Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions

: We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have inﬁnite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d = 1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient.


Introduction
The fourth order linearized Kuramoto-Sivashinsky (LKS) SPDEs are related to the model of pattern formation phenomena accompanying the appearance of turbulence (see [1][2][3][4] for the LKS class and for its connection to many classical and new examples of deterministic and stochastic pattern formation PDEs, and see [5,6] for classical examples of deterministic and stochastic pattern formation PDEs).
The fundamental kernel associated with the deterministic version of this class is built on the Brownian-time process in [3,7,8].In this article, we give exact dimension-dependent asymptotic distributions of the realized power variations in time, for the important class of stochastic equation: where L is the d-dimensional Laplacian operator, (ε, ϑ) ∈ R + × R is a pair of parameters, the noise term ∂ d+1 W/∂t∂x is the space-time white noise corresponding to the real-valued Brownian sheet W on R + × R d , d = 1, 2, 3.The initial data u 0 here is assumed Borel measurable, deterministic, and 2-continuously differentiable on R d whose 2-derivative is locally Hölder continuous with some exponent 0 < γ ≤ 1.
Of course, Equation ( 1) is the formal (and nonrigorous) equation.Its rigorous formulation, which we work with in this paper, is given in mild form as kernel stochastic integral equation (SIE).This SIE was first introduced and studied by [1][2][3][7][8][9][10].We give it below in Section 3, along with some relevant details.
The existence/uniqueness as well as sharp dimension-dependent L p and Hölder regularity of the linear and nonlinear noise version of (1) were investigated in [1,2,9,10].It was studied in [4] that exact uniform and local moduli of continuity for the LKS-SPDE in the time variable t and space variable x, separately.In fact, it was established in [4] that exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for the fourth order the LKS-SPDEs and their gradient.It was studied in [11] that the solution to a stochastic heat equation with the space-time white noise in time has infinite quadratic variation and is not a semimartingale, and also investigated temporal central limit theorems for modifications of the quadratic variation of the stochastic heat equation with space-time white noise in time.
The analysis of the asymptotic behavior of the realized variations is motivated by the study of the exact rates of convergence of some approximation schemes of scalar stochastic differential equations driven by a Brownian motion B (see, e.g., [11,12]), besides, of course, the traditional applications of the realized variations to parameter estimation problems (see, e.g., [13][14][15][16][17][18][19] in which asymptotic distributions for power variations of fractional Brownian motion (FBM) and related Gaussian processes were investigated).
In this paper we show that the realized power variation of the process U and its gradient in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions.It builds on and complements Allouba and Xiao's earlier works on the LKS-SPDEs and builds on the recent works on delicate analysis of variations of Gaussian processes and stochastic heat equations with space-time white noise.Our proof is based on the approach method in [11].We make use of the product-moments of various orders of the normal correlation surface of two variates in [20] to establish exact convergence rates of variances of the realized power variation of the process U and its gradient in time.On one hand, this work builds on the recent works on delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space-time white noise.Moreover, it builds on and complements Allouba's earlier works on the LKS-SPDEs and their gradient.
The rest of the paper is organized as follows.Some notations and main results of this paper are stated in Section 2. In Section 3, we discuss the rigorous LKS-SPDE kernel SIE (mild) formulation and estimate the temporal increments of LKS-SPDEs and their gradient by using the LKS-SPDE kernel SIE formulation and spectral/harmonic analysis.As a consequence of the result obtained, both LKS-SPDEs and their gradient in time have infinite quadratic variation.In Section 4, we prove Theorems 1 and 2 by using the productmoments of various orders of the normal correlation surface of two variates in [20] and the approach method in [11], respectively.In the final section, the results are summarized and discussed.

Exact Convergence Rates of Variances and Temporal CLTs for the Realized Power Variations of LKS-SPDEs
In order to establish our main results we first introduce some notation.We consider discrete Riemann sums over a uniformly spaced time partition t j = j∆t, where ∆t = n −1 .Fix x ∈ R d .Let ∆U x;j = U(t j , x) − U(t j−1 , x) and σ x;j = (E[∆U 2 x;j ]) 1/2 .For any p ∈ N + and n ∈ N + , we define Here and in the sequel, a denotes an integer satisfying a − 1 < a ≤ a for a ∈ R + .Let µ p denote the p-moment of a standard Gaussian random variable following an N (0, 1) law, that is, and Here Γ(s) = ∞ 0 u s−1 e −u du, s > 0, is the Gamma function.We will first show the exact convergence rates of variance for the realized power variation of processes U. Theorem 1. Fix (ε, ϑ) ∈ R + × R and x ∈ R d , and assume d ∈ {1, 2, 3}.Assume that u 0 ≡ 0 and ϑ = 0 in (1).Then for each fixed t > 0 and any p ∈ N + , as n tends to infinity.
By (4), we have the following convergence in probability for the realized power variation of the process U.
uniform convergence in probability in the time interval [0, T] with some T > 0.Moreover, (5) implies that for a fixed point in space, the process U(•, x) has infinite quadratic variation.
Temporal central limit theorems (CLTs) for the realized power variation of processes U is as follows.
Theorem 2. Fix (ε, ϑ) ∈ R + × R and x ∈ R d , and assume d ∈ {1, 2, 3}.Assume that u 0 ≡ 0 and ϑ = 0 in (1).Then for any p ∈ N + , as n tends to infinity, where B = {B(t), t ∈ [0, T]} is a Brownian motion independent of the process U, and the convergence is in the space D([0, T]) We will first show the exact convergence rates of variance for the realized power variation of the gradient processes ∂ x U(t, x).Theorem 3. Fix (ε, ϑ) ∈ R + × R and x ∈ R, and assume d = 1.Assume that u 0 ≡ 0 and ϑ = 0 in (1).Then for each fixed t > 0 and any p ∈ N + , as n tends to infinity.
By (8), we have the following convergence in probability for the realized power variation of the gradient process ∂ x U(t, x).Corollary 2. Fix (ε, ϑ) ∈ R + × R and x ∈ R, and assume d = 1.Assume that u 0 ≡ 0 and ϑ = 0 in (1).Then for each fixed t > 0 and any p ∈ N + , in L 2 and in probability as n tends to infinity.
uniform convergence in probability in the time interval [0, T] with some T > 0.Moreover, (9) implies that for a fixed point in space, the gradient process ∂ x U(•, x) has infinite quadratic variation.
Temperal central limit theorems for the realized power variation of the gradient processes ∂ x U(t, x) is as follows.
Theorem 4. Fix (ε, ϑ) ∈ R + × R and x ∈ R, and assume d = 1.Assume that u 0 ≡ 0 and ϑ = 0 in (1).Then for any p ∈ N + , as n tends to infinity, where B = {B(t), t ∈ [0, T]} is a Brownian motion independent of the process U, and the convergence is in the space D([0, T]) 2 equipped with the Skorohod topology.

Remark 4.
It is natural to expect that ( 6) and (10) hold for x → U(t, x) in d = 1, 2, 3.However, substantial extra work is needed for proving these statements.In particular, in order to apply the method in [11], one will have to establish the property of the increments for U(t, •).Unfortunately the method in [11] does not seem useful anymore and some new ideas may be needed.
Remark 5.By using Lemma 3 below, following the same lines as the proof of Theorem 1, we get Theorem 3. Similarly, following the same lines as the proof of Theorem 2, we get Theorem 4. Therefore, only Theorems 1 and 2 are proved and Theorems 3 and 4 are omitted.

Rigorous Kernel Stochastic Integral Equations Formulations
As in [4], for the LKS-SPDE, we use the LKS kernel to define their rigorous mild SIE formulation.This LKS kernel, as shown in as in [1][2][3], is the fundamental solution to the deterministic version of (12) (a ≡ 0 and b ≡ 0) below, and is given by: where i = √ −1 and Then, the rigorous LKS kernel SIE (mild) formulation is the stochastic integral equation (see p. 530 in [5] and Definition 1.1 and Equation (1.11) in [1]).Of course, the mild formulation of (1.1) is then obtained by setting a ≡ 1 and b ≡ 0 in (13).
Notation 1. Positive and finite constants (independent of x) in Section i are numbered as c i,1 , c i,2 , ....We conclude this section by citing the following spatial Fourier transform of the (ε, ϑ) LKS kernels from Lemma 2.1 in [4].11) is given by Here, the following symmetric form of the spatial Fourier transform has been used:

Estimates on the Temporal Increments of LKS-SPDEs and Their Gradient
Since U(•, x) is a centered Gaussian process, its law is determined by its covariance function, which is given in the following lemma.We also derive some needed estimates on the covariance function and the increment of U(•, x).Lemma 2. Fix (ε, ϑ) ∈ R + × R and x ∈ R d , and assume d ∈ {1, 2, 3}.Assume that u 0 ≡ 0 and ϑ = 0 in (1).For all s, t ∈ (0, T], we have and where K d is given in (3).
To show (17), we introduce the following auxiliary Gaussian random field where a + = max{a, 0} for all a ∈ R. Then the LKS-SPDE solution U may be decomposed as This idea of decomposition originated in [23] in the second order SPDEs setting; and it has been applied in [24,25], also in the second order heat SPDE setting.Fix x ∈ R d .By Theorem 3.1 in [4], one has for any 0 < s < t, Fix x ∈ R d .We apply Parseval's identity to the integral in y to get that for any 0 < s < t: Equation ( 24) becomes Now, we apply Parseval's identity to the inner integral in r.To this end, let Hence, by Parseval's identity, we see that for each 0 < s < t Equation ( 26) becomes Since |1 − e −u | ≤ 2u for all u ≥ 0, one has that for each 0 < s < t Equation ( 27) becomes This yields (17).The proof of Lemma 2 is completed.
Since ∂ x U(•, x) is a centered Gaussian process, its law is determined by its covariance function, which is given in the following lemma.We also derive some needed estimates on the increment of ∂ x U(•, x).Lemma 3. Fix (ε, ϑ) ∈ R + × R and x ∈ R, and assume d = 1.Assume that u 0 ≡ 0 and ϑ = 0 in (1).For all s, t ∈ (0, T], we have where D 0 is given in (7).
Fix x ∈ R. We apply Parseval's identity to the integral in y to get that for any 0 < s < t: Equation ( 34) becomes Now, we apply Parseval's identity to the inner integral in r.To this end, let Hence, by Parseval's identity, we see that for each 0 < s < t Equation (36) becomes Since |1 − e −x | ≤ 2x for all x ≥ 0, one has that for each 0 < s < t Equation (37) becomes  (38) Thus, by using similar argument of the proof of ( 17), (31) holds.The proof of Lemma 3 is completed.

Exact Convergence Rates of Variances for LKS-SPDEs
We need the following product-moment of various orders of the normal correlation surface of two variate, which are Equations (viii) and (ix) in [20].
Proof of Corollary 1. Write Obviously, the third term of (67) tends to zero as n → ∞.It follows from (43) (with r = p) and ( 45) that the second term of (67) tends to zero as n → ∞.Thus, by ( 4), one has This proves (5).

Temporal CLTs for LKS-SPDEs
The following lemma is needed to prove Theorem 2.
Lemma 5. Let X 1 , ..., X 4 be mean zero, jointly normal random variables, such that E[X 2 j ] = 1 and Furthermore, there exists ε > 0 such that Proof.Following the same lines as the proof of Lemma 3.3 in [11] with h j (X j ) = Z j , 1 ≤ j ≤ 4, we get Lemma 5 immediately.
To show that a sequence of cadlag processes {F n } is relatively compact, it suffices to show that for each T > 1, there exist constants β > 0, C > 0, and q > 1 such that for all n ∈ N, all t ∈ [0, T] and all h ∈ [0, t].(See, e.g., Theorem 3.8.8 in [26].)Taking β = 2 and using (71) together with Hölder inequality gives If nh < 1/2, then the right-hand side of this inequality is zero.Assume nh ≥ 1/2.Then nt The other factor is similarly bounded, so that R Θ n r (U(•,x)) (t, h) ≤ c 6,18 h 2 .
Proof.Let {n(j)} ∞ j=1 be any sequence of natural numbers.We will prove that there exists a subsequence {n(j m )} such that Θ (U(•, x)) s converges in law to the given random variable.
For each m ∈ N + , choose n(j m ) ∈ {n(j)} such that n(j m ) > n(j m−1 ) and n(j m ) ≥ Let us now introduce the filtration where λ denotes Lebesgue measure on R d+1 .Let τ k = n(j m ) −1 u k−1 .For each pair (i, k) such that u k−1 < i ≤ u k , define Note that ξ x;i,k is F τ k+1 -measurable and independent of F τ k .Recall that Moreover, given constants 0 ≤ τ ≤ s ≤ t, one has It follows from ( 80) and (81) that This yields that {ξ x;i,k } has the same law as {∆U where This, together with (17), gives Thus, since ∆U x;i − ξ x;i,k is Gaussian, by ( 40) and ( 84), one has Thus, by (85) and Hölder inequality, Similarly, by (84) and Lagrange mean value theorem, Therefore, by (86), (87) and Hölder inequality, But since n(j m ) was chosen so that n(j m ) ≥ m in order to complete the proof.
For this, we will use the Lindeberg-Feller theorem (see, e.g., Theorem 2.4.5 in [27]), which states the following: for each m, let To verify these conditions, recall that {ξ x;i,k } and {∆U x;i−u k−1 } have the same law, so that Hence, by (71), Jensen inequality now gives m , so that by passing to a subsequence, we may assume that (a) holds for some ν ≥ 0.
For (b), let δ > 0 be arbitrary.Then By the Skorohod representation theorem, we may assume that the convergence is a.s.By Proposition 1, the family and the proof is complete.
Proof of Theorem 2. It is sufficient to prove (6) for the even p case since the odd p case can be proved similarly.Let {n(j)} ∞ j=1 be any sequence of natural numbers.By Proposition 1, the sequence {(U(•, x), Θ n(j) p (U(•, x)))} is relatively compact.Therefore, there exists a subsequence {n(j k )} and a cadlag process Y such that (U( As in the proof of Proposition 2, η x;n(j k ) → 0 in probability.It therefore follows that x ].It is easy to see that Σ x is invertible.Hence, we may define the vectors v x;j ∈ R by v x;j = E[Z x ∆U x;j ], and w x;j = Σ −1 x v x;j .Let ξ x;j = ∆U x;j − w T x;j Z x , so that ξ x;j and Z x are independent.Define  Note that by (42) and Hölder inequality, one has |E[U(s i , x)∆U x;j ]| ≤ c 6,40 σ x;j ≤ c 6,41 n −(1−d/4)/2 for all 1 ≤ i ≤ and 1 ≤ j ≤ nt , and that by (15) and Lagrange mean value theorem, for any 1 ≤ i ≤ and 1 ≤ j ≤ nt , E[U(s i , x)∆U x;j ] = K d ((s i + t j ) 1−d/4 − (s i + t j−1 ) 1−d/4 − (s i − t j ) 1−d/4 + (s i − t j−1 ) This finish the proof.

Conclusions
In this paper, we have presented that the realized power variations for the fourth order LKS-SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions.We are concerned with the fluctuation behavior, with delicate analysis of the realized variations, of the sample paths for the above class of equations and their gradient, and complement Allouba's earlier works on the spatio-temporal Hölder regularity of LKS-SPDEs and their gradient.These asymptotic distributions are expressed in terms of the parameters of the problem, and may be used to analyze how the fluctuation behavior depends on those parameters.