Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions
aa r X i v : . [ m a t h - ph ] F e b SUPERBALLISTIC AND SUPERDIFFUSIVE SCALING LIMITS OFSTOCHASTIC HARMONIC CHAINS WITH LONG-RANGEINTERACTIONS
HAYATE SUDA
Abstract.
We consider one-dimensional infinite chains of harmonic oscillators withrandom exchanges of momenta and long-range interaction potentials which have poly-nomial decay rate ∣ x ∣ − θ , x → ∞ , θ > x ∈ Z is the interaction range. The dynamicsconserve total momentum, total length and total energy. We prove that the systemsevolve macroscopically on superballistic space-time scale ( yε − , tε − θ − ) when 1 < θ < ( yε − , tε − √− log ( ε − ) − ) when θ =
3, and ballistic space-time scale ( yε − , tε − ) when θ >
3. Combining our results and the results in [10], we show the existence of twodifferent space-time scales on which the systems evolve. In addition, we prove scalinglimits of recentered normal modes of superballistic wave equations, which are analoguesof Riemann invariants and capture fluctuations around characteristics. The space-timescale is superdiffusive when 2 < θ ≤ θ > Introduction
In recent years, one-dimensional stochastic harmonic chains have been considered asgood approximations of some non-random anharmonic chains and mostly studied fromthe point of view of anomalous heat transport and corresponding superdiffusion of energy[1, 2, 5], see also the review [7, Chapter 5]. Roughly speaking, the model is definedas Hamiltonian dynamics perturbed by momentum-exchange noise which acts only onmomenta locally: ⎧⎪⎪⎪⎨⎪⎪⎪⎩ dq x ( t ) = p x ( t ) dtdp x ( t ) = − ∑ z ∈ Z α x − z q z ( t ) dt + γS ( p x − ( t ) , p x ( t ) , p x + ( t ) ; dw x − ( t ) , dw x ( t ) , dw x + ( t )) . Here p x stands for the momentum of the particle x ∈ Z , and q x for the position of theparticle x ∈ Z . Typical assumptions of interaction potential α x , x ∈ Z are as follows: ( a. ) α x ≤ x ∈ Z ∖ { } , α x ≠ x ∈ Z . ( a. ) α x = α − x for all x ∈ Z . ( a. ) There exists some positive constant C > ∣ α x ∣ ≤ Ce − ∣ x ∣ C for all x ∈ Z . ( a. ) ˆ α ( k ) > k ≠ α ( ) = , ˆ α ′′ ( ) > α is the discrete Fourier transform defined asˆ α ( k ) ∶= ∑ x ∈ Z α x e − π √− xk , k ∈ T , and T = [ − , ) is the one-dimensional torus. Especially, exponential-decay interactionproportional to interaction range is assumed. In addition, { w x ( t ) ; x ∈ Z , t ≥ } are i.i.d.standard Brownian motions, γ > S , and S is added to destroy infinite number of local conservation law, but to conserve to-tal momentum ∑ x p x , total length of the system ∑ x r x , r x ∶= q x − q x − and total energy Key words and phrases. stochastic harmonic chain, long-range interaction, superballistic wave equa-tion, superdiffusion, fractional diffusion equation. x e x , e x ∶= ∣ p x ∣ − ∑ z ∈ Z α x − z ( q x − q z ) formally. We will give rigorous definition of S inSection 3. The physical meaning of r x , x ∈ Z is the inter-particle distance or tensionbetween neighbor particles. We call this model exponential decay model .In this paper, we consider the models which have strong long-range interaction, thatis, our interaction potential is given by α x ∶= − ∣ x ∣ − θ , x ∈ Z ∖ { } α ∶= ∑ x ∈ N ∣ x ∣ − θ θ > . Our model does not satisfy the conditions ( a. ) . In addition, if θ ≤ ( a. ) is notsatisfied because ∑ x ∈ Z ∣ x ∣ − θ = ∞ and we can not define α ′′ as a continuous function on T .From this polynomial decay model, we obtain following new phenomena for stochasticharmonic chains: Two different space-time scales without stochastic perturbation.:
In [6], theauthors pointed out an interesting feature of exponential decay models: there ex-ists two different equilibrium states and corresponding different space-time scalesof the conserved quantities due to the stochastic perturbation. One is called me-chanical equilibrium , which means that the macroscopic profiles of momenta andtensions are constant, and the ohter is called thermal equilibrium , which meansthat the macroscopic profile of energy is constant. At first, the system go tomechanical equilibrium at ballistic scaling. Actually, they prove the weak conver-gence of the scaled empirical measure of ( p x ( t ) , r x ( t ) , e x ( t )) :lim ε → ε ∑ x ∈ Z E ε [⎛⎜⎝ p x ( tε ) r x ( tε ) e x ( tε )⎞⎟⎠] ⋅ ⎛⎜⎝ J ( εx, t ) J ( εx, t ) J ( εx, t )⎞⎟⎠ = ∫ R dy ⎛⎜⎝ ¯ p ( y, t ) ¯ r ( y, t ) ¯ e ( y, t )⎞⎟⎠ ⋅ ⎛⎜⎝ J ( y, t ) J ( y, t ) J ( y, t )⎞⎟⎠ for any test function J i ∈ C ∞ ( R × [ , ∞ )) , i = , , ( ¯ p, ¯ r ) is the solution ofthe linear wave equation ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∂ t ¯ r ( y, t ) = ∂ y ¯ p ( y, t ) ∂ t ¯ p ( y, t ) = ˆ α ′′ ( ) π ∂ y ¯ r ( y, t ) . (1.1)In addition, ¯ e ( y, t ) ∶ = ( ¯ p ( y, t ) + ˆ α ′′ ( ) π ¯ r ( y, t )) + T ( y ) and T ( y ) is called themacroscopic temperature profile. After the system reaches mechanical equilib-rium, then ( p, r ) terms are macroscopically constant, but the energy will evolvein time. At superdiffusive scaling the energy distribution converges weakly,lim ε → ε ∑ x ∈ Z ∫ ∞ dt E ε [ e x ( tε / )] J ( εx, t ) = ∫ R dy ∫ ∞ dt T ( y, t ) J ( y, t ) for any J ∈ C ∞ ( R × [ , ∞ )) and the limit point T ( y, t ) satisfies a 3 / ∂ t T ( y, t ) = − ( α ′′ ( )) / / ( γ ) / ( − ∆ ) T ( y, t ) T ( y, ) = T ( y ) , where γ > ( p, r ) into two terms ( p = p ′ + p ′′ , r = r ′ + r ′′ ) , where ( p ′ , r ′ ) is called thermal terms and ( p ′′ , r ′′ ) phononic terms , andeach term have different initial distributions, called thermal type and phononictype . A phononic type distribution means that there is no thermal energy, and thermal type distribution means that the system is at mechanical equilibrium,see [6, Section 2]. The role of the stochastic perturbation is very crucial for theexponential decay model. Actually, if there is no stochastic perturbation, whichmeans that γ =
0, then the macroscopic energy transport is purely ballistic.For polynomial decay models, in [10] we show that the thermal energy evolveson a superdiffusive space-time scale and the time evolution law of the macroscopicthermal energy W ( y, t ) , y ∈ R , t ≥ θ > Theorem (Theorem 1 in [10]) . Under some initial conditions, the following weakconvergence holds for any test function J ∈ C ∞ ( R × [ , ∞ )) : lim ε → ε ∑ x ∈ Z ∫ ∞ dt E ε [ e x ( tf θ ( ε ) )] J ( εx, t ) = ∫ R dy ∫ ∞ dt W ( y, t ) J ( y, t ) , where f θ ( ε ) is the time scaling which defined as f θ ( ε ) ∶ = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ε − θ < θ < , ∣ h ( ε )∣ θ = ,ε θ > , and h ( ⋅ ) is the inverse function of y ↦ ( y − log ( y ) ) on [ , ) . In addition, W ( y, t ) is the solution of the following fractional diffusion equation: ∂ t W ( y, t ) = ⎧⎪⎪⎨⎪⎪⎩ − C θ,γ ( − ∆ ) − θ W ( y, t ) < θ ≤ , − C θ,γ ( − ∆ ) W ( y, t ) θ > , where C θ,γ is a positive constant defined as C θ,γ ∶ = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
24 csc ( ( − θ ) π − θ ) − θ ( ( θ − ) ) − θ γ − θ − − θ C ( θ ) − θ < θ ≤ , √ γ − C ( θ ) θ > . and C ( θ ) is defined in Appendix A in this paper. In the present paper, we consider the scaling limits for phononic terms of polyno-mial decay models. We explain our main results in the latter half of Introduction.We emphasize that by combining [10, Theorem 1, 2], Theorem 1 and Theorem 2in this paper, we verify that if 2 < θ ≤
3, then there exists two different space-timescales regardless of the existence of stochastic perturbation. Especially, the fasterscaling is superballistic scaling and the other is ballistic or superdiffusive scaling.That is, the long-range interaction decomposes the energy into two terms in thesense of space-time scale.Before we explain Theorem 1, 2 and show the limiting equations with super-ballistic space-time scaling, we will introduce new variable, which is dual variableof the momentum for polynomial decay models.
New conserved quantity for polynomial decay models:
For our model, r x = q x − q x − , x ∈ Z is no longer the proper dual variable for p x , x ∈ Z because r x onlycaptures the tension between neighbor particles. Actually, when θ ≤
3, then wecan not derive the limiting equations for the variables ( p x , r x ) . This leads us to ntroduce the generalized tension, denoted by l x , x ∈ Z , which is formally definedas l x = H Z ( ˇ ω ∗ q ) x , where ˇ ω ∶ Z → R is the inverse Fourier transform of ω ( k ) ∶ = √ α ( k ) , k ∈ T , and H Z is the Hilbert transform on Z defined in Section 2. We see that ∑ x l x is formallyconserved, and if θ > r x , l x are macroscopically equal up to a constantmultiple, see Corollary 4.1. Hence we can think that l x is a natural generalizationof r x . One of our results is the weak convergence of the scaled empirical measureof ( p x ( t ) , l x ( t )) . The time scaling j θ ( ε ) is given by j θ ( ε ) ∶ = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ε θ − , < θ < ,ε ( − log ε ) , θ = ,ε, θ > . (1.2)In Theorem 1 we show the scaling limit of the scaled dynamics {( p x ( tj θ ( ε ) ) , l x ( tj θ ( ε ) )) ; x ∈ Z , t ≥ } with initial distribution which satisfies the phononic type condition (4.1).Especially, Theorem 1 implies the weak convergence of the scaled empirical mea-sure of ( p x ( tj θ ( ε ) ) , l x ( tj θ ( ε ) )) , that is,lim ε → ε ∑ x ∈ Z E ε [( p x ( tj θ ( ε ) ) l x ( tj θ ( ε ) ) )] ⋅ ( J ( εx ) J ( εx )) = ∫ R dy ( ¯ p ( y, t ) ¯ l ( y, t ) ) ⋅ ( J ( y ) J ( y )) (1.3)for any t ≥ J i ∈ C ∞ ( R ) , i = ,
2. The macroscopic dy-namics {( ¯ p ( y, t ) , ¯ l ( y, t )) ; y ∈ R , t ≥ } is the solution of the following superballistic wave equation ⎧⎪⎪⎨⎪⎪⎩ ∂ t ¯ l ( y, t ) = √ C ( θ )D θ ¯ p ( y, t ) ∂ t ¯ p ( y, t ) = √ C ( θ )D θ ¯ l ( y, t ) , (1.4)where C ( θ ) is a positive constant defined in Appendix A, D θ is an operatordefined as (D θ f )( y ) ∶ = ⎧⎪⎪⎨⎪⎪⎩H R ( − ( − ∆ ) θ − f )( y ) < θ ≤ ,f ′ ( y ) θ > , for any f ∈ S ( R ) , and H R is the Hilbert transform on R . The reason why we call(1.4) the superballistic wave equation is that ¯ p ( y, t ) satisfies ∂ t ¯ p ( y, t ) = ⎧⎪⎪⎨⎪⎪⎩ − C ( θ )( − ∆ ) θ − ¯ p ( y, t ) < θ ≤ ,C ( θ ) ∆¯ p ( y, t ) θ > . (1.5)In addition, in Theorem 2 we prove the following scaling limit of the scaled em-pirical measure of the phononic energy :lim ε → ε ∑ x ∈ Z E ε [ e x ( tj θ ( ε ) )] J ( εx ) = ∫ R dy ¯ e ( y, t ) J ( y ) where ¯ e ( y, t ) is given by¯ e ( y, t ) ∶ = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
12 ¯ p ( y, t ) + L θ ( y, t ) < θ < ,
12 ¯ p ( y, t ) +
12 ¯ l ( y, t ) θ ≥ , (1.6) nd L θ ( y, t ) , < θ < L θ ( y, t ) ∶ = ( − ( − ∆ ) θ − (D − θ ¯ l ) )( y, t ) − ( − ( − ∆ ) θ − D − θ ¯ l )D − θ ¯ l ( y, t ) . (1.7)One might think that the value θ = θ > < θ < Non-monotonic superdiffusive behaviors of normal modes:
We consider thefluctuation of (1.4) by considering normal modes , which is an analogue of the Rie-mann invariant for (1.4). For exponential decay models, in [6] the authors show thediffusive fluctuations of the microscopic normal modes corresponding to the Rie-mann invariants of (1.1). For polynomial decay models, we can define the normalmodes of (1.4) and corresponding microscopic normal modes f ± x ( t ) , x ∈ Z , t ≥ ε → ε ∑ x ∈ Z E ε [ f ± x ( tn θ ( ε ) )]( S ± θ ( tm θ ( ε ) ) J )( εx ) = ∫ R dy ¯ F ± ( y, t ) J ( y ) , for any J ∈ C ∞ ( R ) , where ( m θ ( ε ) , n θ ( ε )) are the macroscopic time scaling andmicroscopic time scaling respectively defined as ( m θ ( ε ) , n θ ( ε )) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩( ε − θ , ε − θ ) < θ < , (( log ε − ) − , ε (√ log ε − ) − ) θ = , ( ε θ − , ε θ − ) < θ ≤ , ( ε, ε ) θ > . (1.8)Moreover, S ± θ ( t ) , t ∈ R is a semigroup generated by ± √ C ( θ )D θ and by using thissemigroup we recenter the dynamics. For the definition of the macroscopic scal-ing limit ¯ F ± ( y, t ) , y ∈ R , t ≥ < θ < < θ < θ <
4. Another interesting behavior is the non-monotonic de-pendence on the exponent θ >
2: If 2 < θ <
3, then the time scaling gets fasterwhen θ gets bigger. But if θ >
3, then the time scaling gets faster when θ getssmaller and the fastest scaling appears when θ =
3. This non-monotonic depen-dence is caused by the first and second order of ˆ α ( k ) , k → n θ ( ε ) m θ ( ε ) − = j θ ( ε ) , where j θ ( ε ) is defined in (1.2).We follow the idea presented in [6] to prove above results. Since our dynamics is infinitevolume, we can not use relative entropy method used in [3, 8]. But if we assume that thetotal energy at t = L bound of the configurations ( p x , l x ) and the Fourier transform. In paticular, wedon’t need any information of canonical states for our system.One of the important open problems is to generalize some results for anharmonic chainswith long-range interaction. A main difficulty is that we do not know cannonical stateseven for our harmonic models. As we mentioned above, for our models the variable r x , x ∈ Z is not proper dual of p x , x ∈ Z , so we can expect that canonical states forstochastic harmonic chains may be expressed by using ( p x , l x ) terms. But the definitionof l x , x ∈ Z heavily depends on the concrete form of interaction potential α x , x ∈ Z and he Fourier transform, if chains are anharmonic then it is quite difficult to find the dualvariables of the momenta.Our paper is organized as follows: In Section 2 we prepare some notations. In Section3 we give the rigorous definition of the dynamics. In Section 4 we state our main results,Theorem 1, 2, 3 and 4. Proofs of Theorem 1, 2 are given in Section 5 and 6 respectively.In Section 7 we prove Theorem 3 and 4.2. Notations
Let R be the real line, Z be the set of all integers and T = [ − , ) be the one-dimensionaltorus. Denote by ℓ ( Z ) the space of all complex valued sequences ( f x ) x ∈ Z equipped withthe norm ∥ f ∥ ℓ ( Z ) ∶ = ∑ x ∈ Z ∣ f x ∣ . Denote by L ( T ) the space of all complex valued functions f ( k ) , k ∈ T equipped with thenorm ∥ f ∥ L ( T ) ∶ = ( ∫ T dk ∣ f ( k )∣ ) . Denote by S ( R ) the Schwarz space on R .For f, g ∶ Z → R , h ∈ ℓ ( Z ) we define f ∗ g ∶ Z → R and ˆ h ∈ L ( T ) as ( f ∗ g ) x ∶ = ∑ x ′ ∈ Z f x − x ′ g x ′ , ˆ h ( k ) ∶ = ∑ x ∈ Z e − π √− kx h x . For f ∈ L ( T ) we define ˇ f ∈ ℓ ( Z ) asˇ f x ∶ = ∫ T dk e π √− kx f ( k ) . For J ∈ S ( R ) we define ̃ J ∶ R → C as ̃ J ( ξ ) ∶ = ∫ R dy e − π √− ξy J ( y ) . Denote by sgn ( y ) , y ∈ R the sign function defined assgn ( y ) ∶ = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ y > , y = , − y < . We define the Hilbert transform on R and Z , denoted by H R and H Z respectively, viatheir Fourier transforms : ̃(H R J )( ξ ) ∶ = − √ − ( ξ ) ˜ J ( ξ ) , J ∈ S ( R ) , ̂(H Z f )( k ) ∶ = − √ − ( k ) ˆ f ( k ) , f ∈ ℓ ( Z ) . For two functions f, g defined on common domain A , we write f ≲ g or g ≳ f if thereexists some positive constant C > f ( a ) ≤ Cg ( a ) for any a ∈ A . If f ≲ g and g is a positive constant, then we also write sup a ∈ A f ( a ) ≲ f, g defined on common open subset A ⊂ R and a number y ∈ A ,where A is the closure of A , we write f ( y ) ∼ g ( y ) as y → y or f ( y ) = O ( g ( y )) as y → y if 0 < lim y → y ∣ f ( y ) g ( y ) ∣ < ∞ . n addtion, we write f ( y ) = o ( g ( y )) as y → y iflim y → y ∣ f ( y ) g ( y ) ∣ = . The dynamics
In this section we define harmonic chains with noise and long-range interactions. Sincewe analyze the system with finite total energy, it is appropriate for us to define thedynamics through the wave functions { ˆ ψ ( k, t ) ; k ∈ T , t ≥ } as L ( T ) solution of thestochastic differential equation (3.3). Then we can reconstruct the classical variables { p x ( t ) , q x ( t ) ; x ∈ Z , t ≥ } from { ˆ ψ ( k, t ) ; k ∈ T , t ≥ } , and then we define the energy { e x ( t ) ; x ∈ Z , t ≥ } and the generalized tension { l x ( t ) ; x ∈ Z , t ≥ } . However, it maybe difficult to understand the physical meaning of the important functions such as ˆ a ( k ) and R ( k ) from (3.3). To clarify the meaning of the feature values, we first give a formaldescription of the dynamics in terms of { p x ( t ) , q x ( t ) ; x ∈ Z , t ≥ } in the introduction.3.1. Formal description of the dynamics and corresponding wave function.
Weconsider the Hamiltonian dynamics perturbed by momentum-exchange noise which actsonly on momenta locally, that is, the dynamics is governed by the following stochasticdynamical system: ⎧⎪⎪⎪⎨⎪⎪⎪⎩ dq x ( t ) = p x ( t ) dtdp x ( t ) = ( − ( α ∗ q ) x ( t ) − γ ( β ∗ p ) x ( t )) dt + √ γ ∑ z = − , , ( Y x + z p x ( t )) dw x + z , (3.1)where γ ≥ Y x , x ∈ Z are vector fields defined as Y x ∶ = ( p x − p x + ) ∂ p x − + ( p x + − p x − ) ∂ p x + ( p x − − p x ) ∂ p x + . In addition, { w x ( t ) ; x ∈ Z , t ≥ } are i.i.d. one-dimensional standard Brownian motions,and β x is defined as β x ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ , x = , − , x = ± , − , x = ± , , otherwise . At this time we assume that γ =
0, that is, the system is deterministic harmonic chain.Taking the Fourier transform of both sides of (3.1), we have ddt ( ˆ q ( k, t ) ˆ p ( k, t )) = ( − ˆ a ( k ) ) ( ˆ q ( k, t ) ˆ p ( k, t )) . The eigenvalues of the above matrix are ± √ − ω ( k ) , ω ( k ) ∶ = √ ˆ α ( k ) , and correspondingeigenvectors { ˆ ψ ( k, t ) , ˆ ψ ∗ ( k, t ) ; k ∈ T , t ≥ } , called wave functions, are given byˆ ψ ( k, t ) ∶ = ω ( k ) ˆ q ( k, t ) + √ − p ( k, t ) . (3.2)Actually, we can check that ddt ˆ ψ ( k, t ) = − √ − ω ( k ) ˆ ψ ( k, t ) . or γ >
0, we also define wave functions { ˆ ψ ( k, t ) ; k ∈ T , t ≥ } as (3.2). Then from (3.1),the time evolution of { ˆ ψ ( k, t ) ; k ∈ T , t ≥ } are given by ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ d ˆ ψ ( k, t ) = [ − √ − ω ( k ) ˆ ψ ( k, t ) dt − γR ( k )( ˆ ψ ( k, t ) − ˆ ψ ∗ ( − k, t ))] dt + √ − √ γ ∫ r ( k, k ′ )( ˆ ψ ( k − k ′ , t ) − ˆ ψ ∗ ( k ′ − k, t )) B ( dk ′ , dt ) , (3.3)where r ( k, k ′ ) ∶ = ( πk ) sin ( π ( k − k ′ )) + ( πk ) sin ( π ( k − k ′ )) ,R ( k ) ∶ = ˆ β ( k ) = ( πk ) +
32 sin ( πk ) , and B ( dk, dt ) is a cylindrical Wiener process on L ( T ) defined as B ( dk, dt ) ∶ = ∑ x ∈ Z e πkx dk w x ( dt ) . Rigorous definition of the dynamics and generalized tension.
Now we givethe rigorous definition of our dynamics. Let ( E, F , P ) be a probability space and assumethat the cylindrical Wiener process B ( dk, dt ) is defined on ( E, F , P ) . For any T >
0, weintroduce a Banach space H T defined as H T ∶ = { f ∶ T × [ , T ] × Ω → C ; ∥ f ∥ H ∶ = ( sup ≤ t ≤ T E [∥ f ( t )∥ L ( T ) ]) < ∞ } . Then we define { ˆ ψ ( k, t ) ∈ H T ; k ∈ T , ≤ t ≤ T } as the solution of (3.3) with initialdistribution µ , where µ is an arbitrary probability measure on L ( T ) . The existenceand uniqueness of the solution is proved by using classical fixed point theorem, see [10,Appendix A]. Note that from (3.3) and Itˆo’s formula we have the energy conservationlaw ∫ T dk E µ [∣ ˆ ψ ( k, t )∣ ] = ∫ T dk E µ [∣ ˆ ψ ( k )∣ ] , (3.4)where E µ is the expectation with respect to the dynamics which starts from µ and E µ is the expectation with respect to µ .From { ˆ ψ ( k, t ) ; k ∈ T , t ≥ } , we can reconstruct the classical variables { p x ( t ) , q x ( t ) ; x ∈ Z , t ≥ } as ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ p x ( t ) ∶ = ∫ T dk e π √− kx ˆ p ( k, t ) q x ( t ) ∶ = ∫ T dk e π √− kx ˆ q ( k, t ) , ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ˆ p ( k, t ) ∶ = √ − ( ˆ ψ ( k, t ) − ˆ ψ ( − k, t ) ∗ ) ˆ q ( k, t ) = ω ( k ) ( ˆ ψ ( k, t ) + ˆ ψ ∗ ( − k, t )) . (3.5)Note that ˆ q ( k ) may not be well-defined as an L ( T ) element, because ω ( ⋅ ) − ∉ L ( T ) .However, we do not use q variable to prove our results, so we do not have to worry aboutthe well-definedness. ow we introduce the dual variable of { p x ( t ) ; x ∈ Z , t ≥ } , called the generalizedtension { l x ( t ) ; x ∈ Z , t ≥ } which is given by l x ( t ) ∶ = ∫ T dk e π √− kx ˆ l ( k, t ) , ˆ l ( k, t ) ∶ = √ − ( k ) ( ˆ ψ ( k, t ) + ˆ ψ ( − k, t ) ∗ ) . Remark 3.1. As r x , x ∈ Z is a function of q x , x ∈ Z , we can formally write l x , x ∈ Z as anon-local function of q x , x ∈ Z . Actually, from (3.5) we have ˆ l ( k ) = √ − ω ( k ) sgn ( k ) ˆ q ( k ) and thus we obtain l x = H Z ( ˇ ω ∗ q ) x . We also note that the Fourier transform of the tension ˆ r ( k ) , k ∈ T is defined by usingwave functions as ˆ r ( k ) ∶ = − e − π √− k ω ( k ) ( ˆ ψ ( k, t ) + ˆ ψ ∗ ( − k, t )) , and thus we have ˆ r ( k ) = − e − π √− k √ − ( k ) ω ( k ) ˆ l ( k ) . (3.6)Next we define the energy { e x ( t ) ; x ∈ Z , t ≥ } as a function of { ˆ ψ ( k, t ) ; k ∈ T , t ≥ } .For harmonic chains, e x ( t ) is usually defined as e x ( t ) = p x ( t ) − ∑ z ∈ Z α x − z ( q x ( t ) − q z ( t )) . As mentioned above, { q x ( t ) ; x ∈ Z , t ≥ } may not be well-defined, but we can define ∑ z ∈ Z α x − z ( q x ( t ) − q z ( t )) directly as a function of { ˆ ψ ( k, t ) ; k ∈ T , t ≥ } by using thefollowing argument: suppose that { q x ; x ∈ Z } is an ℓ ( Z ) element, then the Fouriertransform of ∑ z ∈ Z α x − z ( q x − q z ) is equal to ∫ T dk ′ ( ˆ α ( k ) − α ( k − k ′ ) − α ( k ′ )) ˆ q ( k ′ ) ˆ q ( k − k ′ ) = ∫ T dk ′ F ( k − k ′ , k ′ )( ˆ ψ ( k ′ ) + ˆ ψ ( − k ′ ) ∗ )( ˆ ψ ( k − k ′ ) + ˆ ψ ( k ′ − k ) ∗ ) where F ( k, k ′ ) ∶ = ˆ α ( k + k ′ ) − ˆ α ( k ) − ˆ α ( k ′ ) ω ( k ) ω ( k ′ ) . Therefore we can define ∑ z ∈ Z α x − z ( q x − q z ) as the Fourier coefficient of the above integra-tion, that is, ∑ z ∈ Z α x − z ( q x − q z ) ∶ = ∫ T dkdk ′ e π √− kx F ( k − k ′ , k ′ ) × ( ˆ ψ ( k ′ ) + ˆ ψ ( − k ′ ) ∗ )( ˆ ψ ( k − k ′ ) + ˆ ψ ( k ′ − k ) ∗ ) , nd thus the energy of the particle x ∈ Z is given by e x ( t ) ∶ = p x ( t ) − ∫ T dkdk ′ e π √− kx F ( k − k ′ , k ′ ) × ( ˆ ψ ( k ′ , t ) + ˆ ψ ( − k ′ , t ) ∗ )( ˆ ψ ( k − k ′ , t ) + ˆ ψ ( k ′ − k, t ) ∗ ) . Note that from Lemma A.1 and A.2 we see that F ( k, k ′ ) is uniformly bounded, and thus e x ( t ) is well-defined for any t ≥
0. 4.
Main Results
Superballistic scaling limit.
Let ( µ ε ) < ε < be a family of probability measures on L ( T ) . We define { ˆ ψ ( k, t ) = ˆ ψ ε ( k, t ) ; k ∈ T , t ∈ R ≥ } as the solution of (3.3) with initialcondition µ ε . Denote by E ε the expectation with respect to the dynamics which startsfrom µ ε . We assume that {( p x ( t ) , l x ( t )) = ( p x ( ˆ ψ )( t ) , l x ( ˆ ψ )( t )) ; x ∈ Z , t ≥ } satisfiesthe following initial condition, called phononic (or mechanical) type condition (cf. [6,Definition 2.3]): there exists some ¯ p , ¯ l ∈ C ∞ ( R ) such thatlim ε → ε ∑ x ∈ Z E µ ε [∣ p x − ¯ p ( εx )∣ + ∣ l x − ¯ l ( εx )∣ ] = . (4.1)Note that (4.1) is equivalent to the following conditionlim ε → ∫ T ε dk E ε [∣ ε ˆ p ( εk, ) − ˜¯ p ( k )∣ + ∣ ε ˆ l ( εk, ) − ˜¯ l ( k )∣ ] = . (4.2)We check the equivalence in Appendix B. In addition, from (3.4) and (4.1) we have ∫ T dk E ε [∣ ˆ p ( k, t )∣ + ∣ ˆ l ( k, t )∣ ] = ∫ T dk E µ ε [∣ ˆ p ( k )∣ + ∣ ˆ l ( k )∣ ] , and lim ε → ∫ T dk ε E ε [∣ ˆ p ( k, t )∣ + ∣ ˆ l ( k, t )∣ ] = ∫ R dξ ∣ ˜¯ p ( ξ )∣ + ∣ ˜¯ l ( ξ )∣ . Now we state one of our main results. Denote by ( ¯ p ( y, t ) , ¯ l ( y, t )) , y ∈ R , t ≥ ( ¯ p ( y ) , ¯ l ( y )) , y ∈ R . Let us recall that the timescaling j θ ( ε ) is defined in (1.2). Theorem 1.
Suppose that θ > , γ ≥ and (4.1) . For any t ≥ , we have lim ε → ∫ T ε dk E ε [∣ ε ˆ p ( εk, tj θ ( ε ) ) − ˜¯ p ( k, t )∣ + ∣ ε ˆ l ( εk, tj θ ( ε ) ) − ˜¯ l ( k, t )∣ ] = , and consequently we have lim ε → ε ∑ x ∈ Z E ε [∣ p x ( tj θ ( ε ) ) − ¯ p ( εx, t )∣ + ∣ l x ( tj θ ( ε ) ) − ¯ l ( εx, t )∣ ] = . From Theorem 1, (3.6) and Lemma A.1 we obtain the weak convergence of the empiricalmeasure and the scaling limit of the tension r x ( t ) , x ∈ Z , t ≥
0. In addition, we see that if θ > r x and l x coincide macroscopically. Corollary 4.1.
Suppose that θ > , γ ≥ and (4.1) . For any t ≥ and J i ∈ C ∞ ( R ) , i = , we have (1.3) . In addition, we have lim ε → ∫ T ε dk E ε [∣ ε ˆ r ( εk, tj θ ( ε ) )∣ ] = < θ ≤ , lim ε → ∫ T ε dk E ε [∣ ε ˆ r ( εk, tj θ ( ε ) ) − ε √ C ( θ ) ˆ l ( εk, tj θ ( ε ) )∣ ] = θ > . emark 4.1. The fractional Laplacians and the Hilbert transform are Fourier multipli-ers: ̃( − ( − ∆ ) a f )( ξ ) = − ∣ πξ ∣ a ˜ f ( ξ ) < a ≤ , ̃(H R f )( ξ ) = − √ − ( ξ ) ˜ f ( ξ ) . By using the above properties we easily see that ⎧⎪⎪⎨⎪⎪⎩ ∂ t ˜¯ p ( ξ, t ) = √ C ( θ )√ − ( ξ )∣ πξ ∣ θ − ˜¯ l ( ξ, t ) ∂ t ˜¯ l ( ξ, t ) = √ C ( θ )√ − ( ξ )∣ πξ ∣ θ − ˜¯ p ( ξ, t ) < θ ≤ , ⎧⎪⎪⎨⎪⎪⎩ ∂ t ˜¯ p ( ξ, t ) = π √ C ( θ )√ − ξ ˜¯ l ( ξ, t ) ∂ t ˜¯ l ( ξ, t ) = π √ C ( θ )√ − ξ ˜¯ p ( ξ, t ) θ > , and ∂ t ˜¯ p ( ξ, t ) = { − C ( θ )∣ πξ ∣ θ − ˜¯ p ( ξ, t ) < θ ≤ , − C ( θ )∣ πξ ∣ ˜¯ p ( ξ, t ) θ > . (4.3) Note that from (4.3) we can derive (1.5) by using the inverse Fourier transform.
Next we consider the scaling limit of the energy. Recall that ¯ e ( y, t ) is defined in (1.6). Theorem 2.
Suppose that θ > , γ ≥ and (4.1) . For any t ≥ and J ∈ C ∞ ( R ) , we have lim ε → ε ∑ x ∈ Z E ε [ e x ( tj θ ( ε ) )] J ( εx ) = ∫ R dy ¯ e ( y, t ) J ( y ) . (4.4) Remark 4.2.
From Theorem 2 we see that when θ ≥ the quantities e x ( t ) and ∣ ψ x ( t )∣ coincide macroscopically, that is, for any t ≥ and J ∈ C ∞ ( R ) we have lim ε → ∣ ε ∑ x ∈ Z E ε [( e x ( tj θ ( ε ) ) − ∣ ψ x ( tj θ ( ε ) )∣ )] J ( εx )∣ = . (4.5) However, if < θ < , then (4.5) does not hold. On the other hand, in [10, Proposition4.1] , we show that under the thermal type condition, an analogue of (4.5) holds for θ > .This difference is caused by initial conditions and from the above we see that ∣ ψ x ( t )∣ maynot coincide the phononic energy macroscopically. Fluctuations of normal modes.
In this subsection we consider the normal modesof the system (1.4) f ι ( y, t ) , ι = ± , an analogue of the Riemann invariants, which aredefined as f ± ( y, t ) ∶ = ¯ p ( y, t ) ± ¯ l ( y, t ) . The time evolution of the normal modes is given by ∂ t f ( y, t ) = ± √ C ( θ )(D θ f ± )( y, t ) , that is, f ± ( y, t ) are the eigenvectors and ± √ C ( θ )D θ are the eigenvalues of the system(1.4).For the sake of clarity, at first we consider the case θ >
4. Then the characteristics of(1.4) are given by straight lines y ± √ C ( θ ) t and thus we have f ± ( y, t ) = f ± ( y ± √ C ( θ ) t ) here f ± ∶ = f ± ( y, ) . This observation motivates us to study fluctuations of microscopic normal modes around macroscopic characteristics. Considering the above, we introducethe microscopic normal modes f ιx ( t ) , x ∈ Z , t ≥ , ι = ± defined as f ± x ( t ) ∶ = p x ( t ) ± l x ( t ) . From Theorem 1 we see thatlim ε → ε ∑ x ∈ Z E ε [ f ± x ( tε )] J ( εx ) = ∫ R dy f ± ( y, t ) J ( y ) , for any θ > t ≥ J ∈ C ∞ ( R ) . To obtain the fluctuations around the characteristicsat the diffusive time scaling, we need to recenter the dynamics by shifting the origin to ± √ C ( θ )( t / ε ) because the space is scaled by ε . Then we obtain the following scalinglimit, which is the special case of Theorem 3 stated in the end of this subsection. Corollary 4.2.
Assume that θ > , γ > and (4.1) . Then for any t ≥ and J ∈ C ∞ wehave lim ε → ε ∑ x ∈ Z E ε [ f ± x ( tε )] J ( εx ∓ √ C ( θ ) tε ) = ∫ R dy ∫ R dξ e π √− ξy e − γ ξ t ( ˜ f + + ˜ f − )( ξ ) J ( y ) . (4.6) Remark 4.3.
The above result is essentially the same as that of [6, Theorem 3.4] . Notethat the ballistic transport of the phononic terms with diffusive fluctuations and / -superdiffusion of thermal energy agree with the theoretical prediction of [9] . We also notethat for polynomial decay models / -superdiffusion of thermal energy is proved in [10] when θ > . Now we generalize the above argument to the case 2 < θ ≤
4. Let { S ιθ ( t ) ; t ∈ R , ι = ± , θ > } be the semigroups corresponding to ± √ C ( θ )D θ defined via its Fourier transform: ̃( S ± θ ( t ) g )( ξ ) ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ e ±√ C ( θ )√− ( ξ )∣ πξ ∣ θ − t ˜ g ( ξ ) < θ ≤ ,e ±√ C ( θ )√− ( πξ ) t ˜ g ( ξ ) θ > g ∈ S ( R ) . Note that the right-hand side of (4.6) can be rewritten as ε ∑ x ∈ Z E ε [ f ± x ( tε )]( S ± θ ( tε ) J )( εx ) . Therefore by using these semigroups we can recenter the dynamics for any θ >
1, but thecorrect time scaling is not diffusive if θ ≤
4. Before we state the scaling limits of the normal odes when 2 < θ ≤
4, we introduce some functions. Define ¯ F ± ( y, t ) , ( y, t ) ∈ R × R ≥ as ( ¯ F + ¯ F − ) ( y, t ) ∶ = ∫ R dξ e πξy exp { M θ ( ξ ) t } ( ˜¯ F + ˜¯ F − ) ( ξ, ) , ¯ F ± ( y, ) ∶ = ¯ p ( y ) ± ¯ l ( y ) , M θ ( ξ ) ∶ = ( M ( i,j ) θ ( ξ )) i,j = , ,M ( , ) θ ( ξ ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ √ − ( ξ ) C ( θ )√ C ( θ ) ∣ πξ ∣ − θ < θ < , √ − ( ξ )√ C ( )∣ πξ ∣ log ∣ πξ ∣ − + √ − ( ξ ) C ( )√ C ( ) ∣ πξ ∣ θ = , √ − ( ξ ) C ( θ )√ C ( θ ) ∣ πξ ∣ θ − < θ < , √ − ( ξ ) C ( θ )√ C ( θ ) ( πξ ) − γ ( πξ ) θ = , − γ ( πξ ) θ > ,M ( , ) θ ( ξ ) = M ( , ) θ ( ξ ) ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ < θ < , − γ ( πξ ) θ ≥ , M ( , ) θ ( ξ ) ∶ = M ( , ) θ ( ξ ) ∗ . In other words, ( ¯ F + , ¯ F − ) is the solution of the following system of linear differentialequations: ∂ t ¯ F ι ( y, t ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ sgn ( ι ) C ( θ )√ C ( θ ) D − θ ¯ F ι < θ < , sgn ( ι )(√ C ( )D ,l ¯ F ι + C ( )√ C ( ) ∂ y ¯ F ι ) θ = , sgn ( ι ) C ( θ )√ C ( θ ) D θ − ¯ F ι < θ < ( ι ) C ( )√ C ( ) H R ( ∆ ¯ F ι ) + γ ( ¯ F + + ¯ F − ) θ = , γ ( ¯ F + + ¯ F − ) θ > , (4.7)¯ F ± ( y, ) = ¯ p ( y ) ± ¯ l ( y ) for ι = ± where D ,l is defined via its Fourier transform ̃(D ,l f )( ξ ) ∶ = √ − ( ξ )∣ πξ ∣ log ∣ πξ ∣ − ˜ f ( ξ ) . Recall that m θ ( ε ) , n θ ( ε ) is defined in (1.8). Theorem 3.
Assume that θ > , γ > and (4.1) . Then for any t ≥ and J ∈ C ∞ ( R ) wehave lim ε → ε ∑ x ∈ Z E ε [ f ± x ( tn θ ( ε ) )]( S ± θ ( tm θ ( ε ) ) J )( εx ) = ∫ R dy ¯ F ± ( y, t ) J ( y ) . Remark 4.4.
We see that if < θ < then the right-hand side of (4.7) does not dependon the strength of the noise γ , and thus the fluctuations of normal modes are determinedby the interaction potential α . On the other hand, if θ > then the fluctuations do not epend on α . The threshold is θ = and in this case the fluctuations depend on both α and γ . In addition, when 2 < θ < the lawof large numbers for the empirical measure of the recentered dynamics. Theorem 4.
Assume that < θ < , γ ≥ and (4.1) . Then for any t ≥ and J ∈ C ∞ ( R ) we have lim ε → E ε ⎡⎢⎢⎢⎢⎣∣ ε ∑ x ∈ Z f ± x ( tn θ ( ε ) )( S ± θ ( tm θ ( ε ) ) J )( εx ) − ∫ R dy ¯ F ± ( y, t ) J ( y )∣⎤⎥⎥⎥⎥⎦ = . proof of theorem 1 In this section we prove Theorem 1 by following the strategy presented in [6, Section4]. Since if 1 < θ ≤ α ( k ) , k → α ( k ) , k → Mean dynamics.
First we define the scaled mean dynamics { ¯ p ε,x ( t ) , ¯ l ε,x ( t ) ; x ∈ Z , t ≥ } as ¯ p ε,x ( t ) ∶ = E ε [ p x ( tj θ ( ε ) )] , ¯ l ε,x ( t ) ∶ = E ε [ l x ( tj θ ( ε ) )] . We also introduce the Fourier transform of the mean dynamics { ˆ¯ p ε ( k, t ) , ˆ¯ l ε ( k, t ) ; k ∈ T ε , t ≥ } , that is, ˆ¯ p ε ( k, t ) ∶ = ε E ε [ ˆ p ( εk, tj θ ( ε ) )] ˆ¯ l ε ( k, t ) ∶ = ε E ε [ ˆ l ( εk, tj θ ( ε ) )] . The following Proposition states the scaling limit for the mean dynamics.
Proposition 5.1.
Suppose that θ > , γ ≥ and (4.1) . For any t ≥ , we have lim ε → ∫ T ε dk ∣ ˆ¯ p ε ( k, t ) − ˜¯ p ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˜¯ l ( k, t )∣ = , (5.1) and consequently we have lim ε → ε ∑ x ∈ Z (∣ ¯ p ε,x ( t ) − ¯ p ( εx, t )∣ + ∣ ¯ l ε,x ( t ) − ¯ l ( εx, t )∣ ) = . In the rest of this subsection we show that Proposition 5.1 implies Theorem 1. InSection 5.2 we prove Proposition 5.1.5.1.1.
Proof of Theorem 1.
First we define ˆ P ε ( k, t ) , ˆ L ε ( k, t ) , k ∈ T , t ≥ P ε ( k, t ) ∶ = ε ˆ p ( εk, tj θ ( ε ) ) − ˆ¯ p ε ( k, t ) , ˆ L ε ( k, t ) ∶ = ε ˆ l ( εk, tj θ ( ε ) ) − ˆ¯ l ε ( k, t ) . From (3.4), for any t ≥ ∫ T ε dk E ε [∣ ε ˆ p ( εk, )∣ + ∣ ε ˆ l ( εk, )∣ ] = ∫ T ε dk E ε [∣ ε ˆ p ( εk, tj θ ( ε ) )∣ + ∣ ε ˆ l ( εk, tj θ ( ε ) )∣ ] = ∫ T ε dk E ε [∣ ˆ P ε ( k, t )∣ + ∣ ˆ L ε ( k, t )∣ ] + ∫ T ε dk ∣ ˆ¯ p ε ( k, t )∣ + ∣ ˆ¯ l ε ( k, t )∣ . y using (4.1) and Proposition 5.1, we get ∫ R dk ∣ ˜¯ p ( k )∣ + ∣ ˜¯ l ( k )∣ = lim ε → ∫ T ε dk E ε [∣ ˆ P ε ( k, t )∣ + ∣ ˆ L ε ( k, t )∣ ] + ∫ R dk ∣ ˜¯ p ( k, t )∣ + ∣ ˜¯ l ( k, t )∣ . Hence from (3.4) we obtainlim ε → ∫ T ε dk E ε [∣ ˆ P ε ( k, t )∣ + ∣ ˆ L ε ( k, t )∣ ] = . (5.2)Combining (5.1) with (5.2), we havelim ε → ∫ T ε dk E ε [∣ ε ˆ p ( εk, tj θ ( ε ) ) − ˜¯ p ( k, t )∣ + ∣ ε ˆ l ( εk, tj θ ( ε ) ) − ˜¯ l ( k, t )∣ ] = , and thus we established Theorem 1.5.2. Proof of Proposition 5.1.
From (3.3), we have the time evolution law of { ˆ¯ p ε ( k, t ) , ˆ¯ l ε ( k, t ) ; k ∈ T ε , t ≥ } : ddt ( ˆ¯ p ε ( k, t ) ˆ¯ l ε ( k, t ) ) = A ε ( k ) ( ˆ¯ p ε ( k, t ) ˆ¯ l ε ( k, t ) ) , (5.3)where A ε ( k ) ∶ = j θ ( ε ) ( − γR ( εk ) √ − ( k ) ω ( εk )√ − ( k ) ω ( εk ) ) . Then we decompose A ε into two parts as follows: A ε ( k ) = A θ ( k ) + B ε,θ ( k ) , A θ = ( A ( i,j ) θ ) i,j = , , B ε,θ = ( B ( i,j ) ε,θ ) i,j = , where A ( , ) θ ( k ) = A ( , ) θ ( k ) ≡ ,A ( , ) θ ( k ) = A ( , ) θ ( k ) ∶ = ⎧⎪⎪⎨⎪⎪⎩√ C ( θ )√ − ( k )∣ πk ∣ θ − < θ ≤ , √ C ( θ )√ − ( k )∣ πk ∣ θ > ,B ( i,j ) ε,θ ( k ) ∶ = A ( i,j ) ε ( k ) − A ( i,j ) θ ( k ) i, j = , . To prove Proposition 5.1, we need the following Lemma 5.1 and Lemma 5.2.
Lemma 5.1. C ∗ ∶ = sup < ε < sup ( k,t ) ∈ R × R ∥ exp ( A ε ( k ) t )∥ < ∞ ,D ∗ ∶ = sup ( k,t ) ∈ R × R ∥ exp ( A ( k ) t )∥ < ∞ where ∣∣ ⋅ ∣∣ is the matrix norm defined as ∥ A ∥ ∶ = sup x ∈ R ∣ Ax ∣∣ x ∣ , A ∈ M ( R ) . In addition, for any K > we have sup ∣ k ∣ ≤ K ∥ B ε,θ ( k )∥ ≤ c K,θ b θ ( ε ) , here b θ ( ε ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ε < θ < ,ε log ε − θ = ,ε − θ < θ < ( log ε − ) − θ = ,ε θ − < θ < ,ε θ = ,ε θ − < θ < ,ε log ε − θ = ,ε θ > . and c K,θ is a positive constant which depends on K > , θ > . Lemma 5.2.
Suppose that ( ˆ¯ p ( ) ε ( k, t ) , ˆ¯ l ( ) ε ( k, t )) is the solution of the following equation: ddt ( ˆ¯ p ( ) ε ( k, t ) ˆ¯ l ( ) ε ( k, t ) ) = A ( k ) ( ˆ¯ p ( ) ε ( k, t ) ˆ¯ l ( ) ε ( k, t ) ) , ( ˆ¯ p ( ) ε ( k, ) ˆ¯ l ( ) ε ( k, ) ) = ( ˆ¯ p ε ( k, ) ˆ¯ l ε ( k, ) ) . Then for any K > , we have lim ε → ∫ ∣ k ∣ ≤ K dk ∣ ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t )∣ = . From now on we prove Lemma 5.1 and Lemma 5.2, and then we verify Proposition 5.1by using these lemmas.5.2.1.
Proof of Lemma 5.1.Proof.
Since the eigenvalues of A ( k ) are imaginary for every k ∈ T , we have D ∗ < ∞ . Theeigenvalues of A ε ( k ) are − j θ ( ε ) ( γR ( εk ) ± √ γ R ( εk ) − ˆ α ( εk )) . Since R ( ⋅ ) and ˆ α ( ⋅ ) are positive, the real parts of the eigenvalues of A ε ( k ) are negative.Therefore we have C ∗ < ∞ . ext we consider the order of sup ∣ k ∣ ≤ K ∥ B ε,θ ( k )∥ , K >
0. From Lemma A.1 we obtain ∣ B ( , ) ε ( k )∣ = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩∣ C ( θ )∣ πk ∣ θ − − ˆ α ( εk ) j θ ( ε ) − √ − ( k ) ω ( εk ) j θ ( ε ) − + √ C ( θ )√ − ( k )∣ πk ∣ θ − ∣ < θ ≤ , ∣ C ( θ )∣ πk ∣ − ˆ α ( εk ) ε − √ − ( k ) ω ( εk ) ε − + √ C ( θ )√ − ( πk ) ∣ θ > , ≲ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ε < θ < ,ε log ε − θ = ,ε − θ < θ < ( log ε − ) − θ = ,ε θ − < θ < ,ε θ = ,ε θ − < θ < ,ε log ε − θ = ,ε θ > , and ∣ B ( , ) ε ( k )∣ ≲ ε on ∣ k ∣ ≤ K . Hence we complete the proof of this lemma. (cid:3) Proof of Lemma 5.2.Proof.
Fix a positive constant K > < ε < K < ε . From (5.3) and thedecomposition of A ε ( k ) , we have ddt ( ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t ) ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t ) ) = A ( k ) ( ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t ) ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t ) ) + B ε,θ ( k ) ( ˆ¯ p ε ( k, t ) ˆ¯ l ε ( k, t ) ) . By using Duhamel’s formula, we have ( ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t ) ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t ) ) = ∫ t ds exp (( t − s ) A ( k )) B ε,θ ( k ) ( ˆ¯ p ε ( k, s ) ˆ¯ l ε ( k, s ) ) . From (3.4) and Lemma 5.1, we have ∫ ∣ k ∣ ≤ K dk ∣ ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t )∣ ≤ t ∫ ∣ k ∣ ≤ K dk ∫ t ds ∣ exp (( t − s ) A ( k )) B ε,θ ( k ) ( ˆ¯ p ε ( k, s ) ˆ¯ l ε ( k, s ) ) ∣ ≲ t ( b θ ( ε )) ∫ t ds ∫ T dk ε E ε [∣ ˆ ψ ( k, sj θ ( ε ) )∣ ] ≲ t ( b θ ( ε )) . We established this lemma. (cid:3) .2.3. Proof of Proposition 5.1.Proof.
By using Schwarz’s inequality we have ∫ T ε dk ∣ ˆ¯ p ε ( k, t ) − ˜¯ p ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˜¯ l ( k, t )∣ ≤ ∫ T ε dk ∣ ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t )∣ + ∫ T ε dk ∣ ˆ¯ p ( ) ε ( k, t ) − ˜¯ p ( k, t )∣ + ∣ ˆ¯ l ( ) ε ( k, t ) − ˜¯ l ( k, t )∣ ≤ ∫ T ε dk ∣ ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t )∣ + ∫ T ε dk ∥ exp ( A ( k ) t )∥ ∣ ( ˆ¯ p ε ( k, ) ˆ¯ l ε ( k, ) ) − ( ˜¯ p ( k ) ˜¯ l ( k ) ) ∣ . (5.4)From (4.1) and Lemma 5.1, we see that the second term of (5.4) vanishes. Now weestimate the first term. From Lemma 5.2, it suffices to show thatlim K →∞ lim ε → ∫ K < ∣ k ∣ < ε dk ∣ ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t )∣ = . (5.5)From (5.3), we can write ( ˆ¯ p ε ( k, t ) ˆ¯ l ε ( k, t ) ) = exp ( A ε ( k ) t ) ( ˆ¯ p ε ( k, ) ˆ¯ l ε ( k, ) ) . In addition, ( ˜¯ p ( ξ, t ) , ˜¯ l ( ξ, t )) , ξ ∈ R , t ≥ ∂ t ( ˜¯ p ( ξ, t ) ˜¯ p ( ξ, t )) = A ( ξ ) ( ˜¯ p ( ξ, t ) ˜¯ l ( ξ, t ) ) , and hence we obtain ( ˜¯ p ( ξ, t ) ˜¯ p ( ξ, t )) = exp ( A ( ξ ) t ) ( ˜¯ p ( ξ ) ˜¯ l ( ξ ) ) . By using (4.1), Lemma 5.1 and Schwarz’s inequality we obtainlim ε → ∫ K < ∣ k ∣ < ε dk ∣ ˆ¯ p ε ( k, t ) − ˆ¯ p ( ) ε ( k, t )∣ + ∣ ˆ¯ l ε ( k, t ) − ˆ¯ l ( ) ε ( k, t )∣ = lim ε → ∫ K < ∣ k ∣ < ε dk ∣ exp ( A ε ( k ) t ) ( ˆ¯ p ε ( k, ) ˆ¯ l ε ( k, ) ) − exp ( A ( k ) t ) ( ˆ¯ p ε ( k, ) ˆ¯ l ε ( k, ) ) ∣ ≤ ε → ∫ K < ∣ k ∣ < ε dk ∣ exp ( A ε ( k ) t ) ( ˆ¯ p ε ( k, ) ˆ¯ l ε ( k, ) ) ∣ + ∣ exp ( A ( k ) t ) ( ˆ¯ p ε ( k, ) ˆ¯ l ε ( k, ) ) ∣ ≤ ( C ∗ + D ∗ ) lim ε → ∫ K < ∣ k ∣ < ε dk ∣ ˆ¯ p ε ( k, )∣ + ∣ ˆ¯ l ε ( k, )∣ ≤ ( C ∗ + D ∗ ) ∫ ∣ ξ ∣ > K dξ ∣ ˜¯ p ( ξ )∣ + ∣ ˜¯ l ( ξ )∣ , and thus we have (5.5). (cid:3) . Proof of Theorem 2
First we observe that from Theorem 1 it is sufficient to show thatlim ε → ε ∑ x ∈ Z E ε [ e x ( tj θ ( ε ) ) − p x ( tj θ ( ε ) )] J ( εx ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∫ R dy L ( y, t ) J ( y ) < θ < , ∫ R dy ¯ l ( y, t ) J ( y ) θ ≥ , = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ ∫ R dξdk sgn ( k ) sgn ( − k − ξ ) ∣ ξ ∣ θ − − ∣ k ∣ θ − − ∣ k + ξ ∣ θ − ∣ k ∣ θ − ∣ k + ξ ∣ θ − ˜¯ l ( k, t ) ˜¯ l ( − k − ξ, t ) ˜ J ( ξ ) < θ < , ∫ R dξdk ˜¯ l ( k, t ) ˜¯ l ( − k − ξ, t ) ˜ J ( ξ ) θ ≥ , where L ( y, t ) is defined in (1.7). To simplify the notation, we may omit the variable t ≥ ε ∑ x E ε [ e x ( tj θ ( ε ) ) − p x ( tj θ ( ε ) )] J ( εx ) = ε ∑ x ∈ Z ∫ T dkdk ′ e π √− ( k + k ′ ) x sgn ( k ) sgn ( k ′ ) F ( k, k ′ ) E ε [ ˆ l ( k ) ˆ l ( k ′ )] ∫ R dξ e π √− ξεx ˜ J ( ξ ) = ∫ R × T dξdk sgn ( k ) sgn ( − k − εξ ) F ( k, − k − εξ ) ε E ε [ ˆ l ( k ) ˆ l ( − k − εξ )] ˜ J ( ξ ) = ∫ R × T ε dξdk sgn ( k ) sgn ( − k − ξ ) F ( εk, − εk − εξ ) ε E ε [ ˆ l ( εk ) ˆ l ( − εk − εξ )] ˜ J ( ξ ) . From Theorem 1 and the boundedness of F ( k, k ′ ) , we havelim ε → ∫ R × T ε dξdk sgn ( k ) sgn ( − k − ξ ) F ( εk, − εk − εξ ) ε E ε [ ˆ l ( εk, tj θ ( ε ) ) ˆ l ( − εk − εξ, tj θ ( ε ) )] ˜ J ( ξ ) = lim ε → ∫ R × T ε dξdk sgn ( k ) sgn ( − k − ξ ) F ( εk, − εk − εξ ) ˜¯ l ( k, t ) ˜¯ l ( − k − ξ, t ) ˜ J ( ξ ) . Since 1 { k ∈ T ε } F ( εk, − εk − εξ ) is uniformly bounded andlim ε → { k ∈ T ε } sgn ( k ) sgn ( − k − ξ ) F ( εk, − εk − εξ ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ sgn ( k ) sgn ( − k − ξ ) ∣ ξ ∣ θ − − ∣ k ∣ θ − − ∣ k + ξ ∣ θ − ∣ k ∣ θ − ∣ k + ξ ∣ θ − < θ < , θ ≥ , almost every ( ξ, k ) ∈ R , we obtainlim ε → ∫ R × T ε dξdk sgn ( k ) sgn ( − k − ξ ) F ( εk, − εk − εξ ) ˜¯ l ( k, t ) ˜¯ l ( − k − ξ, t ) ˜ J ( ξ ) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ∫ R dξdk sgn ( k ) sgn ( − k − ξ ) ∣ ξ ∣ θ − − ∣ k ∣ θ − − ∣ k + ξ ∣ θ − ∣ k ∣ θ − ∣ k + ξ ∣ θ − ˜¯ l ( k, t ) ˜¯ l ( − k − ξ, t ) ˜ J ( ξ ) < θ < , ∫ R dξdk ˜¯ l ( k, t ) ˜¯ l ( − k − ξ, t ) ˜ J ( ξ ) θ ≥ . Proof of Theorem 3 and 4
In this section we show Theorem 3 and 4 by using the strategy which is similar to thatused in Section 5. First we observe thatlim ε → ε ∑ x ∈ Z E ε [ f ± x ( tn θ ( ε ) )]( S ± θ ( tm θ ( ε ) ) J )( εx ) = lim ε → ∫ T ε dk ˆ¯ F ± ε ( k, t ) ˜ J ( − k ) , where ˆ¯ F ± ε ( k, t ) , k ∈ T ε , t ≥ F ± ε ( k, t ) ∶ = ε E ε [ ˆ F ± ε ( k, t )] , ˆ F ± ε ( k, t ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ exp ( ∓ √ C ( θ )√ − ( k )∣ πk ∣ θ − tm θ ( ε ) ) ˆ f ± ε ( k, t ) < θ ≤ , exp ( ∓ √ C ( θ )√ − ( πk ) tm θ ( ε ) ) ˆ f ± ε ( k, t ) θ > , and ˆ f ± ε ( k, t ) ∶ = ε ˆ f ± ( εk, tn θ ( ε ) ) . The main subject of this section is to show the following Proposition, which impliesTheorem 3.
Proposition 7.1.
Suppose that θ > , γ > and (4.1) . For any t ≥ , we have lim ε → ∫ T ε dk ∣ ⎛⎝ ˆ¯ F + ε ( k, t ) ˆ¯ F − ε ( k, t )⎞⎠ − ( ˜¯ F + ( k, t ) ˜¯ F − ( k, t )) ∣ = . We prove Proposition 7.1 in Section 7.1. In Section 7.2 we show Theorem 4 by usingProposition 7.1. In the rest of this subsection, we consider the time evolution law of { ˆ¯ F ± ε ( k, t ) ; k ∈ T ε , t ≥ } and prepare two lemmas to show Proposition 7.1.From (3.3) we have ddt ⎛⎝ ˆ¯ f + ε ˆ¯ f − ε ⎞⎠ = n θ ( ε ) (√ − ( k ) ω ( εk ) − γR ( εk ) − γR ( εk ) − γR ( εk ) − √ − ( k ) ω ( k ) − γR ( εk )) ⎛⎝ ˆ¯ f + ε ˆ¯ f − ε ⎞⎠ . Thus the dynamics { ˆ¯ F ± ε ( k, t ) ; k ∈ T ε , t ≥ } is given by ddt ⎛⎝ ˆ¯ F + ε ˆ¯ F − ε ⎞⎠ ( k, t ) = M ε,θ ( k ) ⎛⎝ ˆ¯ F + ε ˆ¯ F − ε ⎞⎠ ( k, t ) , M ε,θ ( k ) = ( M ( i,j ) ε,θ ( k )) i,j = , ,M ( , ) ε,θ ( k ) ∶ = √ − ( k ) ω ( εk ) n θ ( ε ) − √ C ( θ )√ − ( k )∣ πk ∣ θ − ∧ m θ ( ε ) − γR ( εk ) n θ ( ε ) ,M ( , ) ε,θ ( k ) = M ( , ) ε,θ ( k ) ∶ = − γR ( εk ) n θ ( ε ) , M ( , ) ε,θ ( k ) ∶ = ( M ( , ) ε,θ ) ∗ ( k ) . Then we have the following decomposition of M ε,θ ( k ) : M ε,θ ( k ) = M θ ( k ) + Rem ε,θ ( k ) , M θ = ( M ( i,j ) θ ) i,j = , , Rem ε,θ = ( Rem ( i,j ) ε,θ ) i,j = , , Rem ( i,j ) ε,θ ( k ) ∶ = M ( i,j ) ε,θ ( k ) − M ( i,j ) θ ( k ) i, j = , , where the matrix M θ is defined right before Theorem 3. We can obtain the followingestimates of the matrix norm of M θ , M ε,θ and Rem ε,θ . emma 7.1. C ′ ∗ ∶ = sup < ε < sup ( k,t ) ∈ R × R ≥ ∥ exp ( M ε,θ ( k ) t )∥ < ∞ ,D ′ ∗ ∶ = sup ( k,t ) ∈ R × R ≥ ∥ exp ( M θ ( k ) t )∥ < ∞ . In addition, for any K > , we have sup ∣ k ∣ ≤ K ∥ Rem ε,θ ( k )∥ ≤ C K,θ r θ ( ε ) , where r θ ( ε ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ε θ − < θ ≤ ε − θ < θ < , ( log ( ε − )) − θ = ,ε θ − < θ ≤ ,ε − θ < θ < ,ε log ( ε − ) θ = ,ε θ − < θ < ,ε log ( ε − ) θ = ,ε θ > . Proof. M ε and M have the form (√ − a + b bb − √ − a + b ) a, b ∈ R , and the eigenvalues of the matrix are b ± √ b − a . Since R ( k ) is non-negative, we seethat the eigenvalues of M ε and M are non-positive and thus we obtain C ′ ∗ , D ′ ∗ < ∞ .Next we consider the order of sup ∣ k ∣ ≤ K ∥ Rem ε,θ ( k )∥ , K >
0. Since ∣ Rem ( , ) ε,θ ( k )∣ ≲ ∣ Rem ( , ) ε,θ ( k )∣ , it is sufficient to estimate ∣ Rem ( , ) ε,θ ( k )∣ . From Lem A.1, we have ω ( εk ) n θ ( ε ) − √ C ( θ )∣ πk ∣ θ − ∧ m θ ( ε ) = ˆ α ( εk ) n θ ( ε ) − − C ( θ )∣ πk ∣ ( θ − )∧ m θ ( ε ) − ω ( εk ) n θ ( ε ) − + √ C ( θ )∣ πk ∣ θ − ∧ m θ ( ε ) − = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ C ( θ )∣ πk ∣ ω ( εk ) j θ ( ε ) − + √ C ( θ )∣ πk ∣ θ − + O ( ε θ − ∣ k ∣ θ + ) < θ < , − C ( )∣ πk ∣ log ( πk ) + C ( )∣ πk ∣ ω ( εk ) j θ ( ε ) − + √ C ( )∣ πk ∣ + O ( ε ∣ k ∣ ) θ = ,C ( θ )∣ πk ∣ θ − ω ( εk ) j θ ( ε ) − + √ C ( θ )∣ πk ∣ + O ( ε ∣ k ∣ θ − ) < θ < ,C ( )∣ πk ∣ ω ( εk ) j θ ( ε ) − + √ C ( )∣ πk ∣ + O ( ε log ( ε − )∣ k ∣ ) θ = , nd ω ( εk ) n θ ( ε ) − √ C ( θ )∣ πk ∣ m θ ( ε ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ O ( ε θ − ∣ k ∣ θ − + ε ∣ k ∣ ) < θ < ,O ( ε log ( ε − )∣ k ∣ ) θ = ,O ( ε ∣ k ∣ + ε θ − ∣ k ∣ θ − ) < θ < ,O ( ε ∣ k ∣ + ε log ( ε − )∣ k ∣ ) θ = ,O ( ε ∣ k ∣ + ε ∣ k ∣ ) θ > . In addition, we have R ( εk ) n θ ( ε ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ O ( ε n θ ( ε ) − ∣ k ∣ + ε n θ ( ε ) − ∣ k ∣ ) < θ < , ( πk ) + O ( ε ∣ k ∣ ) θ ≥ . Thus we obtainRem ( , ) ε,θ ( k ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ O ( ε − θ ∣ k ∣ − θ + ε θ − ∣ k ∣ θ + + ε θ − k ) < θ < ,O (( log ( ε − )) − ∣ k ∣ + ε ∣ k ∣ + ε (√ log ( ε − )) k ) θ = ,O ( ε θ − ∣ k ∣ θ − + ε ∣ k ∣ θ − + ε − θ k ) < θ < ,O ( ε log ( ε − )∣ k ∣ + ε ∣ k ∣ ) θ = ,O ( ε θ − ∣ k ∣ θ − + ε ∣ k ∣ + ε ∣ k ∣ ) < θ < ,O ( ε log ( ε − )∣ k ∣ + ε ∣ k ∣ ) θ = ,O ( ε ∣ k ∣ + ε θ − ∣ k ∣ θ − + ε ∣ k ∣ ) < θ < ,O ( ε ∣ k ∣ + ε log ( ε − )∣ k ∣ + ε ∣ k ∣ ) θ = ,O ( ε ∣ k ∣ + ε ∣ k ∣ + ε ∣ k ∣ ) θ > . Hence on ∣ k ∣ ≤ K we have ∣ Rem ( , ) ε,θ ( k )∣ ≲ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ε θ − < θ ≤ ε − θ < θ < , ( log ( ε − )) − θ = ,ε θ − < θ ≤ ,ε − θ < θ < ,ε log ( ε − ) θ = ,ε θ − < θ < ,ε log ( ε − ) θ = ,ε θ > . (cid:3) e introduce the dynamics { ˆ¯ F ± , ( ) ε ( k, t ) ; k ∈ T ε , t ≥ } , which is generated by M ( k ) withthe same initial condition as { ˆ¯ F ± ε ( k, t ) ; k ∈ T ε , t ≥ } , that is, ddt ⎛⎝ ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − , ( ) ε ( k, t )⎞⎠ ( k, t ) = M θ ( k ) ⎛⎝ ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − , ( ) ε ( k, t )⎞⎠ , ⎛⎝ ˆ¯ F + , ( ) ε ( k, ) ˆ¯ F − , ( ) ε ( k, )⎞⎠ = ⎛⎝ ˆ¯ F + ε ( k, ) ˆ¯ F − ε ( k, )⎞⎠ . Lemma 7.2.
For any K > and T > , we have lim ε → ∫ ∣ k ∣ ≤ K dk ∥ ⎛⎝ ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − , ( ) ε ( k, t )⎞⎠ − ⎛⎝ ˆ¯ F + ε ( k, t ) ˆ¯ F − ε ( k, t )⎞⎠ ∥ = . Proof.
Since ddt ⎛⎝ ˆ¯ F + ε ( k, t ) − ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − ε ( k, t ) − ˆ¯ F − , ( ) ε ( k, t )⎞⎠ = M θ ( k ) ⎛⎝ ˆ¯ F + ε ( k, t ) − ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − ε ( k, t ) − ˆ¯ F − , ( ) ε ( k, t )⎞⎠ + Rem ε,θ ( k ) ⎛⎝ ˆ¯ F + ε ( k, t ) ˆ¯ F − ε ( k, t )⎞⎠ , by using Duhamel’s formula we obtain ⎛⎝ ˆ¯ F + ε ( k, t ) − ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − ε ( k, t ) − ˆ¯ F − , ( ) ε ( k, t )⎞⎠ = ∫ t ds exp ( M ( k )( t − s )) Rem ε,θ ( k ) ⎛⎝ ˆ¯ F + ε ( k, t ) ˆ¯ F − ε ( k, t )⎞⎠ . Thanks to (3.4), (4.1) and Lemma 7.1, we have ∫ ∣ k ∣ ≤ K dk ∥ ⎛⎝ ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − , ( ) ε ( k, t )⎞⎠ − ⎛⎝ ˆ¯ F + ε ( k, s ) ˆ¯ F − ε ( k, s )⎞⎠ ∥ ≤ t ∫ ∣ k ∣ ≤ K dk ∫ t ds ∥ exp ( M θ ( k )( t − s )) Rem ε,θ ( k ) ⎛⎝ ˆ¯ F + ε ( k, s ) ˆ¯ F − ε ( k, s )⎞⎠ ∥ ≲ r θ ( ε ) ∫ ∣ k ∣ ≤ K dk ∫ t ds ∥ ⎛⎝ ˆ¯ F + ε ( k, s ) ˆ¯ F − ε ( k, s )⎞⎠ ∥ = r θ ( ε ) ∫ ∣ k ∣ ≤ K dk ∫ t ds ∥ ⎛⎝ ˆ¯ f + ε ( k, s ) ˆ¯ f − ε ( k, s )⎞⎠ ∥ = r θ ( ε ) ∫ ∣ k ∣ ≤ K dk ∫ t ds ∥ ( ˆ¯ p ε ( k, s ) ˆ¯ l ε ( k, s ) ) ∥ ≲ r θ ( ε ) ∫ T dk ∫ t ds ε E ε [∣ ˆ ψ ( k, sn θ ( ε ) )∣ ] ≲ r θ ( ε ) . (cid:3) Proof of Proposition 7.1. roof. Thanks to Schwarz’s inequality, we get ∫ T ε dk ∣ ⎛⎝ ˆ¯ F + ε ( k, t ) ˆ¯ F − ε ( k, t )⎞⎠ − ( ˜¯ F + ( k, t ) ˜¯ F − ( k, t )) ∣ ≤ ∫ T ε dk ∣ ⎛⎝ ˆ¯ F + ε ( k, t ) ˆ¯ F − ε ( k, t )⎞⎠ − ⎛⎝ ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − , ( ) ε ( k, t )⎞⎠ ∣ + ∫ T ε dk ∣ ⎛⎝ ˆ¯ F + , ( ) ε ( k, t ) ˆ¯ F − , ( ) ε ( k, t )⎞⎠ − ( ˜¯ F + ( k, t ) ˜¯ F − ( k, t )) ∣ = ∫ T ε dk ∣[ exp ( M ε,θ ( k ) t ) − exp ( M θ ( k ) t )] ⎛⎝ ˆ¯ F + ( k, ) ˆ¯ F − ( k, )⎞⎠ ∣ + ∫ T ε dk ∣ exp ( M θ ( k ) t ) ⎛⎝ ˆ¯ F + ε ( k, ) − ˜ F + ( k, ) ˆ¯ F − ε ( k, ) − ˜ F − ( k, )⎞⎠ ∣ . (7.1)From (4.1) and Lemma 7.1, we see that the second term of (7.1) vanishes as ε →
0. Nowwe estimate the first term of (7.1). From Lemma 7.2, it is sufficient to show thatlim K →∞ lim ε → ∫ K < ∣ k ∣ < ε dk ∣[ exp ( M ε,θ ( k ) t ) − exp ( M θ ( k ) t )] ⎛⎝ ˆ¯ F + ( k, ) ˆ¯ F − ( k, )⎞⎠ ∣ = . By using (4.1) and Lemma 7.1 we havelim ε → ∫ K < ∣ k ∣ < ε dk ∣[ exp ( M ε,θ ( k ) t ) − exp ( M θ ( k ) t )] ⎛⎝ ˆ¯ F + ( k, ) ˆ¯ F − ( k, )⎞⎠ ∣ ≲ lim ε → ∫ K < ∣ k ∣ < ε dk ∣ ⎛⎝ ˆ¯ F + ( k, ) ˆ¯ F − ( k, )⎞⎠ ∣ = lim ε → ∫ K < ∣ k ∣ < ε dk ∣ ⎛⎝ ˆ¯ f + ε ( k, ) ˆ¯ f − ε ( k, )⎞⎠ ∣ = ε → ∫ K < ∣ k ∣ < ε dk ∣ ˆ¯ p ε ( k, )∣ + ∣ ˆ¯ l ε ( k, )∣ = ∫ ∣ ξ ∣ > K dξ ∣ ˜¯ p ( ξ )∣ + ∣ ˜¯ l ( ξ )∣ . Hence we complete the proof of this lemma. (cid:3)
Proof of Theorem 4.
Defineˆ F ± ε ( k, t ) ∶ = ˆ F ± ε ( k, t ) − ˆ¯ F ± ε ( k, t ) . We see that ∣ ˆ F ± ε ( k, t )∣ = ∣ ˆ f ± ε ( k, t )∣ and from (3.4) we have ∫ T ε dk E ε [∣ ˆ f + ε ( k, t )∣ + ∣ ˆ f − ε ( k, t )∣ ] = ∫ T ε dk E ε [∣ ε ˆ p ( εk, tn θ ( ε ) )∣ + ∣ ε ˆ l ( εk, tn θ ( ε ) )∣ ] = ∫ T ε dk E ε [∣ ε ˆ p ( εk, )∣ + ∣ ε ˆ l ( εk, )∣ ] for any t ≥
0. Thus we obtain2 ∫ T ε dk E ε [∣ ε ˆ p ( εk, )∣ + ∣ ε ˆ l ( εk, )∣ ] = ∫ T ε dk E ε [∣ ˆ F + ε ( k, t )∣ + ∣ ˆ F − ε ( k, t )∣ ] + ∫ T ε dk ∣ ˆ¯ F + ε ( k, t )∣ + ∣ ˆ¯ F − ε ( k, t )∣ . y using (4.1) and Proposition 7.1 we getlim ε → ∫ T ε dk E ε [∣ ˆ F + ε ( k, t )∣ + ∣ ˆ F − ε ( k, t )∣ ] = ∫ R dξ ∣ ˜¯ p ( ξ )∣ + ∣ ˜¯ l ( ξ )∣ − ∫ R dξ ∣ ˜¯ F + ( ξ, t )∣ + ∣ ˜¯ F − ( ξ, t )∣ . If 2 < θ < t ≥ ∣ ˜¯ F ± ( ξ, t )∣ = ∣ ˜¯ F ± ( ξ, )∣ = ∣ ˜¯ p ( ξ ) ± ˜¯ l ( ξ )∣ and thus we obtain lim ε → ∫ T ε dk E ε [∣ ˆ F + ε ( k, t )∣ + ∣ ˆ F − ε ( k, t )∣ ] = . (7.2)Proposition 7.1 and (7.2) imply Theorem 4 becauselim ε → E ε ⎡⎢⎢⎢⎢⎣∣ ε ∑ x ∈ Z f ± x ( tn θ ( ε ) )( S ± θ ( tm θ ( ε ) ) J )( εx ) − ∫ R dy ¯ F ± ( y, t ) J ( y )∣⎤⎥⎥⎥⎥⎦ = lim ε → E ε ⎡⎢⎢⎢⎢⎣∣ ∫ T ε dk ( ˆ F ± ε ( k, t ) − ˜¯ F ± ( k, t )) ˜ J ( − k )∣⎤⎥⎥⎥⎥⎦ = lim ε → E ε ⎡⎢⎢⎢⎢⎣∣ ∫ T ε dk ˆ F ± ε ( k, t ) ˜ J ( − k )∣⎤⎥⎥⎥⎥⎦ ≤ ∥ J ∥ L ( R ) lim ε → ⎛⎝ ∫ T ε dk E ε [∣ ˆ F + ε ( k, t )∣ + ∣ ˆ F − ε ( k, t )∣ ]⎞⎠ = . Acknowledgement
We are grateful to Professor Stefano Olla for insightful discussions. HS was supportedby JSPS KAKENHI Grant Number JP19J11268 and the Program for Leading GraduateSchools, MEXT, Japan.
Appendix A. asymptotic behavior of ˆ α ( k ) , k → . Lemma A.1. ˆ α ( k ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ C ( θ )( π ) θ − ∣ k ∣ θ − + O (∣ k ∣ θ ) < θ < ,C ( ) π ∣ k ∣ + O ( k log ∣ k ∣ − ) θ = ,C ( θ )( π ) θ − ∣ k ∣ θ − + C ( θ )( π ) ∣ k ∣ + O (∣ k ∣ θ ) < θ < ,C ( )( π ) ∣ k ∣ log ∣ πk ∣ − + C ( )( π ) ∣ k ∣ + O (∣ k ∣ ) θ = ,C ( θ )( π ) ∣ k ∣ + C ( θ )( π ) θ − ∣ k ∣ θ − + O (∣ k ∣ θ ) < θ < ,C ( θ )( π ) ∣ k ∣ + C ( θ )( π ) θ − ∣ k ∣ θ − + O (∣ k ∣ log ∣ k ∣ − ) θ = ,C ( θ )( π ) ∣ k ∣ + C ( θ )( π ) θ − ∣ k ∣ θ − + O (∣ k ∣ ) < θ < ,C ( )( π ) ∣ k ∣ + C ( )( π ) ∣ k ∣ log ∣ k ∣ − + O (∣ k ∣ ) θ = ,C ( θ )( π ) ∣ k ∣ + C ( θ )( π ) ∣ k ∣ + O (∣ k ∣ θ − ) < θ < ,C ( θ )( π ) ∣ k ∣ + C ( θ )( π ) ∣ k ∣ + O (∣ k ∣ log ∣ k ∣ − ) θ = ,C ( θ )( π ) ∣ k ∣ + C ( θ )( π ) ∣ k ∣ + O (∣ k ∣ ) θ > s k → , where C ( θ ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ∫ ∞ dy − yy θ < θ < , θ = , ∑ x ≥ ∣ x ∣ − θ θ > , and C ( θ ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ − ∫ dy y θ − + ∫ ∞ dy ([ y ] −( θ − ) − y −( θ − ) ) < θ < , θ = , ∫ ∞ dy − y − y y θ < θ < , − θ = , − ∑ x ≥ x θ − θ > . Moreover, [ y ] , y ∈ R is the greatest integer less than or equal to y .Proof. First we observe that ˆ α can be devided into three parts:ˆ α ( k ) = ∑ x ∈ Z α x e − πkx = ∫ ∞ dy − cos 2 πk [ y ][ y ] θ = α ( k ) + α ( k ) + α ( k ) , here α ( k ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩( π ) θ − ∣ k ∣ θ − ∫ ∞ dy − yy θ < θ < , ( π ) ∣ k ∣ log ∣ πk ∣ − θ = , ( π ) k ∑ x ≥ ∣ x ∣ θ − θ > ,α ( k ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ < θ ≤ , − ( π ) θ − ∣ k ∣ θ − ∫ π ∣ k ∣ dy y θ − + ( π ) ∣ k ∣ ∫ ∞ dy [ y ] θ − − y θ − < θ < , ( π ) ∣ k ∣ { ∫ ∞ dy [ y ] − y + ∫ ∞ dy − yy + ∫ dy − y − y y } θ = , ( π ) θ − ∣ k ∣ θ − ∫ ∞ dy − y − y y θ < θ < , − ( π ) ∣ k ∣ log ∣ k ∣ − θ = , − ( π ) ∣ k ∣ ∑ x ≥ x θ − θ > ,α ( k ) ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∫ ∞ dy − πk [ y ][ y ] θ − − cos 2 πkyy θ − ( π ) θ − ∣ k ∣ θ − ∫ π ∣ k ∣ dy − yy θ < θ ≤ , ∫ ∞ dy − πk [ y ] − ( π ) k [ y ] [ y ] θ − − πky − ( π ) k y y θ − ( π ) θ − ∣ k ∣ θ − ∫ π ∣ k ∣ dy − y − y y θ < θ < , ∫ ∞ dy − πk [ y ] − ( π ) k [ y ] [ y ] − − πky − ( π ) k y y + ( π ) ∣ k ∣ ∫ ∞ dy − y − y y + ( π ) ∣ k ∣ ∫ π ∣ k ∣ dy −
24 cos y − y + y y + ( π ) ∣ k ∣ log 2 π θ = , ∑ x ≥ −
24 cos 2 πkx − ( π ) k x + ( π ) k x x θ θ > . Each α i corresponds to i -th order term of ˆ α and we can easily see the asymptotic behaviorof α , α . In the following subsections we compute the order of the remainder term α .Since in the case θ > < θ ≤
2, 2 < θ < θ =
5. Before showing the asymptotic behavior of α , we note hat if θ = α ( k ) = k because ∫ ∞ dy [ y ] − y = γ E , ∫ ∞ dy − yy = − cos 1 + ∫ ∞ dy sin yy = − cos 1 + sin 1 − Ci ( ) , ∫ dy − y − y y = − + cos 1 + ∫ dy sin y − yy = + cos 1 − sin 1 + ∫ dy cos y − y = + cos 1 − sin 1 − γ E + Ci ( ) , where γ E is the Euler’s constant and Ci is the cosine integral. At the last line we use thefollowing equationCi ( x ) ∶ = − ∫ ∞ x dy cos yy = γ E + log x + ∫ x dy cos y − y . A.1.
When < θ ≤ . Define f ( y ) ∶ = − cos 2 πkyy θ , y >
0. Since f ([ y ]) = f ( y ) + f ′ ( y )([ y ] − y ) + f ′′ ( y ′ )([ y ] − y ) for some [ y ] ≤ y ′ ≤ y , and ∣ f ′ ( y )∣ ≲ ∣ k ∣ ∣ sin 2 πky ∣ y θ , ∣ f ′′ ( y ′ )∣ ≲ ∣ k ∣ [ y ] θ , we have ∣ ∫ ∞ dy f ([ y ]) − f ( y )∣ ≲ ∣ k ∣ ∫ ∞ dy ∣ sin 2 πky ∣ y θ + ∣ k ∣ ∫ ∞ dy [ y ] θ ≲ ∣ k ∣ θ { ∫ ∣ k ∣ dy ∣ sin 2 πy ∣ y θ + ∫ ∞ dy ∣ sin 2 πy ∣ y θ } + ∣ k ∣ ≲ {∣ k ∣ θ < θ < , ∣ k ∣ log ∣ k ∣ − θ = . In addition, we get ∣ k ∣ θ − ∣ ∫ π ∣ k ∣ dy − cos yy θ ∣ ≲ ∣ k ∣ θ − ∫ ∣ k ∣ dy y θ − ≲ ∣ k ∣ . A.2.
When < θ < . Define f ( y ) ∶ = − πky −( π ) k y y θ , y >
0. Since f ([ y ]) = f ( y ) + f ′ ( y )([ y ] − y ) + f ′′ ( y ′ )([ y ] − y ) or some [ y ] ≤ y ′ ≤ y , we have ∣ f ′ ( y )∣ ≲ ∣ k ∣ ∣ sin 2 πky − πky ∣ y θ , ∣ f ′′ ( y ′ )∣ ≲ ∣ k ∣ { − cos 2 πky ′ ( y ′ ) θ + π ∣ k ∣ y ′ − sin 2 π ∣ k ∣ y ′ ( y ′ ) θ + + π k ( y ′ ) − sin πky ′ ( y ′ ) θ + } ≲ ⎧⎪⎪⎨⎪⎪⎩∣ k ∣ [ y ] − θ < θ ≤ , ∣ k ∣ [ y ] − θ θ > , and ∫ ∞ dy ∣ f ′ ( y )∣ ≲ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩∣ k ∣ θ { ∫ π ∣ k ∣ dy y − sin yy θ + ∫ ∞ dy y − sin yy θ } < θ ≤ , ∣ k ∣ ∫ ∞ dy y θ − < θ < , ≲ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∣ k ∣ θ < θ < , ∣ k ∣ log ∣ k ∣ − θ = , ∣ k ∣ θ > . In addition, we get ∣ k ∣ θ − ∣ ∫ π ∣ k ∣ dy − y − y y θ ∣ ≲ ∣ k ∣ θ − ∫ π ∣ k ∣ dy y θ − ≲ ∣ k ∣ . A.3.
When θ = . Since ∣ k ∣ ∣ ∫ π ∣ k ∣ dy −
24 cos y − y + y y ∣ ≲ ∣ k ∣ , we can see that α ( k ) = O (∣ k ∣ ) . (cid:3) Lemma A.2. ˆ α ( k + k ′ ) − ˆ α ( k ) − ˆ α ( k ′ ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩( π ) θ − C ( θ )(∣ k + k ′ ∣ θ − − ∣ k ∣ θ − − ∣ k ′ ∣ θ − ) + O (∣ k ∣ θ − ∣ k ′ ∣ + ∣ k ∣∣ k ′ ∣ θ − ) < θ < , πC ( )(∣ k + k ′ ∣ − ∣ k ∣ − ∣ k ′ ∣) + O (∣ k ∣∣ k ′ ∣ log ∣ k ∣ − + ∣ k ∣∣ k ′ ∣ log ∣ k ′ ∣ − ) θ = , ( π ) θ − C ( θ )(∣ k + k ′ ∣ θ − − ∣ k ∣ θ − − ∣ k ′ ∣ θ − ) + O (∣ kk ′ ∣) < θ < , ( π ) (∣ k + k ′ ∣ log (∣ k + k ′ ∣ − ) − ∣ k ∣ log (∣ k ∣ − ) − ∣ k ′ ∣ log (∣ k ′ ∣ − )) + O (∣ kk ′ ∣) θ = , ( π ) C ( θ ) kk ′ + o (∣ kk ′ ∣) θ > , for any k, k ′ ∈ T . In addition, if θ = then we obtain ∣ ˆ α ( k + k ′ ) − ˆ α ( k ) − ˆ α ( k ′ )∣ ≤ ( π ) ∣ kk ′ ∣√ log (∣ k ∣ − ) log (∣ k ′ ∣ − ) + O (∣ kk ′ ∣√ log (∣ k ∣ − ) + ∣ kk ′ ∣√ log (∣ k ′ ∣ − )) . roof. First we observe thatˆ α ( k + k ′ ) − α ( k ) − α ( k ′ ) = ∑ x ≥ cos ( πkx ) + cos ( πk ′ x ) − cos ( π ( k + k ′ ) x ) − ∣ x ∣ θ = ∑ x ≥ sin ( πkx ) sin ( πk ′ x ) − ( − cos ( πkx ))( − cos ( πk ′ x ))∣ x ∣ θ . A.4.
When < θ < . Define f ( y ) ∶ = ( πky ) sin ( πk ′ y ) − ( − cos ( πky ))( − cos ( πk ′ y ))∣ y ∣ θ . Then we have α ( k ) + α ( k ′ ) − ˆ α ( k + k ′ ) = ∫ ∞ dy f ( y ) + ∫ ∞ dy f ([ y ]) − f ( y ) . For any y ≥
1, there exists some y ′ ≥ , [ y ] ≤ y ′ ≤ y such that f ([ y ]) = f ( y ) + f ′ ( y ′ )([ y ] − y ) , and we have ∣ f ′ ( y ′ )∣ ≲ ∣ k ∣ ∣ sin 2 πk ′ y ∣∣ y ∣ θ + ∣ k ′ ∣ ∣ sin 2 πky ∣∣ y ∣ θ where we use an inequality [ y ] − ≤ y − on { y ≥ } . Since ∫ ∞ dy ∣ sin 2 πky ∣∣ y ∣ θ ≲ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∣ k ∣ θ − < θ < , ∣ k ∣ log ∣ k ∣ − θ = , ∣ k ∣ θ > , and ∫ ∞ dy f ( y ) = ∫ ∞ dy − ( π ( k + k ′ ) x )∣ y ∣ θ − ∫ ∞ dy − ( πkx )∣ y ∣ θ − ∫ ∞ dy − ( πk ′ x )∣ y ∣ θ = ( π ) θ − C ( θ )(∣ k + k ′ ∣ θ − − ∣ k ∣ θ − − ∣ k ′ ∣ θ − ) − ∫ dy sin ( πky ) sin ( πk ′ y ) − ( − cos ( πky ))( − cos ( πk ′ y ))∣ y ∣ θ = ( π ) θ − C ( θ )(∣ k + k ′ ∣ θ − − ∣ k ∣ θ − − ∣ k ′ ∣ θ − ) + O (∣ kk ′ ∣) , we haveˆ α ( k + k ′ ) − α ( k ) − α ( k ′ ) = ( π ) θ − C ( θ )(∣ k + k ′ ∣ θ − − ∣ k ∣ θ − − ∣ k ′ ∣ θ − ) + ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ O (∣ k ∣ θ − ∣ k ′ ∣ + ∣ k ∣∣ k ′ ∣ θ − ) < θ < ,O (∣ kk ′ ∣ log ∣ k ∣ − + ∣ kk ′ ∣ log ∣ k ′ ∣ − ) θ = ,O (∣ kk ′ ∣) θ > . .5. When θ = . In this case, from the following decomposition, ∫ ∞ dy f ( y ) = ( π ) ∣ k + k ′ ∣ log (∣ π ( k + k ′ )∣ − ) − ( π ) ∣ k ∣ log (∣ πk ∣ − ) − ( π ) ∣ k ′ ∣ log (∣ πk ′ ∣ − ) + ( π ) (∣ k + k ′ ∣ − ∣ k ∣ − ∣ k ′ ∣ ) ∫ ∞ dy − ( y )∣ y ∣ + ( π ) (∣ k + k ′ ∣ − ∣ k ∣ − ∣ k ′ ∣ ) ∫ dy − ( y ) − y ∣ y ∣ − ∫ dy sin ( πky ) sin ( πk ′ y ) − π kk ′ y ∣ y ∣ + ∫ dy ( − cos ( πky ))( − cos ( πk ′ y ))∣ y ∣ , we obtainˆ α ( k + k ′ ) − α ( k ) − α ( k ′ ) = ( π ) (∣ k + k ′ ∣ log (∣ k + k ′ ∣ − ) − ∣ k ∣ log (∣ k ∣ − ) − ∣ k ′ ∣ log (∣ k ′ ∣ − )) + O (∣ kk ′ ∣) . Note that by using Schwarz’s inequality we obtain the following boundedness : ∣ ∫ ∞ dy f ( y )∣ = ∣ ∫ ∞ sin ( πky ) sin ( πk ′ y ) − ( − cos ( πky ))( − cos ( πk ′ y ))∣ y ∣ ∣ = ∣ ∫ ∞ sin ( πky ) sin ( πk ′ y )∣ y ∣ ∣ + O (∣ kk ′ ∣) ≤ ( ∫ ∞ ∣ sin ( πky )∣ ∣ y ∣ ) ( ∫ ∞ ∣ sin ( πk ′ y )∣ ∣ y ∣ ) + O (∣ kk ′ ∣) = (( π ) ∣ k ∣ log (∣ k ∣ − ) + O (∣ k ∣)) (( π ) ∣ k ′ ∣ log (∣ k ′ ∣ − ) + O (∣ k ∣)) + O (∣ kk ′ ∣) = ( π ) ∣ kk ′ ∣√ log (∣ k ∣ − ) log (∣ k ′ ∣ − ) + O (∣ kk ′ ∣√ log (∣ k ∣ − ) + ∣ kk ′ ∣√ log (∣ k ′ ∣ − )) . A.6.
When θ > . In this case we can easily see the order because2 ∑ x ≥ sin ( πkx ) sin ( πk ′ x ) − ( − cos ( πkx ))( − cos ( πk ′ x ))∣ x ∣ θ = ( π ) C ( θ ) kk ′ + ∑ x ≥ sin ( πkx ) sin ( πk ′ x ) − ( π ) kk ′ x ∣ x ∣ θ − ∑ x ≥ ( − cos ( πkx ))( − cos ( πk ′ x ))∣ x ∣ θ = ( π ) C ( θ ) kk ′ + o (∣ kk ′ ∣) . (cid:3) ppendix B. On the equivalence of (4.1) and (4.2)
First we observe that ε ∑ x ∈ Z ∣ p x − ¯ p ( εx )∣ ≤ ε ∑ x ∈ Z ∣ p x − ∫ T ε dk e π √− kεx ˜¯ p ( k )∣ + ε ∑ x ∈ Z ∣ ¯ p ( εx ) − ∫ T ε dk e π √− kεx ˜¯ p ( k )∣ = ∫ T ε dk ∣ ε ˆ p ( εk ) − ˜¯ p ( k )∣ + ε ∑ x ∈ Z ∣ ¯ p ( εx ) − ∫ T ε dk e π √− kεx ˜¯ p ( k )∣ and ∫ T ε dk ∣ ε ˆ p ( εk ) − ˜¯ p ( k )∣ = ε ∑ x ∈ Z ∣ p x − ε ∫ T dk e π √− kx ˜¯ p ( kε )∣ ≤ ε ∑ x ∈ Z ∣ p x − ¯ p ( εx )∣ + ε ∑ x ∈ Z ∣ ¯ p ( εx ) − ∫ T ε dk e π √− kεx ˜¯ p ( k )∣ , where we use Parseval’s identity. Therefore to check the equivalence of (4.1) and (4.2),it is sufficient to show thatlim ε → ε ∑ x ∈ Z ∣ ¯ p ( εx ) − ∫ T ε dk e π √− kεx ˜¯ p ( k )∣ = . (B.1)Since ¯ p ∈ C ∞ ( R ) , we have ∣ ¯ p ( εx ) − ∫ T ε dk e π √− kεx ˜¯ p ( k )∣ = ∣ ∫ ∣ k ∣ > ε dk e π √− kεx ˜¯ p ( k )∣ = RRRRRRRRRRR π √ − εx ⎛⎝( e − π √− x ˜¯ p ( − ε ) − e π √− x ˜¯ p ( ε )) − ∫ ∣ k ∣ > ε dk e π √− kεx ∂ k ˜¯ p ( k )⎞⎠RRRRRRRRRRR ≲ ε ∣ x ∣ , and thus we obtain (B.1). References [1]
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