law, as γ → +∞, to a white noise process (ẇ(t)) t≥0 on 2 (Z), i.e.ẇ(t) = (ẇ y (t)) y∈Z where (ẇ y ) y∈Z are i.i.d. Gaussian, S (R)-valued processes satisfying for all y, y ∈ Z, J 1 , J 2 ∈ S(R). The model of harmonic oscillators with the conservative noise introduced in [6] is therefore the limiting case of the present model, as the mixing rate (spectral gap size) γ of the Ornstein-Uhlenbeck process tends to +∞.

4.2.
Multiple stochastic integrals. Using the spectral measureV (dk) we can define a multiple spectral integral Computations using objects of this type appear in the proofs of Lemma 5.1 (see steps 2 and 3) and Lemma 5.2 below. We let I n be a linear mapping defined on the space L Here I(ψ j ) is the stochastic integral given by (75). By a density argument we can extend I n to the space L 2 n (σ) obtained as the completion of L n with respect to the Hilbert pseudo-norm The summation extends over all possible pairings F made between elements of the set {1, . . . , 2n}. The multiple stochastic integral in (78) is then defined on L 2 n (σ) as the resulting extension of I n , see appendix of [17] for details of this construction.
Given n ≥ 1 the subspace p n of L 2 (π) is defined as the closure of the linear span over all I m (ψ), ψ ∈ L 2 m (σ), where m ≤ n. It is called the space of the polynomials of n-th degree, cf Def. 2.1, p. 17 of [12]. It is clear that p n−1 ⊂ p n for each n ≥ 1, with p 0 := span(1).

4.3.
Pseudogenerator. We recall the notion of the pseudo-generator of a process following Section 2.2 p. 38 of [11] (see also [20]). Suppose that η = (η(t)) t≥0 is a complex valued stochastic process over (Ω, F, P) that is progressively measurable with respect to a filtration (F t ) t≥0 and such that We say that the process Lη = (Lη(t)) t≥0 is the pseudo-generator of η if it is progressively measurable, subordinated with respect to (F t ) t≥0 , satisfies (79) and It is well known, see Theorem 2.2.1, p. 39 of [11], that then the process In what follows we assume that the functional J : E → H m is continuous for some m > 0 (cf (17)) and such thatJ y,y (y − y ) belongs to D(Q) -the domain of the generator Q for each y, y ∈ Z. In addition, we suppose that P t J : E → H m , where (P t J) y,y (·; k) := P t (J y,y (·; k)). A similar notational convention shall be used in the case of the generator Q.
We suppose that the functional J satisfies also the following  Under the hypotheses made above the processes (Φ (t)) and (Ψ (t)) satisfy (79) for each > 0. In addition, the respective pseudo-generators equal The proof of this result is presented in Section B of the Appendix.

5.1.
Outline of the proof of Theorem 2.5. We would like to sketch briefly and somewhat informally the method of the proof of the theorem. It follows from (69) -(71) that for any test function G ∈ S(R × T) we can write that d dt where o(1) → 0+, as → 0+, G is given by (67) and K ι , ι ∈ {−, +, o} are the random scattering operators introduced in Section 3. It is worthwhile to compare this equation with the corresponding one for ∂ t W (t), G appearing in [7], see (42) p. 187. All terms in the right hand side of (42) are of order 1. The elimination of the terms containing Y ,± (t) is then possible due to the fact that from the equation desciribing the dynamics of Y ,± (t), G , see (44) ibid., we can easily conclude that | Y ,± (t), G | → 0, as → 0+, provided G(x, k)/ω(k) stays bounded. The identification of the limit of W (t), G follows then from a fairly direct calculation, see Section 4.3.1 of [7]. This argument cannot be applied in the present case since the terms appearing in the right hand side of (88), except the one corresponding to the dispersion, are of order of magnitude −1/2 . To deal with the large terms we use the perturbed test method that comes from the stochastic averaging theory, see [20,11]. Let us describe briefly this technique. To keep our discussion as simple as possible we shall omit in our sketch the terms in (88) containing Y ,± (t). In fact, as we shall show rigorously in what follows, see (100) and (101), these terms do not contribute significantly to the asymptotics, as → 0+.
The main idea is to replace the test function G in (88) by a fieldḠ : E → S, cf (15), such thatḠ (ξ (t)) = G + √ G 1, (ξ (t)) + G 3, (ξ (t)), where for a given G ∈ S(R × T) (deterministic) test function G is defined by (67) and G j, : E → S, j = 1, 2 (called correctors) are yet to be determined. Substituting into (88) the expression forḠ (ξ (t)) and using the Markov property of ξ, see Section 4.1, we obtain that where, . Here Q is the generator of the Ornstein-Uhlenbeck process introduced in Section 4.1. The actual calculation performed in Section 5.2 below, (see Step 1 of the proof of Lemma 5.1) involves a bit more complex random processes Ψ j (t), j = 1, 2, 3, since we also have to account for the terms Y ,± (t).
Next, cf (99) below, we choose G 1, in such a way that ψ 1 (t) ≡ 0 (this choice eliminates the large term in (89)). This is achieved, by solving in explicit terms, the corrector equation (Q − D)G 1, = −iK o G . The solution exists, due to the fact that E K o G dπ = 0. To see this, note that from the definition of Q − D constants are in the null space of its adjoint, so the latter condition is indeed necessary for solvability of the corrector equation. Thanks to the spectral gap assumption (ii) (that holds due to γ(·) ≥ γ * > 0) the operator Q−D is invertible on the subspace of π-zero mean functions so the solution of the corrector equation exists and is unique in that space.
Having chosen G 1, we turn our attention to the term ψ 2 (t), which is stochastic and of order of magnitude O(1), as → 0+. It can be replaced by a deterministic term, if we recenter the random scattering operator, i.e. choose G 2, in such a way that It turns out, see Step 2 of the proof of Lemma 5.1 and Section 5.1, that LG = 0, withL and L the approximate and limiting scattering operators described in (97) and (8) respectively. This allows us to conclude the proof of Theorem 2.5.

Approximate kinetic equation for the Wigner function.
We introduce the following shorthand notation: for given a bounded measurable g : T 4 → C we define the (random) operators K (±) g that generalize the operators K (±) introduced in (63). Let for π a.s. f and J ∈ S. e 2πik (y −y) r(k + q/2, k )r(k − q/2, k ) − r 2 (k + ι q/2, k ) and a.s. in ξ (·). Here G is given by (67). In addition, for every T > 0 Proof. The proof of the lemma is divided it into three steps. We start with the two scale expansion scheme for the test function in the formula for the pseudo-generator, see (87). The requirement that no large terms arise as a result of this expansion leads to equations for the resulting terms (whose solutions we call the correctors).
In step 2 we solve these equations and in consequence compute quite explicitly the terms of the expansion and the scattering operatorL appearing in (94). Finally, in step 3 of the proof we prove the validity of the expansion claimed in (94).
To simplify the notation we shall writeḠ (t) instead ofḠ (ξ (t)). A similar convention shall be used for the other functions appearing in (95). We also use the following convention: suppose that K is any one of the operators K ι , ι ∈ {−, +, o} and J is a random S-valued element. We let To calculate the pseudo-generator of the process Ψ (t) we use Proposition 4.2. Here we do it somewhat formally (without verifying the assumptions (83)-(85)). The remark on the applicability of the proposition is made in Section 5.3, after determining formulas for the fields G i, j , i = 1, 2, j ∈ {−, o, +}. Using formula (87) we obtain then

TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ
where we have grouped terms that stand by the respective powers of j/2 , j = −1, 0, 1. We have

Let alsoL
Here G j, ±,o,y,y (k) := G j, ±,y,y (k) + G j, ±,y ,y (−k). The term Ψ (1) (t) corresponds to an expression in (96) of an apparent order of magnitude O( −1/2 ), see (96). To make it vanish we assume that G 1, ι are chosen in such a way that Having solved the above system for G 1, ι , our next step is to eliminate the term Ψ 2 (t) that corresponds to random operatorsK ι . We stipulate therefore that G 2, ι , ι ∈ {−, +, 0} are such that The above facts combined allow us to conclude (94).
Step 2: Computation of the corrector terms and scattering operators. Since the right hand sides of the equations (99) are fields of the form where H y,y (k, k ) are some determnistic functions, we are looking for solutions that take the form for some determnistic J y,y (k, k ). Note that then Comparing the left and right hand sides of the first equation of (99) we obtain with Similarly, where Concerning functions G 2, ι we conclude that they are of the form

5.2.1.
Proof of (101). We shall explain only the fact that lim sup as the remaining terms arising in the proof of (101) can be dealt with in a similar fashion. The random field K + G 2, 0 is of the form where ρ ( ) j ∈ {0, 1} and each f is a complex valued function that belongs to W 1,∞ (T 4 × R) (i.e. it is bounded with all its first partial derivatives). The latter is the result of the fact that γ(k) ≥ γ * for all k ∈ T. Therefore the term under the expectation in (107) can be written as the sum of 8 terms of the form for some σ j k j and using the bound (42) we arrive at the estimate of (109) by the following expression Thanks to the Sobolev embedding we can further estimate this expression by for some constant C > 0 independent of > 0. The expectation of this expression stays bounded for → 0+, so (107) follows.
We have where K is as in (28) and R y, := y |J y,y (y − y)|, R y, := y |J y,y (y − y)|.
Note that for any y = y we havẽ where ∆∂ k H (q, k, q ) := ∂ k H (q, k, q +) − ∂ k H (q, k, q −) . Therefore, using the fact that γ(k) is bounded away from 0, we conclude that sup >0,y R y, + R y, < +∞, and lim Using the definition of the operator K o ξ (t) we conclude, see the proof of Proposition 3.1, that for some coefficients c z1,z2,z3 , c z1,z2,z3 and |z 1 |, |z 2 |, |z 3 | ≤ 2, where z j -s are integers. We show that lim We prove (122) for j = 1, as the argument for j = 2 follows analogously. We can write We shall deal only with the term z 1 = z 2 = z 3 = 0. The other cases are similar. Suppose that m ∈ (0, 1/2). The right hand side of (123) is then estimated by K[sup y (R y, +R y, ) +R ], with R y, := y |J * y,y (y − y)|E |ξ y (t)| , R y, := y |J * y ,y (y − y )|E sup The same argument as in the case of I can be used to prove that sup >0,yR y, < +∞.
The only thing yet to be shown is

TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ
Using formula (120) and integrating by parts in the q variable we conclude that there exists C > 0 for which for > 0. Here the termsR ,1 andR ,2 correspond to the summation over |y − y| ≥ |y|/4 and |y − y| < |y|/4. Let ρ ∈ (0, 1). We can write then that for some constants C , C (cf (20)) Adjusting ρ > 0 in such a way that (1+m)(1−ρ) > 1 we conclude that lim →0+R ,1 = 0. Finally, we can write that for some constant C > 0 we havẽ Since y ≥ −m we have y ≥ ρ y ρ , when ρ ∈ (0, 1/2). We obtain therefore that for some constant C > 0 independent of we havẽ Hence lim →0+ √ R ,2 = 0. We have shown that lim The proof that also lim →0+ III = 0 is analogous. Thus we conclude the proof of (100) and finish in this way the demonstration of Lemma 5.1.

Remarks about derivation of formula (96).
Here we explain how to derive formula (96), used in step 1 of the proof of Lemma 5.1. The issue consists in verifying that the functionals given in (95) satisfy hypotheses (83)-(85) so we can apply Proposition 4.2. Let us focus our attention onḠ 0 , as the arguments for the other functionals are very similar. In case of the term G that appears in the definition ofḠ 0 the verification is trivial because it is constant over E, clearly takes values in H m (as G ∈ S(R × T)) and QG y,y (k) ≡ 0 for each y, y ∈ Z and k ∈ T. Consider therefore the term G 1, 0 . In fact we can take = 1 (as in this case the value of is fixed) and omit writing the superscript in our notation. We can use formula (102) together with (70) to conclude that G 1 0,y,y (k) takes values in the space of the first degree polynomials p 1 (see Section 4.2) for each y, y ∈ Z and k ∈ T. Applying formula (76) we can verify that whereũ ± are given bỹ Calculating the second moment of the · L2 norm of the first term on the right hand side of (125) we conclude that it is of the form T |e −γ(k)t − 1| 2 |f (k)| 2 dk for some function f ∈ L 2 (T). This would prove the condition (83). The verification of the remaining conditions (84) and (85) can be done in an analogous manner. Similarly we can perform the calculations for the second degree polynomials G 2 0,y,y (k), y, y ∈ Z and k ∈ T.

5.4.
The evolution of the energy density in the frequency domain. Let E (t, k) := (1/2) |ψ (t, k)| 2 µ , whereψ (t, k) is the Fourier transform of the wave function ψ y (t) . Recall that the scattering operator L is defined by (45). We have the following analogue of Lemma 5.1.
In the proof of Lemma 5.1 we can use function G λ in place of G and obtain in this way the formula (94) for G λ . Letting λ → 0+, for a fixed > 0, we obtain (127) with Here Ψ 3 ,0 (t) is obtained from (98) by letting q = 0 and collapsing the integration over q. The term Ψ 3 ,0 (t) is therefore a combination of a finite number of the terms of the form: either where functions f are complex valued and belong to W 1,∞ (T 4 ) (due to the fact that γ(k) ≥ γ * ). The argument used in Section 5.  On the other hand, with Concerning the term given by (129), its expectation can be written as a sum of the terms in the form where each J is given by: either or and H,H are given by (114) and (115)   Proof. Suppose that η > 0 is arbitrary. We can find δ > 0 such that for any J ∈ C ∞ (T) such that J(k) ≡ 1 for |k| ≤ δ, J(k) ≡ 0 for |k| ≥ 2δ and 0 ≤ J(k) ≤ 1 otherwise we have Using (127) and then (128) we obtain, cf (28), Thanks to (50) we conclude that lim δ→0+ lim sup and, since η > 0 has been arbitrarily chosen, we conclude (133).

5.5.
The end of the proof of Theorem 2.5. Thanks to the bound (43) we know that (W (t)) is * -weakly compact in A for each t ≥ 0 fixed. Suppose that n → 0+. We prove that lim n→+∞ W n (t) = U (t) * -weakly in A , with U (t) described in the statement of the theorem. It suffices to show that any convergent subsequence of the sequence (which we denote for convenience by the same symbol) has U (t) as its limit. In fact, choosing an appropriate subsequence we may assume that (W n (t)) converges for any non-negative rational t. Suppose that G ∈ S(R × T). Condition (43) implies that for any T > 0 Thanks to (5.1) the bound (136), in turn, implies that the sequence is equicontinuous in C[0, T ]. Due to (43) it is also uniformly bounded in the t variable. Using the above one can easily argue that (W n (t)) converges * -weakly in A for any t ≥ 0. We show that its limit U (t) is the solution of (44) in the sense of definition given in Section 2.7. Thanks to Theorem B. 4, p. 154 of [22] we have U (t) ∈ M + (R × T) for each t ≥ 0. It is an immediate consequence of the construction of U (t) that conditions i) and ii) of Definition 2.2 are satisfied. To prove iii) it suffices only to show that We consider first the acoustic case, i.e. when (a4') holds (then condition (133) holds). Given δ > 0 we let J 0 ∈ C ∞ (T) be such that J 0 (k) ≡ 1 for |k| ≤ δ, J 0 (k) ≡ 0 for |k| ≥ 2δ and 0 ≤ J 0 (k) ≤ 1 otherwise. Let J 1 (k) = 1 − J 0 (k). We can write then where Choose an arbitrary ρ > 0. Thanks to (43) and (133) one can adjust δ > 0 in such a way that lim sup n→+∞ W n (s), J (n) ι1,ι2 provided (ι 1 , ι 2 ) = (1, 1). Due to the assumed C 2 -regularity of σ(·), γ(·) and C ∞ regularity of ω(·) on T \ {0} we conclude that J Since δ > 0 can be adjusted in such a way that we conclude therefore that for any ρ > 0 lim sup for G ∈ A, ι = ±. Thus, This ends the demonstration of (137) concluding in this way the proof of the fact that (U (t)) is the solution of (44), thus identifying the limit of (W n (t)).
In the case when the chain is pinned, i.e. (a4) holds, we can essentially repeat the argument for the convergence in (137) given above. In fact, things get simpler due to the fact that the dispersion relation is C ∞ (T) smooth and we do not need to introduce the partition as in (138). For a given integer N ≥ 1 let θ N y,z (t) be a sequence formed according to (26) from ξ N y (t), given by ξ N y (t) = ξ y (t) when |y| ≤ N 0 when |y| ≥ N + 1.
We define A N (t; ξ(·)) using an analogue of (144), where θ y,z (t) is replaced by θ N y,z (t). Then A N (t)f ∈ C for f ∈ 2 (Z) and A (N ) (t) t∈R are operators acting on 2 (Z) that are uniformly bounded on compact intervals. One can easily check that for each N ≥ 1, all realizations of (ξ(t)) and all g, h ∈ 2 (Z), t ∈ R.
As a consequence of the above lemma and Young's inequality the operator Ω : 2 (Z) → 2 (Z), given by Ωf := −iω * f is bounded. Using a standard theorem on the existence and uniqueness of solutions of ordinary differential equations with a Lipschitz right hand side we conclude that for any N ≥ 1 and ψ 0 ∈ 2 (Z) there exists a unique ψ N ∈ C(R; 2 (Z)) that is differentiable in 2 (Z), such that Our main goal in this section is to show the following.
A.1. Some auxiliary evolution families. Consider the following linear system of equations on 2 (Z) Denote by U N (t, s; ξ(·))Ψ 0 := Ψ N (t). The following result is a simple consequence of the existence and uniqueness result for solutions of equations with Lipschitz coefficients and anti-symmetry of A N (t).
Proposition A.3. For a.s. realization of (ξ(t)) and an arbitrary N ≥ 1 we have In addition, i) U N (t, t) is the identity map on 2 , ii) U N (t, s)U N (s, u) = U N (t, u) for any t, s, u ∈ R, iii) the mapping (t, s) → U N (t, s) is uniformly continuous.
Here |Λ n (R, −R)| is the volume of the symplex, that equals (2R) n /n!. The expectation of the mixed Gaussian 2n-th moment can be expressed as a sum of (2n − 1)!! terms. Each is a product of n suitable covariances between θ y+z1,j−1,zj (τ j ) and θ y+z 1,j −1 ,z j (τ j ) that can be estimated by 2Eξ 2 0 (0). Therefore, the expression in the right hand side of (159) can be estimated by for some constant C > 0 independent of n, and (158) follows.
Using the same argument one can also conclude that E sup |s|,|t|≤R where the constant C > 0 does not depend on N . To prove (157) it suffices therefore to show that for a fixed n we have From the definition of the truncated field θ N y,z it is clear that the expectation under the limit can be estimated by θ N y+z1,j−1,zj (τ j ) where the constant C > 0 may depend on n. Letting N → +∞ we conclude that the expression on the utmost right hand side tends to 0 and (162) follows.
Corollary A.5. For a.s. realization of ξ we can define a strongly continuous evolution family U (t, s) on 2 (Z), i.e. i) U (t, t) is an identity map, ii) U (t, s)U (s, u) = U (t, u) for any t, s, u ∈ R, iii) the mapping (t, s) → U (t, s) is strongly continuous and satisfies U (t, s)Ψ = Ψ(t, s) for all t, s ∈ R, iv) for P a.s. ξ we have U (t + h, s + h; ξ) = U (t, s; θ h ξ), for any t, s, h ∈ R.
Proof. Suppose that G ⊂ 2 (Z) is dense and countable. For each Ψ ∈ 2 (Z) we can find a set N Ψ such that P[N Ψ ] = 0 and the family Ψ(t, s) is defined by (155) for all s, t ∈ R outside N Ψ . Using (157) we can adjust the set so that Thanks to (166) we conclude that (164) holds for all Ψ ∈ G outside N . Operators U (t, s) can be therefore extended to the entire 2 (Z) outside N . In addition, (165) holds ouside N . This and (154) imply properties i)-iv) of U (t, s).
To prove the convergence part of the assertions made in Theorem A.2 it suffices to show that an analogous assertion holds for z N (t) = u N (t) v N (t) .
Let z 0 = z N (0). Note that z N (t) satisfies the following mild formulation of (168) For any J ∈ S and h > 0 define a function t → J h (t) whose Fourier transform (in the first variable) equals J h (t, p, k) := E exp 2πip t 0 ω h (K s (k)) ds J (p, K t (k)) .
It belongs to C 1 ([0, +∞), S). A simple calculation, using (47), shows that The absolute value of the expression on the right hand side of (183) can be estimated by which clearly tends to 0, as h → 0+. As a result we again obtain equality (180), which identifies U (t).
In addition, (49) is a consequence of the fact that, according to the existence part, U (t) is the joint law of a random vector. If, on the other hand, U 0 ∈ L 1 + (R × T) then U (t, x, k) = EU 0 x − t 0 ω (K s (k))ds, K t (k) .
Due to the invariance of the Lebesgue measure under the dynamics of K t (k) we conclude that R×T |U (t, x, k)|dxdk ≤ T dkE R U 0 x − t 0 ω (K s (k))ds, K t (k) dx = T dkE R |U 0 (x, K t (k))| dx = U 0 L 1 (R×T) , which proves that U (t) ∈ L 1 + (R × T).