Temporal asymptotics for fractional parabolic Anderson model

In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u}{\partial t}=-(-\Delta)^{\frac{\alpha}{2}}u+u\dot W(t,x)$, where $-(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha\in(0,2]$ is a fractional Laplacian and $\dot W$ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by $\alpha$-stable process. As a byproduct, we obtain the critical values for $\theta$ and $\eta$ such that $\mathbb{E}\exp\left(\theta\left(\int_0^1 \int_0^1 |r-s|^{-\beta_0}\gamma(X_r-X_s)drds\right)^\eta\right)$ is finite, where $X$ is $d$-dimensional symmetric $\alpha$-stable process and $\gamma(x)$ is $|x|^{-\beta}$ or $\prod_{j=1}^d|x_j|^{-\beta_j}$.

If we abuse the notation β = d i=1 β i , the spatial covariance function has the following scaling property γ(cx) = |c| −β γ(x) (1.1) for both cases. In this paper, we shall study the following fractional parabolic Anderson model,    ∂u ∂t = −(−∆) α 2 u + uẆ (t, x), t > 0, x ∈ R d u(0, x) = u 0 (x), x ∈ R d , (1.2) where −(−∆) α 2 with 0 < α ≤ 2 is the fractional Laplacian and where the initial condition satisfies 0 < δ ≤ |u 0 (x)| ≤ M < ∞. Without loss of of generality, we assume u 0 (x) ≡ 1 when we study the long-term asymptotics of u(t, x). The product uẆ (t, x) appearing in the above equation will be understood in the sense of Skorohod and in the sense of Stratonovich.
Let us recall some results from [28] for the SPDE (1.2).
(i) Theorem 5.3 in [28] implies that, under the following condition: Eq. (1.2) in the Skorohod sense has a unique mild solutionũ(t, x), and its n-th moment can be represented as (see [28,Theorem 5.6]) (1.4) where X 1 , . . . , X n are n independent copies of d-dimensional symmetric α-stable process and are independent of W , and E X denotes the expectation with respect to (X x t , t ≥ 0).
(ii) Under a more restricted condition: αβ 0 + β < α (1.5) the following Feynman-Kac formula is a mild Stratonovich solution to (1.2) (see [28,Theorem 4.6]), where δ 0 (x) is the Dirac delta function. Consequently, Theorem 4.8 in [28] provides a Feynman-Kac type representation for n-th moment of u(t, x) (1.7) The more restricted condition (1.5) is to ensure that the "diagonal" terms, i.e., the sum n k=1 t 0 t 0 |r − s| −β 0 γ(X k r − X k s )drds appearing in (1.7) are exponentially integrable (see Lemma 2.1 and Theorem 2.3, or [28,Theorem 3.3] in a more general setting). To deal with the moments given by (1.4) and that given by (1.7) simultaneously, we introduce, under the condition (1.5), for a positive ρ ∈ [0, 1], (1.8) When ρ = 0, u ρ (t, x) is the Stratonovich solution u(t, x) to (1.2), and when ρ = 1, u ρ (t, x) is the Skorohod solutionũ(t, x) to (1.2). The n-th moment of u ρ (t, x) for a positive integer n is given by The goal of this article is to obtain the precise asymptotics, as t → ∞, of the p-th moment E [|u ρ (t, x)| p ] for any (fixed) positive real number p. To describe our main result, we recall the definition of Fourier transform and introduce some notations. Denote by S(R d ) the Schwartz space of smooth functions that are rapidly decreasing on R d , and let S ′ (R d ) be its dual space, i.e., the space of tempered distributions. Let f (ξ) or (F f )(ξ) denote the Fourier transform of f , for f in the space S ′ (R d ) of tempered distributions. In particular, we set f (ξ) = R d e −2πix·ξ f (x)dx, for f ∈ L 1 (R d ).
Theorem 1.1 Let ρ ∈ [0, 1] and assume the condition (1.5) (and when ρ = 1, the condition (1.5) is replaced by the condition (1.3)). Let p ≥ 2 be any real number or p = 1. Then We conclude this introduction with some remarks on the motivation of our work and a brief literature review for the related results. The following three points motivate us to obtain the above asymptotics.
(i) The limit related to the long-term asymptotics is known as the moment Lyapunov exponent in literature and the problem is closely related to the issue of intermittency (see, e.g., [23]). To illustrate our idea, write the limit in Theorem 1.1 in the following form: lim The system satisfies the usual definition of intermittency, i.e., the function Λ(p)/p is strictly increasing on [2, ∞). By the large deviation theory (see, e.g., Theorem 1.1.4 in [11] and its proof for the lower bound), for any sufficiently large l > 0 and there is p l > 0 such that This observation shows that as in other cases of intermittency, it is rare for the solution u(t, x) to take large values but that the impact of taking large values should not be ignored.
(ii) When the noiseẆ is the space-time white noise with dimension one in space, the parabolic Anderson model (1.2) is the model for the continuum directed polymer in random environment (see [1] for the case α = 2 and [6] for the case α < 2), where (1.2) is understood in the Skorohod sense, the solutionũ(t, x) is the partition function for the polymer measure, and logũ(t, x) is the free energy for the polymer (see, e.g., [15]).
Similarly, if we consider an α-stable motion X in the random environment modelled byẆ , one may consider the Hamiltonian x) ] is the partition function for the polymer measure, and log u ρ (t, x) is the free energy for the polymer.
(iii) The equation (1.2), as one of the basic SPDEs, describes a variety of models, such as the parabolic Anderson model (see, e.g. [7]) and the model for continuum directed polymer in random environment (see, e.g., [1]), in which the long-term asymptotic property of the solution is desirable. In the recent publication [8], the space-time fractional diffusion equation of the form has been studied, where ∂ β is the Caputo derivative in time t. It is highly non-trivial to obtain precise asymptotics in general case. Our model (1.2) corresponds to the case β = 1, and our result may provide some perspective for the general situation.
The moment Lyapunov exponent has been studied extensively with vast literature. To our best knowledge, however, the investigation in the setting of white/fractional space-time Gaussian noise started only recently, especially at the level of precision given in this paper. When the driving processes are Brownian motion instead of stable process, i.e., the operator in (1.2) is 1 2 ∆ instead of the fractional Laplacian, the long-term asymptotic lower and upper bounds for the moments of the solution were studied in [4] for the Skorohod solution and in [29] for the Stratonovich solution; the precise moment Lyapunov exponents were obtained in recent publications [9,10] for the Skorohod solutions, and [12] for the Stratonovich solution. In [3], the authors obtained the intermittency property for the fractional heat equation in the Skorohod sense, by studying the lower and upper asymptotic bounds of the solution.
In the present paper, we aim to obtain the precise p-th moment Lyapunov exponents for both Stratonovich solution and Skorohod solution to the fractional heat equation in a unified way, for any real positive number p ≥ 2. The mathematical challenges and/or the originality of this work come from the following aspects. First, compared with case of the heat equation, the fact that the fractional Laplacian is not a local operator makes the computations and analysis more sophisticated. New ideas and methodologies are required. In particular, Fourier analysis is involved in a more substantial way. Second, the Feynman-Kac large deviation result for stable process (Theorem 3.1) is a key to our approach. However, the method used to derive a similar result for Brownian motion in [12] can no longer be applied, as the behavior of stable process is totally different from the behavior of Brownian motion. Third, we obtain the precise long-term asymptotics for u ρ (t, x) with ρ ∈ [0, 1], which enables us to get the precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution simultaneously. Finally, the existing results on precise Lyapunov exponents were mainly for n-th moment with n a positive integer, due to the fact that the Feynman-Kac type representation is valid only for the moment of integer orders. We are able to extend the result from positive integers to real numbers p ≥ 2. The idea is to use the variational inequality and the hypercontractivity of the Ornstein-Uhlenbeck semigroup operators.
The paper is organized as follows. In Section 2, we establish some rough bounds for the long-term asymptotics of the Stratonovich solution by comparison method. The rough bounds will be used in the derivation of the precise upper bound in Section 6. The critical exponential integrability of 1 0 1 0 |r − s| −β 0 γ(X r − X s )drds is also studied. In Section 3, we obtain an Feynman-Kac type large deviation result for α-stable processes, which plays a critical role in obtaining the variational representation for the precise moment Lyapunov exponent. In Section 4, we establish a lower bound for the p-th moment of u ρ (t, x) which is also valid if the α-stable process is replaced by some general symmetric Lévy process. In Sections 5 and 6, we validate the lower bound and the upper bound in Theorem 1.1, respectively. Finally, in Appendix, the well-posedness of the variation given in (1.12) which appears in Theorem 1.1 is justified, and the proof of a technical lemma that is used in Section 6 is provided.

Asymptotic bounds by comparison method
In this section we derive long-term asymptotic bounds by comparison method for log E[u(t, Note that by the self-similarity property of the stable process X, the integral inside the above exponential has the following scaling property, First, we present the following integrability result. Proof Using the self-similarity of X, and the scaling property of γ(x), we have E[γ(X r − X s )] = |r −s| − β α E[γ(X 1 )], noting that 0 < E[γ(X 1 )] < ∞, under the condition of this lemma. Hence, we have which concludes the proof.
has a continuous version.
Proof We shall use the notation F p = (E[|F | p ]) 1/p . For any 0 ≤ s < t ≤ ∞, we have for any p ≥ 1, By scaling property, when 1 < p < α β , Thus, This means I p 1 ≤ Ct p(1−β 0 − β α ) |t − s| p . Similar estimates for I 2 and I 3 can also be obtained. Thus for 0 ≤ s, t ≤ T, there is a constant C T depending only on (α, β, β 0 , T ) such that It follows from Kolmogorov continuity criterion that {Y t , t ≥ 0} has a continuous version.

Remark 2.4
The inequality (2.3) is a consequence of Theorem 3.3 in [28] which was proved by using moment method. Below we will provide another approach to prove (2.3) by using the techniques from the theory of large deviations, and it turns out that this approach enables us to get a stronger result (see Remark 2.6).
Proof Denote First we shall show that Z t is sub-additive and hence exponentially integrable by [11, theorem 1.3.5].
The following identity holds where C 0 > 0 depends on β 0 only. Similarly, for the function γ(x) we also have where C(γ) > 0 is a constant and (2.7) With these identities, we can rewrite Z t as For t 1 , t 2 > 0, by the triangular inequality The translation invariance of the integral on R d+1 implies that Therefore, the process Z t is sub-additive, which means that for any t 1 , , and hence for all θ > 0, Chebyshev inequality implies that exp(θt)P(Z t ≥ t) ≤ E exp(θZ t ) and then θt + log P(Z t ≥ t) ≤ log E exp(θZ t ) .

12)
and Furthermore, and Proof Recall that Z t is defined in (2.4). Theorem 1.1 implies that, when p = 1 and ρ = 0, By the scaling property (2.1) of Z 2 t and the change of variable s = t Then the Gärtner-Ellis theorem implies and hence C 0 is a finite positive constant. Then we have For any fixed σ ∈ (0, 1), there exists T σ > 0 such that when t > T σ , where the right-hand side is finite when θ < σC 0 . Since σ ∈ (0, 1) can be arbitrarily chosen, the first result (2.12) is obtained. Finally the inequalitys (2.13), (2.14), (2.15) can be proved in a similar way by using (2.17), and the proof is concluded.
To obtain the optimal asymptotics for E exp LetX be an independent copy of X. Then under the condition This is because that d j=1 |x j | −β j ≥ |x| −β and hence C 1 is greater than or equal to the constant C > 0 in (2.18) when γ(x) = |x| −β . This means that if C 1 < ∞ satisfies (2.20), then it will be automatically positive.
From now on the generic constant C may be different in different places.
We claim that (2.20) is equivalent to for some constant C ∈ (0, ∞), which can be proved in the same way as we did to get (2.9) in the proof of the Theorem 2.
Now we prove (2.19). The upper bound can be obtained by (2.18)

and the observation that
For the lower bound, if suffices to consider the case where a is a positive constant. By the scaling property (2.1), the above equality is equivalent to Then by Varadhan's integral lemma, we have Based on the above result, we shall derive the following asymptotics for E exp θ t 0 t 0 |r − s| −β 0 γ(X r − X s )drds by comparison method. Proposition 2.9 Under the condition 1.5, there is 0 < C 1 < C 2 < ∞ such that for any θ > 0, (2.23) Remark 2.10 By the scaling property (2.1), the above asymptotics (2.22) is equivalent to respectively. We also have a similar result for (2.23).
Proof The proof is similar to [12, Proposition 2.1], but we include details for the reader's convenience. First we prove the lower bound in (2.22). Note that where the term on the right-hand side has the same distribution as by the scaling property (2.1). Then the lower bound is an immediate consequence of Lemma 2.8.
Now we show the upper bound of (2.22). By the symmetry of the integrand function, we have Thus, the inequality (2.22) is equivalent to Compared with lower bound, the estimation (2.25) is more difficult to obtain because |r−s| −β 0 is unbounded when r and s are close. We shall decompose the integral [0≤s≤r≤t] |r− s| −β 0 γ(X r − X s )drds and then apply Hölder inequality to obtain the desired result. More drds are mutually independent and are equal in law, by the Hölder inequality, Furthermore, by the scaling property (2.1), where 1/p + 1/q = 1 and p, q > 0 are to be determined later. Since X has stationary increments and by (2.1), we have Now let us choose p = 2 2α−β−αβ 0 α , and the above identity combined with (2.27) yields The lower bound in (2.23) can be obtained in a similar way as for the lower bound in (2.22), by using the second half of Lemma 2.8. Now we show the upper bound. Noting that the stable process has stationary increments which are independent over disjoint time intervals, we have By Remark 5.7 in [28], under the condition (1.3), Hence (2.26) still holds under the condition (1.3), and therefore the upper bound in (2.23) is obtained. The proof is concluded.

Feynman-Kac large deviation for stable process
In this section, we will obtain a Feynman-Kac large deviation result (Proposition 3.1 below) for symmetric α-stable process, which is a space-time extension of Lemma 6 in [13] and will play a critical role in the derivation of our main result. In [12] a similar result for Brownian motion was obtained (Proposition 3.1 in that paper) in order to get the precise moment Lyapunov exponent for the Stratonovich solution of heat equation. The approach in [12] heavily depends on the local property of the Laplacian operator and the property of Brownian motion such as the continuity of paths and the Gaussian tail probability, and hence cannot be adapted to our situation, as the fractional Laplacian is a non-local operator, the stable process is a pure jump process, and the stable distribution is fat-tailed. Inspired by the idea in [13], instead of considering the stable process itself, we shall consider the stable process restricted in bounded domains by taking its image of quotient map, which will be elaborated below. Then, X M is a Markov process with independent increments on T d M , and its associated Dirichlet form is given by where f denotes the usual Fourier transform for functions on T d M , i.e., for k ∈ Z d , Here the function f on T d M is considered as an M-periodic function (with the same symbol f ) on R d , which means that f (x + kM) = f (x) for any k ∈ Z d . Let is the L 2 -norm on T d M endowed with the Lebesgue measure.
Proof Let {0 = s 0 < s 1 < · · · < s n−1 < s n = 1} be a uniform partition of the interval [0, 1]. First, we consider the functions of the form By the Markov property, we have where E x denotes the expectation with respect to the stable process staring from x.
Repeating the above procedure, we can get Similarly, we have By boundedness of f i and the Markov property, we have wherep(y) is the density function of X M 1 . Note thatp(y) is strictly positive and continuous on T d M , and Then, there exists ε > 0 such that inf y∈R Mp(y) ≥ ε and consequently On the other hand, for any g ∈ F α,M , where in the last step T α,M is the self-adjoint operator associated with the Dirichlet form E α,M , V f is the operator of the multiplication of the function f , and the equality follows from [13,Lemma 5]. By spectral representation theory, there exists a probability measure µ g (dλ) such that and .
By spectral representation, for any g ∈ F α,M , Combining In the meantime, the lower bound in (3.3) also holds for the original stable process X. where λ(f ) = sup g∈Fα g, f g 2,R d − E α (g, g) .

Proof
The proof is similar to the lower bound part of the proof for Proposition (3.3). We shall only sketch the idea.
We still start with the functions of the form f (s, x) = n−1 i=0 f i (x)I [s i ,s i+1 ) (s)+f n−1 (x)I {1} (s). Fix a compact set D ⊂ R d , Then, there exists a positive ε such that the density function p(y) of X 1 is bigger than ε for all y ∈ D. For any g ∈ F α with support inside D, using a similar argument as (3.8) -(3.12), we can get lim inf Therefore, for any g ∈ F α with compact support, we have lim inf and hence lim inf Finally, (3.15) follows from a limiting argument.

A variational inequality
In this section, we will establish a lower bound for u ρ (t, x) p for p ≥ 1, ρ ∈ [0, 1], where u ρ is given by (1.8) when ρ ∈ [0, 1) under the condition (1.5) and u 1 (t, x) is the Skorohod solutionũ(t, x) under the condition (1.3). This will be used to obtain the lower bound in Theorem 1.1.
First let us introduce some notations by recalling the Dalang's approach (see [16]) of defining stochastic integral with respect to the Gaussian noiseẆ . Let D(R d+1 ) be the set of smooth functions on R d+1 with compact support, and H be the Hilbert space spanned by D(R d+1 ) under the inner product ϕ, ψ H := In the probability space (Ω, F , P), let W = {W (h), h ∈ H} be an isonormal Gaussian process with covariance function give by E[W (h)W (g)] = h, g H . We also write, for h ∈ H, Denote the Fourier transforms of |s| −β 0 and γ(x) by µ 0 (dτ ) and µ(dξ), respectively, then 3) The Parseval's identity provides an alternative representation for the inner product, With the above notations (1.3) is equivalent to the following general form of the Dalang's condition and (1.5) is equivalent to (4.5) Now we recall the approximation procedure used in [20,21,28], which we shall use in the proof of the main result in this section. Denote g δ (t) : is a symmetric probability density function and its Fourier transform p(ξ) ≥ 0 for all ξ ∈ R d . For positive numbers ε and δ, definė . Consider the following approximation of (1.2) (4.7) Then, Feynman-Kac formula for the Stratonovich solution u ε,δ is ε,δ (r, X x t−r )dr , and the Feynman-Kac formula for the Skorohod solutionũ ε,δ (t, x) is by stochastic Fubini's theorem.
The following is the main result in this section.

Remark 4.2
The result still holds if the α-stable process X in u ρ (t, x) is replaced by a general symmetric Lévy process with characteristic function E[e iξ·Xt ] = e −tΨ(ξ) . In this case, the conditions (1.5) and (1.

On the lower bound
In this section, we establish the lower bound in Theorem 1.1 for all p ≥ 1.
and for any h ∈ S H (R d+1 ) denote Then, by (5.1), change of variables and the self-similarity of the α-stable process, we have Clearly, h t ∈ S H (R d+1 ). Proposition 4.1 and the above two identities imply By Proposition 3.2, lim inf where A α,d is given by (1.11). Therefore, Since S H (R d+1 ) is dense in L 2 (R d+1 , µ 0 ⊗ µ) (see, e.g., [24]), and Γ(·, g) is continuous with respect to the L 2 (R d+1 , µ 0 ⊗ µ)-norm, we have Summarizing the computations starting from (5.3), we have and the lower bound is established.

On the upper bound
In this section, we provide a proof for the upper bound in Theorem 1.1. In Subsections 6.1 and 6.2, we shall obtain the upper bound for any positive integer n ≥ 1, i.e., lim sup The proof for real number p ≥ 2 is inspired by the idea in [26]. We shall compare u ρ (t, x) p with u ρ (t, x) 2 by using the Mehler's formula and hypercontractivity of the Ornstein-Uhlenbeck semigroup operators. First, we address the case when ρ ∈ [0, 1], under the condition (1.5).
Finally, for the case ρ = 1 under the condition (1.3), in which u ρ (t, x) is the Skorohod solution to (1.2), we can apply the approach in [26] and obtain the upper bound for all real numbers p ≥ 2.

Upper bound under the condition (1.5).
In this subsection, we deal with the case ρ ∈ [0, 1] under the condition (1.5). The proof will be split into four steps.
Step 1. In this step, we will reduce the study of n-th moment to the study of first moment. Recall that (2.5) and (2.6) imply Therefore, by the inequality ( n j=1 a j ) 2 ≤ n n j=1 a 2 j , we have n j,k=1 Consequently, to obtain the upper bound in Theorem 1.1, it suffices to show lim sup By the scaling property (2.1), we see Now, to obtain the upper bound, it suffices to prove (6.5). To this goal, we shall use the representations (2.5) and (2.6) for the covariance functions. But in these two representations, the integrals are over infinite domains. We shall approximate them by bounded, continuous, and locally supported functions, and this will enable us to apply Hahn-Banach theorem in Step 4.
Step 2. In this step, we will replace the temporal covariance function by a smooth function with compact support. Let the function ̺ : R + → [0, 1] be a smooth function such that ̺(u) = 1, u ∈ [0, 1], ̺(u) = 0 for u ≥ 2, and −1 ≤ ̺ ′ (u) ≤ 0. Define the following truncated functions with A > 0 being a large number and a > 0 being a number close to zero.
Combining (2.5) and (6.8), for the second term in (6.7), we have lim sup where the last step follows from Hölder's inequality and (2.24). Therefore, for fixed (ε, q), this term can be as small as we wish if we choose A sufficiently large and a sufficiently small. On the other hand, we can choose ε arbitrarily close to 0 and p arbitrarily close to 1. Consequently, to prove (6.5), it suffices to prove lim sup ≤ M(α, β 0 , d, γ). (6.10) Step 3. In this step, we will replace the spatial covariance function by a smooth function with compact support. Similarly to the truncation for the temporal covariance function, for 0 < b < B < ∞, we let where K(x) is given in (2.7). Then, 0 ≤ K B,b (x) ≤ K(x) and K B,b (x) → K(x) when B → ∞ and b → 0. Now the left-hand side of (6.10) can be estimated in the similar way as in (6.7), i.e., and is uniformly bounded (say, by L), we have Using that (a+b) 2 t+s ≤ a 2 t + b 2 s , we have where the last equality follows from a change of variable for s and the fact that the Lebesgue measure on R d is invariant under the translation x → x + X t . Hence, by the independent and stationary properties of the increments of Lévy processes, we have Therefore, lim sup for any θ > 0. Now letting B → ∞ and b → 0, by the dominated convergence theorem we see that the term on the right-hand side of (6.11) goes to 0. Now combining all the inequalities after (6.10), noting that we can choose ε arbitrarily close to 0, and p arbitrarily close to 1, we have that (6.10) can be reduced to lim sup Step 4. Summarizing the arguments in Step 2 and Step 3, we see that to obtain the upper bound in Theorem 1.1, it suffices to show lim sup In this final step, we will prove the above inequality. Fix positive constants A, a, B, b and choose arbitrarily M > 2 max{A, B}.
are M-periodic functions. Note that the summations in (6.14) are well-defined, since the supports of k A,a (·) and K B,b (·) are bounded domains. The process can be considered as a process taking values in the Hilbert space L 2 ([0, M] d+1 ) with the norm denoted by · . Since k M and K M are bounded, smooth functions with bounded derivatives, there is a constant C > 0, such that for all t and (u 1 , x 1 ), (u 2 , x 2 ) ∈ [0, M] d+1 . Let K be the closure of the following set in Then, φ t defined in (6.15) belongs to K, and it follows from [19,Theorem IV8.21] that K is compact in L 2 ([0, M] d+1 ).
Let δ > 0 be fixed. For any g ∈ K, noting that the set of bounded and continuous functions are dense in L 2 ([0, M] d+1 ), the Hahn-Banach theorem ([30]) implies that there is a bounded and continuous function f ∈ L 2 ([0, M] d+1 ) such that g 2 < − f 2 +2 f, g +δ. By the finite cover theorem for compact sets, one can find finitely many bounded and continuous functions f 1 , · · · , f m such that g 2 < δ + max 1≤i≤m {− f j 2 + 2 f i , g } for all g ∈ K. In particular, we have, noting that φ t ∈ K, Therefore, lim sup Notice that, for i = 1, . . . , m, Since K M is a periodic function and It is easy to check thatf i satisfies the condition in Proposition 3.1. Hence, Since δ in (6.16) can be arbitrarily small and M in (6.17) can be arbitrarily large, the desired inequality (6.12) follows from inequalities (6.13) -(6.17) and Lemma 7.3.

When ρ = 1 under the condition (1.3)
In this subsection, we consider the Skorohod case, i.e., ρ = 1, under the condition (1.3), by applying the methodology used in Section 6.1. However, under condition (1.3), there will be a technical issue in step 1, since the left-hand side of (6.5) is infinity if condition (1.5) is violated. To deal with this issue, we will first, do step 2 for n-th moments which reduces |s| −β 0 to a smooth function with compact support, and then, we do step 1 to reduce the n-th moment to first moment.
By a similar argument used in Step 1, in order to show (6.18) that has been reduced to (6.20), it suffices to prove lim sup The left-hand side now is finite under condition (1.3) since ψ A,a is a bounded function. Noting that (6.23) is identical to (6.10), we may prove it in the exact same way as in Step 3 and Step 4 in Subsection 6.1.
Summarizing the above computations, we obtain γ(x − y)g 2 (x)g 2 (y)dxdy where the variation on the right-hand side is finite by Lemma 7.1.
The following lemma was used in the proof of upper bound.
where C α,d = R d