Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

We consider the long time, large scale behavior of the Wigner transform $W_\eps(t,x,k)$ of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $\gamma_0>0$ such that for any $\gamma\in(0,\gamma_0]$ the weak limit of $W_\eps(t/\eps^{3/2\gamma},x/\eps^{\gamma},k)$, as $\eps\ll1$, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ with $\hat c>0$. In the pinned case an analogous result can be claimed for $W_\eps(t/\eps^{2\gamma},x/\eps^{\gamma},k)$ but the limit satisfies then the usual heat equation.


Introduction
In the present paper we are concerned with the asymptotic behavior of the Wigner transform of the wave function corresponding to a discrete wave equation on a one dimensional integer lattice with a weak multiplicative noise, see (2.1) below. This kind of an equation arises naturally while considering a stochastically perturbed chain of oscillators with harmonic interactions, see [2] and also [18]. It has been argued in [2] that, due to the presence of the noise conserving both the energy and the momentum (in fact the latter property is crucial), in the low dimensions (d = 1, or 2) conductivity of this explicitly solvable model diverges as N 1/2 in dimension d = 1 and log N, when d = 2, where N is the length of the chain. This complies with numerical results concerning some anharmonic chains with no noise, see e.g. [17] and [16]. We refer an interested reader to the review papers [7,16] and the references therein for more background information on the subject of heat transport in anharmonic crystals.
It has been shown in [4] that in the weakly coupled case, i.e. when the coupling parameter ǫ is small, the asymptotics of the Wigner function W ǫ (t, x, k) (defined below by (2.8), with γ = 0), that describes the resolution of the energy in spatial and momentum coordinates (x, k) at time t ∼ ǫ −1 , is given by a linear Boltzmann equation, see (2.9) below. Furthermore, since in the dimension d = 1 the scattering rate of a phonon is of order k 2 for small wavenumber k, in the unpinned case (then the dispersion relation satisfies ω ′ (k) ≈ sign k, for |k| ≪ 1) the long time, large space asymptotics of the solution of the transport 1 2 TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ equation can be described by a fractional (in space) heat equation for someĉ > 0. The initial condition W (0, x) is the limit of the average of the initial Wigner transform over the wavenumbers, see [13] and also [3,19]. Note that the above equation is invariant under time-space scalings t ∼ t ′ /ǫ 3γ/2 , x ∼ x ′ /ǫ γ for an arbitrary γ > 0. This suggests that the fractional heat equation is the limit of the Wigner transform in the above time space scaling. In our first main result, see part 1) of Theorem 2.1 below, we prove that it is indeed the case when γ ∈ (0, γ 0 ] for some γ 0 > 0. On the other hand, in the pinned case, i.e. when the dispersion relation satisfies ω ′ (k) ≈ 0, as |k| ≪ 1, one can show that the solution of the Boltzmann equation approximates the regular heat equation In part 2) of Theorem 2.1 below we assert that if the Wigner transform is considered under the scaling (t, x) ∼ (t ′ /ǫ 2γ , x ′ /ǫ γ ), for γ ∈ (0, γ 0 ] and some γ 0 > 0, then it converges to W (t, x), as ǫ ≪ 1. The coefficientĉ in the heat equation (1.1) differs from the thermal conductivity coefficient calculated explicitly in the pinned case with the help of the Green-Kubo formula in [1], see Theorem 1. In fact, its computation, see formulas (6.12) -(6.13) below, requires solving a Poisson equation (6.14) and does not lead to an explicit formula. Finally, we mention also the results concerning the diffusive limits for the Wigner transform of a solution of the wave equation on a lattice with a random local velocity in the weak coupling regime (see [18]), for the geometric optics regime for the wave equation in continuum (see [15]) and in the case of Schrödinger equation in the radiative transport regime (see [12]). The proof of Theorem 2.1 is made of two principal ingredients: the estimates of the convergence rate for the Wigner transform towards the solution of the kinetic equation, see Theorem 3.2 below in the unpinned case (resp. Theorem 3.5 for the pinned case), and the respective rate of convergence estimates for the solutions of the scaled kinetic equation, see Theorem 3.3 (resp. Theorem 3.6). To prove the latter we show two probabilistic results: Theorems 5.5 and 5.8 that are of interest on their own. They provide estimates of the rate of convergence of the characteristic functions corresponding to a scaled additive functional of a stationary Markov chain towards the characteristic function of an appropriate stable limit and the respective result in the continuous time case.
CONVERGENCE RATES 3 2. Description of the model and preliminaries 2.1. Discrete wave equation with a noise. We consider a discrete wave equation with the multiplicative noise on a one dimensional integer lattice, see [2], x (t). (2.1) Here (p, q) = {(p x , q x ), x ∈ Z}, where the component labelled by x corresponds to the one dimensional momentum p x and position q x . The Hamiltonian corresponds to an infinite chain of harmonic oscillators and is given by The interaction potential {α x , x ∈ Z} will be further specified later on. The noises {ξ (ǫ) x (t), x ∈ Z} are defined by the following stochastic differentials dξ (ǫ) x (t) = √ ǫ k=−1,0,1 (Y x+k p x ) • dw x+k (t), (2.2) understood in the Stratonovich sense. Here and {w x (t), t ≥ 0}, x ∈ Z are i.i.d. standard, one dimensional Brownian motions over a certain probability space (Ω, F , P). Note that the vector field Y x is tangent to the surfaces and p x−1 + p x + p x+1 ≡ const (2.4) therefore the system (2.1) conserves the total energy and momentum. System (2.1) can be rewritten formally in the Itô form: dq y (t) = p y (t)dt (2.5) dp y (t) = −(α * q(t)) y − ǫ 2 (β * p(t)) y dt, (Y y+k p y (t))dw y+k (t), y ∈ Z.

2.2.
Formulation of the main results. To describe the distribution of the energy of the chain over the position and momentum coordinates it is convenient to consider the Wigner transform of the wave function corresponding to the chain. Adjusting the time variable to the macroscopic scale it is defined as Hereω is the inverse Fourier transform, see (2.17), of the dispersion relation function given by ω(k) = α(k), withα(k) the direct Fourier transform of the potential, defined on T-the one dimensional torus, see (2.16). Suppose that the initial condition in (2.1) is random, independent of the realizations of the noise and such that for some γ > 0 lim sup Here · ǫ denotes the average with respect to the probability measure µ ǫ corresponding to the randomness in the initial data. In fact, since the total energy of the system y∈Z |ψ (ǫ) y (t)| 2 is conserved in time, see Section 2 of [4], an analogue of condition (2.7) holds for any t > 0.
The (averaged) Wigner transform of the wave function, see [4], is a distribution defined as follows for anyJ belonging to S -the Schwartz class of functions on R × T, see Section 2.4. Here E ǫ is the average with respect to the product measure µ ǫ ⊗P. It has been shown in [4], see Theorem 5, that, under appropriate assumptions on the potential α(·), see conditions a1) and a2) below, the respective Wigner transforms W ǫ,0 (t) converge in a distribution sense, CONVERGENCE RATES 5 as ǫ → 0+, to the solution of the linear kinetic equation where L is the scattering operator defined in (2.35). Our principal result concerning this model deals with the limit of the Wigner transform in the longer time scales, i.e. when γ > 0. It is a direct consequence of Theorems 3.1 and 3.4 formulated below that contain also the information on the convergence rates. Before its statement let us recall the notion of a solution of a fractional heat equation. Assume that W 0 is a function from the Schwartz class on R. The solution of the Cauchy problem for the fractional heat equation is the Fourier transform of W 0 (x).

15)
where W (t, x) is the solution of the ordinary heat equation, i.e.
The above assumptions imply that both y → α y and k →α(k) are real valued, even functions. In additionα ∈ C ∞ (T) and ifα(0) = 0 then α(k) = k 2 φ(k 2 ) for some strictly positive φ ∈ C ∞ (T). This in particular implies that, in the latter case, the dispersion relation ω(k) = α(k) belongs to C ∞ (T \ {0}). It can be easily checked that under the hypotheses made about the potential α y the mapping given by (2.19) is Lipschitz from L 2 C (T) to itself and r∈Z Q[g](e r ) 2 L 2 ≤ C g 2 L 2 for some C > 0 and all g ∈ L 2 C (T) so Q[g] is Hilbert-Schmidt. Using Theorem 7.4, p. 186, of [9] one can show that for any L 2 there exists a unique solution to (2.18) understood as L 2 C (T) -valued, continuous trajectory, adapted process {ψ (ǫ) (t), t ≥ 0} a.s. satisfying (2.18). In addition, see Section 2 of [4], for every initial dataψ Some function spaces. Denote by S the set of functions J : R × T → C that are of C ∞ class and such that for any integers l, m, n we have sup p,k (1 + p 2 ) n/2 |∂ l p ∂ m k J(p, k)| < +∞. For any a ∈ R we introduce the norm for a given J ∈ S. By A ′ a we denote the completion of S in the norm · A ′ a . Note that A ′ a is dual to A a defined as the completion of S in the norm (2.23) We use a shorthand notation A := A 0 and A ′ := A ′ 0 . With some abuse of notation by ·, · we denote the scalar product in L 2 C (T) and the extension of We shall also use the space B a,b obtained by completion of S in the norm Random and average Wigner transform. For a given ǫ > 0 letψ (ǫ) (t) be a solution of (2.18) with a random initial conditionψ (ǫ) (0) distributed according to a probability measure µ ǫ on L 2 C (T). Define where, as we recall, · ǫ is the average with respect to the initial condition. Using (2.7) and (2.21) we conclude that both W ǫ (t) and Y ǫ (t) belong to L 1 (P; A) -the space of A-valued random elements possessing the absolute moment, for any t ≥ 0. We also introduce the average objects W ǫ (t, p, k) and Y ǫ (t, p, k) using formulas analogous to (2.25) and (2.26) with · ǫ replaced by E ǫ corresponding to the average over both the initial data and realization of Brownian motion.
The (averaged) Wigner transform W ǫ,γ (t) is defined as The anti-transform Y ǫ,γ (t) is defined by an analogous formula, with W ǫ replaced by Y ǫ .
In order to guarantee that the stochastic integrals defined above are martingales, not merely local ones, we need to make an additional assumption that µ ǫ has the 4-th absolute moment. Taking the expectation of both sides of (2.28) with respect to the realizations of the Brownian motion we conclude that W ǫ (t, p, k) = E W ǫ (t, p, k) satisfies and Formula (2.30) remains valid also when only the second absolute moment exists. This can be easily argued by an approximation of the initial condition by random elements that are deterministically bounded.
Since the momentum, that is the inverse Fourier transform ofp (ǫ) (t, k), is real valued, the expression under the expectation appearing on the right hand side of (2.30) is an even function of k ′ , thus the last term appearing on the right hand side of the equation can be replaced by Note that The following relation holds We conclude therefore that W ǫ (t, p, k) satisfies the following , for any f ∈ S. We let ω ′ (0) := 0 in case ω is not differentiable at 0. In addition, where for each ǫ ∈ [0, 1] operator L (ǫ) acts on S according to the formula and extends to a bounded operator on either A, or A ′ . Finally,

Probabilistic interpretation of the kinetic linear equation.
Denote by U (t, p, k) the Fourier transform of the solution of (2.9) in the x variable. Let K t (k) be a T-valued, Markov jump process, defined over (Ω, F , P), starting at k, with the generator L. Suppose also that is understood as a continuous A-valued function U (t) such that for all J ∈ A ′ . It is well known that this solution admits the following probabilistic representation Here E is the expectation over P. For given J ∈ A ′ we let Therefore J(t) ∈ A ′ . Using the reversibility of the Lebesgue measure under the dynamics of K t , we conclude that the law of Likewise, using the definition of R ǫ (t) (see (2.37)), from (2.34) and the Duhamel formula we get

Convergence of the Wigner transform
For a given a ∈ R define the norm Recall also that µ ǫ is the distribution of the initial data for equation (2.18). We assume that: A a ) for a given a > 0 part 1) of Theorem 2.1 is a consequence of the following result.
Theorem 3.1. Assume that t 0 > 0,α(0) = 0 and a ∈ (0, 1] is such that (3.1) holds. Then, for any γ ∈ (0, 2a/3), and b > 1 one can find C > 0 such that Here Denote the terms appearing above by J 1 and J 2 respectively. The proof is made of two principal ingredients: the estimates of the convergence rates of the averaged Wigner transform of the wave function given by (2.18) to the solution of the linear equation (2.40) (that correspond to the estimates of J 1 ) and further estimates for the long time-large scale asymptotics of these solutions (these will allow us to estimate J 2 ). To deal with the first issue we formulate the following.

TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ
for all J ∈ S, ǫ ∈ (0, 1] and t ≥ 0. We postpone the proof of this theorem until Section 4.1, proceeding instead with estimates of J 1 . Using (3.7) we can write that for every ǫ ∈ (0, 1] and b ≥ 0, we obtain the last two terms on the right hand side of (3.8) account for the last two terms on the right hand side of (3.3).
Denote the first term on the right hand side of (3.8) by I. We can write that I ≤ I 1 + I 2 , where Here for any function f : Term I 1 accounts for the first term on the right hand side of (3.3).
Using the reversibility of the Lebesgue measure under the dynamics of K t (see (2.44)), we conclude that To estimate I 2 (and then further to estimate J 2 ) we need a bound on the convergence rate of the scaled functionals of the form (2.42). Let The proof of this result shall be presented in Section 6.1. Using the above theorem we can estimate The above estimates account for the second term on the right hand side of (3.3), thus concluding the proof of the estimate in (3.3).
3.2. Pinned case. Part 2) of Theorem 2.1 is a direct consequence of the following result.
Proof. We proceed in the same fashion as in the proof of Theorem 3.1 so we only outline the main points of the argument. First, we estimate the left hand side of (3.12) by an expression corresponding to (3.6). The first term is estimated by an analogue of Theorem 3.2 that in this case can be formulated as follows.
Theorem 3.5. Assume that conditions a1)-a2) and A 0 ) hold. In addition, we letα(0) > 0. Then, the average Wigner transform W ǫ (t) satisfies the following: there exists a constant C > 0 such that The proof of this result is presented in Section 4.2. In the next step we can estimate the rate of convergence of the functional appearing in the formula for the probabilistic solution of the linear Boltzmann equation towards the solution of the heat equation given in the following theorem. Let for all ǫ ∈ (0, 1], t ≥ t 0 , W ∈ B a and J ∈ A ′ 4 ∩ B a,b . The proof of the above theorem is contained in Section 6.2. The remaining part of the argument follows the argument of Section 3.1. with R ǫ (t) given by (2.37). Estimates of the last term on the right hand side above shall be done separately for each term appearing on the right hand side of (2.37).

TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ
for some constant C > 0. Using the above estimate we obtain By virtue of (4.1) we get that Taking the Taylor expansion of R ǫ (p, k, k ′ ), up to terms of order ǫ, it is also straightforward to conclude that Summarizing, (4.3), (4.5) and (4.6) together imply ǫ . Straightforward computations, taking the Taylor expansions ofβ(k − ǫp/2) and R ǫ (p, k, k ′ ) up to ǫ, show that From (2.39) we conclude the following estimate.
Lemma 4.2. Suppose that φ ǫ : R × T → R is such that Then, there exists C > 0 such that for all J ∈ S, ǫ ∈ (0, 1], t > 0. Proof. The left hand side of (4.9) can be rewritten as where Γ ǫ (p, k) := φ ǫ (p, k)ω −1 ǫ (p, k). Using (2.39) we can estimate this expression by (4.11) Thanks to (2.21) and (4.8) the second term is bounded by ǫΓ * J A ′ K γ t. On the other hand, integration by parts allows us to estimate the first one by by virtue of (2.21) and (4.8). Since from the above lemma we conclude that This ends the proof of (3.7).

4.2.
Proof of Theorem 3.5. We maintain the notation from the argument made in the previous section. Estimate (4.4) can be improved since in this case there exists C > 0 such that |∆ω ǫ (p, k)| ≤ Cǫ|p| for all (p, k), ǫ ∈ (0, 1]. Thus, we can write for all t ≥ 0. With this improvement in mind we conclude that there is a constant C > 0 such that for all t ≥ 0, ǫ ∈ (0, 1]. Repeating the argument from the proof of Theorem 3.2 for the term corresponding to R Since P is also a contraction in L 1 (π) and L ∞ (π) we conclude, via Riesz-Thorin interpolation theorem, that for any β ∈ [1, +∞): for all f ∈ L β 0 (π) -the subspace of L β 0 (π) consisting of functions satisfying f dπ = 0, with κ(β) Furthermore, we assume the following regularity property of transition of probabilities: Condition 5.2. (existence of bounded probability densities w.r.t. π) transition probability is of the form P (w, dv) = p(w, v)π(dv), where the kernel p(·, ·) belongs to L ∞ (π ⊗ π).

5.2.
Convergence of additive functionals. Suppose that Ψ : E → R satisfies the tail estimate for some C > 0 and all λ ≥ 1 and We wish to describe the behavior of tail probabilities P Z (N ) t ≥ N κ when κ > 0 for the scaled partial sum process Z (N ) t To that end we represent Z (N ) t as a sum of an L β integrable martingale for β ∈ [1, α) and a boundary term vanishing with N. Let χ be the unique solution, belonging to L β 0 (π) for β ∈ [1, α), of the Poisson equation In fact using Condition 5.2 we conclude that P χ ∈ L ∞ (π). Therefore the tails of χ and Ψ under π are identical. We introduce an L β integrable martingale letting: M 0 := 0, and the respective partial sum process M (N ) t , t ≥ 0. Using the dual version of Burkholder inequality for L β integrable martingales, when β ∈ (1, 2), see Corollary 4. 22, p. 101 of [20] (and also [14]) we conclude that there exists C > 0 such that Proof. Choose β ∈ [1, α). We can write From (5.10) and Chebyshev's inequality we can estimate the left hand side of (5.9) by The last inequality follows from (5.8). Choosing β sufficiently close to α we conclude the assertion of the lemma.

(5.13)
In what follows we shall prove the following.
Proof. In the first step we replace an additive functional of the chain by a martingale partial sum process. Using (5.10) we conclude that there exists C > 0 such that p ∈ R and (5.14) shall follow as soon as we can show that for any δ ∈ (0, α/(α + 1)) there exist C > 0 such that for all p ∈ R, t ≥ 0, N ≥ 1. The remaining part of the argument is therefore devoted to the proof of (5.15). Denote Z N,n := N −1/α Z n with Z n defined in (5.7) and χ the solution of Poisson equation (5.6) with the right hand side equal to Ψ. Introduce also Lemma 5.6. There exists a constant C > 0 such that In addition, for any bounded set ∆ ⊂ R and β ∈ [1, α) there exists Proof. DenoteΨ(w, v) := Ψ(v) + P χ(v) − P χ(w). With this notation the expression on the left hand side of (5.17) can be rewritten as Here F N (λ) := π(Ψ ≤ N 1/α λ). The first term on the right hand side can be estimated by where ∆ is a bounded interval containing all possible values of P χ(v) − P χ(w). This expression can be further estimated by As for the second term on the right hand side of (5.19), using integration by parts, we conclude that it can be bounded by The first estimate follows from (5.11), while the last one follows upon the change of variables λ ′ := λp and the fact that 1 < α + α 1 < 2.

Concerning (5.18) expression appearing there can be estimated by
The first term is estimated by for some C > 0 and all p ∈ R, N ≥ 1. Since ∆ is bounded the second term on the other hand is smaller than Hence, (5.18) follows. In fact, in light of the above lemma to prove the theorem it suffices only to show that for any δ ∈ (0, α/(α + 1)) one choose C > 0 so that for all p ∈ R, t ≥ 0, N ≥ 1. To shorten the notation we let M j,N := M j /N 1/α . Since {M n , n ≥ 0} is adapted and E[Z N,n+1 | F n ] = 0, we can write To derive a recursive formula for Since M 0 = 0, adding up from j = 0 up to [Nt] − 1 and then dividing both sides of obtained equality by exp{ψ (N ) (p)[Nt]/N} we obtain that We denote the terms appearing on the right hand side of (5.24) by I and II and examine each of them separately. As far as II is concerned we bound its absolute value by Here [a] stands for the integer part of a ∈ R. To simplify the notation we shall assume that K ′ = K. This assumption does not influence the asymptotics.
We need to estimate the absolute value of The conditional expectation in the formula for I 1 equals The supremum of its absolute value can be estimated by for some constant C > 0. Here ∆ is a bounded set containing 0 and all possible values of P χ(w) − P χ(z) for z, w ∈ E. The last inequality follows from Lemma 5.6. On the other hand, since the real part of log e N,j is non-positive we have According to (5.18) the first term in parentheses can be estimated by C|p|(1+|p|)K/N. Choose β ∈ (1, α). From (5.8) and Doob's inequality we can estimate the second term by CK 1/β /N 1/α . Summarizing we have shown that On the other hand,
From the above we get, that the optimal rate of convergence is obtained when β is as close to α as possible, and for each δ 1 < α/(α + 1) we can choose then C > 0 so that |I| ≤ C(t + 1) N δ 1 (1 + |p|) 5 . Taking into account the above and (5.26) we conclude (5.23).

Convergence rates in the central limit theorem regime.
We maintain assumptions made about the chain in the previous section. This time however we assume instead of Condition (5.4) the following.
We define the martingale approximation, using (5.7) from the previous section. Decomposition (5.10) remains in force. Instead of (5.8) we have then an inequality where C > 0 is independent of N ≥ 1. Let and ψ(p) := σ 2 p 2 /2.

TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ
Define the partial sum process Z (N ) t by (5.5) with α replaced by 2. Our main result concerning the convergence of characteristic functions reads as follows.
Theorem 5.8. There exist C > 0 such that Proof. Denote Z N,n := N −1/2 Z n with Z n defined in (5.7) and M j,N := M j /N 1/2 . Using this notation as well as the notation from the previous section we can write that Let h p (x) be given by (5.16). We denoteh where e N,j := exp{(σp) 2 (j + 1 − [Nt])/(2N)}e ipM j /N 1/2 . Denote the absolute value of the terms appearing on the right hand side by I and II respectively. We can easily estimate II ≤ C tp 4 N for some constant C > 0 and all t ≥ 0, N ≥ 1 and p ∈ R.
To estimate I we invoke the block argument from the previous section. Fix K ≥ 1 and divide the set Λ N = {0, . . . , N − 1} in ℓ = [[Nt]/K] + 1 contiguous subintervals, ℓ of size K and the last one of size K ′ ≤ K. In fact for simplicity sake we just assume that all intervals have the same size and maintain the notation introduced in the previous section. Estimate (5.27) remains in force with the obvious adjustments needed for α = 2. Repeating the argument leading to (5.28) with (5.29) used in place of (5.8) we conclude that On the other hand, Choosing K = N 1/3 in the above estimates we conclude (5.31).

5.4.
Convergence rates for additive functionals of jump processes. Assume that {τ n , n ≥ 0} are i.i.d. exponentially distributed random variables with Eτ 0 = 1 that are independent of the Markov chain {ξ n , n ≥ 0} considered in the previous section. Suppose furthermore that V, θ are Borel measurable functions on E such that: there exist C * > 0 and α 2 > 1 such that We shall also assume that either Ψ(w) := V (w)θ(w) satisfies Condition 5.4, or it satisfies Condition 5.7.
We also define processes with β = α when this condition holds and β = 2, in case Condition 5.7 is in place.
Consider the stable process {Z t , t ≥ 0} whose Levy exponent is given by (5.12) with c * (λ) replaced bỹ Let {B t , t ≥ 0} be a zero mean Brownian motion whose variance equalŝ The aim of this section is to prove the following.
Theorem 5. 10. In addition to the assumptions made in Section 5.1 suppose that Condition 5.9 holds. Then, for any 0 < δ < δ * := min[α/(α + 1), (α 2 − 1)/(αα 2 + 1)] (5.41) there exists C > 0 such that If, on the other hand the assumptions made in Section 5.3 and Condition 5.9 hold then for any 0 < δ < δ * := (α 2 − 1)/(1 + 2α 2 ) (5.43) there exists C > 0 such that The proof of this result is carried out below. It relies on Theorem 5.5. The principal difficulty is that definition of Y (N ) t involves a random sum instead of a deterministic one as in the previous section. This shall be resolved by replacing n N t appearing in the upper limit of the sum in (5.37) by the deterministic limit [N t], whereN := N/θ, that can be done with a large probability, according to the lemma formulated below.
5.4.1. From a random to deterministic sum.
Lemma 5.11. Suppose that κ > 0. Then, for any δ ∈ (0, α 2 κ) there exists C > 0 such that Let κ ′ ∈ (0, κ) be arbitrary and We adopt the convention of summing up to the largest integer smaller than, or equal to the upper limit of summation. Note that on A + N we have Hence,

TOMASZ KOMOROWSKI AND LUKASZ STȨ PIEŃ
To estimate the probability appearing on the utmost right hand side we apply Lemma 5.3 for the Markov chain {ξ n , n ≥ 0} and Ψ(w, τ ) := θ(w)τ −θ. We conclude therefore that for any δ ∈ (0, α 2 κ ′ ) there exists C > 0 such that Since Eτ 0 = 1 and κ ′ ∈ (0, κ) for any x ∈ (0, 1) we can find C > 0 such that The last inequality follows from the large deviations estimate of Cramer, see e.g. Theorem 2.2.3 of [10]. Using this and (5.47) we get Probability P[A − N ] can be estimated in similar way. Instead of C N we consider the event and carry out similar estimates to the ones done before.

5.4.2.
Proof of (5.42). Choose any κ > 0. We can write with A N defined in (5.46). The last term on the right hand side can be estimated by the expression appearing on the right hand side of (5.42), by virtue of Theorem 5.5.
The first term on the right hand side can be estimated by C(t+1)N −δ for some δ ∈ (0, α 2 κ) and C > 0. The second term is less than, or equal to |p|(EI N + EJ N ), where and Proof. First we prove (5.53). SinceΨ(ξ 0 ) is L β integrable we can write for any β ∈ (1, α) Choosing β sufficiently close to α we conclude (5.53). Now we prove (5.52). Again we can use martingale decomposition andχ(·) is unique, zero mean, solution ofχ −P χ =Ψ, withP the transition operator for the chain {ξ n , n ≥ 0}. Using stationarity we can bound Denote the terms on the right hand side by I We use again (5.8) and conclude that Summarizing from the above estimates we get where δ is as in the statement of the lemma.

5.4.3.
Proof of (5.44). In this case we can still write inequality (5.49). With the help of Lemma 5.11, for any κ > 0 and δ ∈ (0, α 2 κ) we can find C > 0 such that where I N , J N are defined by (5.50) and (5.51) respectively, with α = 2. repeating the estimates made in the previous section we obtain that for some C > 0. Using the above estimates and (5.30) we conclude (5.44).
6. Proofs of Theorems 3.3 and 3.6 6.1. Proof of Theorem 3.3. Let N := ǫ −3γ/2 and J ∈ A be a real valued function. Define (6.1) where {K t (k), t ≥ 0} is the Markov jump process starting at k, introduced in Section 2.6. It can be easily verified that the Lebesgue measure on the torus is invariant and reversible for the process and we denote the respective stationary process by {K t , t ≥ 0}. Its generator L is a symmetric operator on L 2 (T) given by e ι , f e −ι (k), ∀ f ∈ L 2 (T).
The process K t (k) is a jump Markov process of the type considered in Section 2.6. The mean jump time and the transition probability operator of the skeleton Markov chain {ξ n , n ≥ 0} are given by θ(k) = R −1 (k) and respectively. Probability measure π(dk) = (1/2)r(k)dk is reversible under the dynamics of the chain. It is clear that Condition 5.2 holds. It has been shown in Section 3 of [13] that Condition 5.1 is satisfied. Let {Q t , t ≥ 0} be the semigroup corresponding to the generator L. It can easily be argued that Q t is a contraction on L p (T) for any p ∈ [1, +∞]. We shall need the following estimate.
Theorem 6.1. For a given a ∈ (0, 1] there exists C > 0 for all f ∈ B a such that T f dk = 0. The proof of this result shall be presented in Section 6.3. We proceed first with its application in the proof of Theorem 3.3. The additive functional appearing in (6.1) shall be denoted by in case it corresponds to the stationary process K t . It is of the type considered in Section 5.4 with α = 3/2 and Ψ(k) := ω ′ (k)θ(k). (6.3) Since the dispersion relation satisfies we have and some C * > 0. Condition 5.4 is therefore satisfied with α = 3/2 and arbitrary α 1 < 1/2. Since Ψ(k) is odd we have c − * = c + * . On the other hand, jump mean time θ(k) satisfies (5.35) with α 2 = 3/2.
We can apply to Y (N ) t the conclusion of part 1) of Theorem 5.10. In this casec * (λ) ≡ĉ, wherê Since the integral on the right hand side equals 4 √ π/3 we obtain (3.5). Let Z t be the corresponding symmetric 3/2-stable process. From (3.9) we obtain W (t, p) = W (p)Ee −ipZt . Therefore, for any J(·) ∈ S Let 1 > β > 1/3. The left hand side of (6.5) is estimated by dpdk , and W p (k) := W (p, k) − W (p). The first term can be estimated as follows To estimate the second term note that by the Markov property we can write Term E 2 can be therefore estimated by Invoking Theorem 6.1 we obtain As a result we conclude immediately that To deal with E 3 note that by reversibility of the process K t it equals whereJ (p) := T J(p, k)dk. We obtain that From this point on handle this term similarly to what has been done before and obtain that and provided b > 1. Term E 33 can be handled with the help of Theorem 5.10 with α = α 2 = 3/2 and an arbitrary α 1 < 1/2. Therefore, for any δ < 2/13 we can find a constant C > 0 such that We have reached therefore the conclusion of Theorem 3.3 with the exponent γ ′ as indicated in (3.2).
The asymptotics of I 4,1 for t ≫ 1 is, up to a term of order f L 1 (T) /t, the same asĨ for sufficiently large t (the support of F is contained in (−̺, ̺)). The first term on the utmost right hand side can be estimated by |ĝ 0 (4i(ν + π/t)/3)| dν ≤ C t 2̺ −2̺ T |f |dνdk r a |ν + π/t| 1−a dk ≤ C t f Ba .