Asymptotic Distributions for Power Variations of the Solution to the Spatially Colored Stochastic Heat Equation

Wang, Wensheng; Chang, Xiaoying; Liao, Wang

doi:https://doi.org/10.1155/2021/8208934

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Stability and Bifurcation Analysis of Discrete Dynamical Systems 2021

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Research Article | Open Access

Volume 2021 | Article ID 8208934 | https://doi.org/10.1155/2021/8208934

Asymptotic Distributions for Power Variations of the Solution to the Spatially Colored Stochastic Heat Equation

Wensheng Wang,¹Xiaoying Chang,¹and Wang Liao¹

Academic Editor: Mehmed Nurkanovic

Received30 May 2021

Accepted15 Sept 2021

Published12 Oct 2021

Abstract

Let be the solution to the stochastic heat equations (SHEs) with spatially colored noise. We study the realized power variations for the process , in time, having infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. We use the underlying explicit kernels and spectral/harmonic analysis, yielding temporal central limit theorems for SHEs with spatially colored noise. This work builds on the recent works on delicate analysis of variations of general Gaussian processes and SHEs driven by space-time white noise.

1. Introduction

Throughout this work, we will consider the following -dimensional stochastic heat equation (SHE):with and Gaussian space-time colored noise . The noise is assumed to have a particular covariance structure (see [1]):wherewith . The initial condition, , is taken to be bounded and -Hölder continuous. We will also assume to be Lipschitz continuous, and there exists such that and . Stochastic PDEs (SPDEs) such as (1) have been studied in [1–6] and others.

It is known (see [1, 7–10]) that (1) admits a unique mild solution if and only if , and this mild solution is interpreted as the solution of the following integral equation:for , where the above integral is a Wiener integral with respect to the noise (see, e.g., [2] for the definition) and is the Green kernel of the heat equation given by

Bezdek [11] investigated weak convergence of probability measures corresponding to the solution of (1) in . He showed that probability measures corresponding to weakly converge to those corresponding to the solution to the SHE with white noise when , that is, the solution of (1) converges in the appropriate sense to the solution of the same equation, but with white noise instead of colored noise as . By that, we mean the solution towhere denotes white noise. SPDEs such as (6) have been studied in [1, 2, 7, 10, 12, 13] and others.

Among others, Tudor and Xiao [14] investigated the exact uniform and local moduli of continuity and Chung-type laws of the iterated logarithm of the process in time. In fact, they investigated these path properties for a more wide class, namely, the solution to the linear SHE driven by a fractional noise in time with correlated spatial structure. Swanson [13] showed that the solutions of the SHEs in (6) with , in time, have infinite quadratic variation and are not semimartingales and also investigated central limit theorems (CLTs) for modifications of the quadratic variations of the solutions of the SHEs with white noise. Pospíšil and Tribe [12] investigated the quartic variations of the solutions of the SHEs in (6) with , in time, having Gaussian asymptotic distributions. Inspired by Swanson [13] and Pospíšil and Tribe [12], in this work, we show that the realized power variations of the solutions of the SHEs in (1) with colored noise, in time, have infinite quadratic variation and Gaussian asymptotic distributions.

For , the -power variation of a process , with respect to a subdivision of , is defined to be the sum

For simplicity, consider from now on the case where , for and . In this work, we wish to point out some interesting phenomena when is the solution to a SHE with colored noise. In fact, we will also drop the absolute value (when is odd). More precisely, we will considerwhere denotes the increment .

The analysis of the asymptotic behavior of quantities of type (8) is motivated, for instance, by the study of the exact rates of convergence of some approximation schemes of scalar stochastic differential equations driven by a Brownian motion (BM) (see, e.g., [15–17]), besides, of course, the traditional applications of quadratic variations to parameter estimation problems.

Now, let us recall some known results concerning the -power variations (for ), which are today more or less classical. First, assume that is the standard BM. Let denote the -moment of a standard Gaussian random variable following an law, that is, and for all . By the scaling property of the BM and using the CLT, it is immediate that (see, e.g., [17]), as :

Assume that , that is, the case where the fractional Brownian motion (FBM) has no independent increments anymore. Then, (9) has been extended by Corcuera et al. [15], Nourdin [17], Dobrushin and Major [18], Taqqu [19], Breuer and Major [20], Giraitis and Surgailis [21], Wang [22], and Wang and Wang [23]. Swanson [13] extended (9) to modifications of the quadratic variation of the solutions of SHE driven by space-time white noise. Motivated by (9), in this work, we show that (9) with different mean and variance also holds for the solution to SHE with colored noise.

Our proofs are based on the method of Swanson [13]. We make use of the product moments of various orders of the normal correlation surface of two variates in Pearson and Young [24] to establish exact convergence rates of variances of the realized power variation of the process with respect to time. This work builds on the recent works on delicate analysis of variations of general Gaussian processes and SHEs driven by space-time white noise.

2. Results

In order to state our results, we first introduce some notations. Let , where is fixed. We consider discrete Riemann sums over a uniformly spaced time partition , where . Let and . For any and , we define

Here and in the sequel, denotes an integer satisfying for .

Let . For , let . For real number , define . It follows from (44) below that is a positive and finite constant depending on , and . For any , we putwherewhere , , is the gamma function.

We will first show the exact convergence rate of variance for the realized power variation of the process .

Theorem 1. Fix and and assume . Assume that and in (1). Then, for each fixed and any ,as tends to infinity.

By (13), we have the following convergence in probability for the realized power variation of the process .

Corollary 2. Fix and and assume . Assume that and in (1). Then, for each fixed and any ,in and in probability as tends to infinity.

Remark 3. Since is monotone, (14) implies that uniform convergence in probability in the time interval with some . Moreover, (14) implies that the process has infinite quadratic variation.

Example 4. If and , the 4-th variation, namely, in (14), the corresponding constant of the right-hand side of (14) is equal to .
The CLT for the realized power variation of the process is as follows.

Theorem 2. Fix and and assume . Assume that and in (1). Then, for any ,as tends to infinity, where is a BM independent of the process , and the convergence is in the space equipped with the Skorokhod topology.

Remark 6. Comparing (15) and (9), we have that the realized power variations of the process for share similar Gaussian asymptotic properties with those of BM.
Throughout this paper, positive and finite constants are numbered as c_2,1, c_2,2, … or c_3,1, c_3,2, ….

3. Proofs

3.1. Preliminaries

We need the following product moment of various orders of the normal correlation surface of two variates, which are equations (9) and (12) in Pearson and Young [24].

Lemma 7. Suppose that , where . Then,

We also derive some needed estimates on the covariance function and the variance function of increments of .

Lemma 8. Fix and and assume . Assume that and in (1). Then, for all ,andwhere is given in (12).

Proof. By Proposition 2.3 of Tudor [10], one has that (17) holds withBy the following integral formula (see Corollary on page 23 in [25]):the constant becomesThis is (12) and yields (17).
Equation (18) is cited from Theorem 2.2 in Tudor [10]. It remains to show (19). To show (19), we define the following pinned string process in time byNote that and can be expressed asIn the above, . Now for every , one has the following decomposition:whereFollowing the same lines as the Proof of Theorem 1 of Tudor and Xiao [14], for any ,Denote by the tempered non-negative measure on . Let denote the Fourier transform of the function and be the Riesz kernel defined in (3). Then, for any (see, e.g., [10, 14]),It follows from (28) that for any ,Since for all , one has for all and ,Thus, by (21), for any ,Combining (29) and (31), one hasIt follows from the argument of (29) thatThis, together with (27) and (32), yields (19). The proof of Lemma 8 is completed.

3.2. Proof of Theorem 1

Proof. of Theorem 1. It is sufficient to prove (13) for the even case since the odd case can be proved similarly. For , define . Note that for a random variable following an law,By (16) and (34), one hasIt follows from (18) thatBy (19), (36), and Lagrange mean value theorem, it holds that for any real number and ,Note that since , one has . Thus,It follows from (37) (with ) and (38) thatHence,It follows from (17) thatwhich simplifies towhere . Thus, by binomial expansion, for every and ,If we write , where , then for each , the Lagrange mean value theorem gives for some . This yields that for all ,and hence for any ,with some as .
Note that since , one hasNote that (44) gives and for all . Thus, by (36) and (46), for every and ,which tends to zero as since .
We now consider the term in (43). Let be a FBM with index , which is a centered Gaussian process with for . Then, for ,Thus,This yieldsBy (36) and (44), for every and any ,This, together with (45), yieldswhich tends to zero by letting .
By (37) (with ), (36), and (48), for every ,which tends to zero as since . Hence, for every ,Similarly, for every ,For every and any ,as and .
Note that for every and ,Hence, by (54)–(57), for every ,as and . It follows from (36) thatThis, together with (43), (47), and (58), yields for every ,Therefore, by (35), (40), and (60), one hasThis proves (13). The Proof of Theorem 1 is completed.

Proof. of Corollary 2. WriteObviously, the third term of (62) tends to zero as . It follows from (37) (with ) and (38) that the second term of (62) tends to zero as . Thus, by (13),This proves (14).

3.3. Proof of Theorem 2

The following lemma is needed to prove Theorem 2.

Lemma 9. Let be normal random variables with mean zero, and . Put . Then, for any ,whenever . Moreover,

Furthermore, there exists such thatwhenever for all .

Proof. Following the same lines as the proof of Lemma 3.3 in Swanson [13] with , , we get Lemma 9 immediately.

Proposition 10. Fix and and assume . Assume that and in (1). Fix . Put

Then, for all and all ,

The sequence is therefore relatively compact in the Skorokhod space .

Proof. We follow the method of Proposition 3.5 in Swanson [13] to prove (68). Let . For and , define and let . Define , and for , let . Further define and , where “med” denotes the median function. For , defineObserve thatand thatLet andThen,By (42) and (44), for all ,It follows from (36) and (74) thatSuppose . Fix and let be arbitrary. If , then . If , then . In either case, by (65), (36), (73), and (75), one hasIf , then and . Hence, by (64), (36), (73), and (75),Now choose such that . With given, is determined by . Since there are two possibilities for and possibilities for , . Therefore,For the second summation, suppose . In this case, if , then , so that by (66), (36), (73), and (75),Since and , one hasThus, using (70), (71), (78), and (80), one haswhich is (68).
To show that a sequence of càdlàg processes is relatively compact, it suffices to show that for each , there exist constants , , and such thatfor all , all , and all (see, e.g., Theorem 3.8.8 in [26]). Taking and using (68) together with Hölder inequality givesIf , then the right-hand side of this inequality is zero. Assume . Then,The other factor is similarly bounded, so that .

Proposition 11. Fix and and assume . Assume that and in (1). Then, for any and ,as , where is a standard normal random variable.

Proof. Let be any sequence of natural numbers. We will prove that there exists a subsequence such that converges in law to the given random variable.
For each , choose such that and . Let . For , define , so thatLet us now introduce the filtrationwhere denotes Lebesgue measure on . Let . For each pair such that , defineNote that is -measurable and independent of . Recall thatAlso, given constants , one hasIt follows from (89) and (90) thatThis yields that has the same law as .
Now define andso that , , are independent andwhereSince and are independent, one hasThis, together with (19), givesThus, since is Gaussian, by (34) and (96),Note that (34) and (36) give and . By Lagrange mean value theorem,Thus, by (97) and Hölder inequality,Similarly, by (96) and Lagrange mean value theorem,Therefore, by (99), (100), and Hölder inequality,Since , this givesBut since was chosen so that , one has and in and in probability. Therefore, by (93), one needs only to show thatin 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 , let , be independent random variables with . Suppose(a).(b)For all , .Then, as .
To verify these conditions, recall that and have the same law, so thatHence, by (68),Jensen inequality now gives , so that by passing to a subsequence, one may assume that (a) holds for some .
For (b), let be arbitrary. Then,which tends to zero as .
It therefore follows that as and it remains only to show that . For this, observe that the continuous mapping theorem implies that . By the Skorokhod representation theorem, one may assume that the convergence is a.s. By Proposition 10, the family is uniformly integrable. Hence, in , which implies . But by Theorem 1, , so and the proof is complete.

Proof. of Theorem 2It is sufficient to prove (15) for the even case since the odd case can be proved similarly. Let be any sequence of natural numbers. By Proposition 10, the sequence is relatively compact. Therefore, there exists a subsequence and a càdlàg process such that . Fix . With notation as in Proposition 11, letand defineAs in the proof of Proposition 11, in probability. It therefore follows thatNote that and are independent. Hence, , and are independent, which implies that and are independent. This yields that the process has independent increments.
By Proposition 11, the increment is normally distributed with mean zero and variance . Also, since for all . Hence, is equal in law to , where is a standard BM. It remains only to show that and are independent.
Fix . Let and . It is easy to see that is invertible. Hence, one may define the vectors by , and . Let , so that and are independent.
DefineThen,By (34), binomial expansion, and Hölder inequality,Note that by (36) and Hölder inequality, one has for all and , and note that by (17) and Lagrange mean value theorem, for any and ,where . Then, for any and ,which tends to zero as since . Thus,Since and are independent, this gives that and are independent.
We now can complete the proof. Note that by (37) and (38),This finishes the proof.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of Zhejiang Province (LY20A010020) and the National Natural Science Foundation of China (11671115).

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Copyright © 2021 Wensheng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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