Complexity
Volume 2018 (2018), Article ID 7402764, 17 pages
https://doi.org/10.1155/2018/7402764

Research Article

Moderate Deviations for Stochastic Fractional Heat Equation Driven by Fractional Noise

Xichao Sun,¹ Ming Li,² and Wei Zhao³

¹College of Science, Bengbu University, 1866 Caoshan Rd., Bengbu 233030, China
²School of Information Science and Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China
³American University of Sharjah, P.O. Box 26666, Sharjah, UAE

Correspondence should be addressed to Xichao Sun; sunxichao626@126.com

Received 28 January 2018; Revised 6 April 2018; Accepted 2 May 2018; Published 10 July 2018

Academic Editor: Maricel Agop

Copyright © 2018 Xichao Sun 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.

Abstract.

We consider a class of stochastic fractional heat equations driven by fractional noises. A central limit theorem is given, and a moderate deviation principle is established.

1. Introduction

Since the work of Freidlin and Wentzell [1], the large deviation principle (LDP) has been extensively developed for small noise systems and other types of models (such as interacting particle systems) (see [2–7]). Cardon-Weber [2] proved a LDP for a Burgers-type SPDE driven by white noise. Marquez-Carreras and Sarra [3] proved a LDP for a stochastic heat equation with spatially correlated noise, and Mellali and Mellouk [4] extended Marquez-Carreras and Sarra’s [3] to a fractional operator. Jiang et al. [5] proved a LDP for a fourth-order stochastic heat equation driven by fractional noise. Budhiraja et al. [6] studied large deviation properties of systems of weakly interacting particles. Budhiraja et al. [7] proved a large deviation for Brownian particle systems with killing.

Similar to the large deviation, the moderate deviation problems also come from the theory of statistical inference. Using the moderate deviation principle (MDP), we can get the rate of convergence and an important method to construct asymptotic confidence intervals, for example, Liming [8], Guillin and Liptser [9], Cattani and Ciancio [10], and other references therein. There are also many works about MDP about stochastic (partial) differential equations; some surveys and literatures could be found in Budhiraja et al. [11], Wang and Zhang [12], Li et al. [13], Yang and Jiang [14], and the references therein. On the other hand, fractional equations have attracted many physicists and mathematicians due to various applications in risk management, image analysis, and statistical mechanics (see Droniou and Imbert [15], Bakhoum and Toma [16], Levy and Pinchas [17], Mardani et al. [18], Niculescu et al. [19], Paun [20], and Pinchas [21] for a survey of applications). Stochastic partial differential equations involving a fractional Laplacian operator have been studied by many authors; see Mueller [22], Wu [23], Liu et al. [24], Wu [25], and the references therein.

Motived above, we investigated the moderate deviations about the stochastic fractional heat equation with fractional noise as follows:

where , , is the fractional Laplacian operator which is defined in Appendix, and denotes a fractional noise which is fractional in time and white in space with Hurst parameter ; that is, is a mean zero Gaussian random field on with covariance.

Assume that the coefficients satisfy the following.

Assumption 1. Function is Lipschitz; that is, there exist an satisfying

Under the conditions of Assumption 1, (1) possesses a unique solution in the sense of Walsh [26] as follows:

As the parameter , the solution of (1) will tend to which is the solution to the following equation:

This paper mainly devotes to investigate the deviations of from the deterministic solution , as , that is, the asymptotic behavior of the trajectories.

where is the same deviation scale that strongly influences the asymptotic behavior of .

If , we are in the domain of large deviation estimate, which can be proved similarly to Jiang et al. [5].

The case provides the central limit theorem. As , we will prove that converges to a random field in this paper.

To fill the gap between scale and scale , we mainly devote to the moderate deviation when the scale satisfies the following:

This paper is organized as follows. In Section 2, the definition of the fractional noise is given. In Section 3, the main result is given and proved. In Appendix, some results about the Green kernel are given.

2. Fractional Noise

Let , and is a centered Gaussian family of random variables with the covariance satisfying

with and covariance kernel , where denotes the Lebesgue measure of the set and denotes the class of Borel sets in .

We denote as the set of step functions on . Let be the Hilbert space defined as the closure of with respect to the scalar product.

According to Nualart and Ouknine [27], the mapping can be extended to an isometry between and the Gaussian space associated with and denoted by

Define the linear operator by

where is defined by

with , and one can get

Moreover, satisfies the following:

Then, since

one can get

where and in are any step functions. So the operator gives an isometry between the Hilbert space and . Hence, defined by

is a space-time white noise, and has the following form:

Therefore, the mild formulation of (4) has the following form:

That is, the last term of (4) is equal to

The following embedding proposition is given by Nualart and Ouknine [27].

Lemma 1. Set , then, we have

3. Main Results and Their Proof

3.1. Main Results

For any function defined on , let

where , , and . Let

Let be the functions which satisfy , endowed with the .

Define

which is a Cameron-Martin space endowed with the norm

Suppose . Now, let be the solution of the following deterministic equation:

where

satisfies the following:

The above (26) shows a unique mild solution.

Similar to Jiang et al. [5], one can get the following:

Theorem 1. Let . Under Assumption¹, the law of the solution to (1) satisfies a deviation principle on with the good rate function:

with the convention , where .
More precisely, for any Borel measurable subset B of ,

where and denote the interior and the closure of , respectively.

We furthermore suppose that the coefficients satisfy the following.

Assumption 2. is differentiable, and the derivative of is Lipschitz. That is to say, there exist positive constant and which satisfy the following:

Together with the Lipschitz of , we conclude that

Now, we give the following central limit theorem.

Theorem 2. Let and its derivative satisfy Assumptions 1 and 2. Then, for , converges in to a random field on with , determined by

for all, .

Let the function be the solution to the following partial differential equation:

Under Assumptions 1 and 2, by Theorem 1, one can get which satisfies large deviation principles on with the speed and the good rate function satisfies the following:

Now, the second result is given as follows:

Theorem 3. In moderate deviation principle, let . Under the Assumptions¹and², then, the random field satisfies a large deviation principle on the space with speed and the good rate function defined by (36), where .

3.2. Convergence of the Solution

Lemma 2. Let . Under Assumption¹, then, there exists a unique solution to (1). Moreover, for any ,

As , we get the convergence of as follows:

Proposition 1. Let and . By Assumption 1, there exists a constant which satisfies the following:

Proof. Note that

One can get

Set

Together with Hölder’s inequality, the Lipschitz condition (C) and (5) of Lemma A.1, for , we have

where .
For any , , , by (21) and (A.9), there exist which satisfies the following:

Similarly,

By (21), (A.10), and (A.11), for , one can get

and

Together with (43), (44), (45), and (46), one can get for any and , there exists a constant such that

where is independent of . For , by Garsis-Rodemich-Rumser’s Lemma, there exist a constant C and a random variable satisfying

and

If we choose in (48), we can get

By Gronwall’s inequality and (40), (42), and (48), there exists a constant satisfying

The proof is completed.

3.3. The Proof of the Main Results

We first prove Theorem 1. Our proof is based on the following proposition (see Doss and Priouret [28]).

Proposition 2. Suppose and are two families of random variables and and are two Polish spaces. Suppose that

(1)There exists a map such that is continuous for any .(2) satisfies with the rate functions .(3)For any , there exist and satisfying and for all , and

Then, satisfies a LDP with the rate functions

To prove Theorem 1, one only needs to prove

(i)Under some topology, is continuous for any .(ii)In Freidin-Wentzell inequality, for any , , and , there exist a satisfying

Theorem 4. When the level set is endowed with the topology of uniform convergence on ,

is the continuous map for any .

Proof. One only needs to prove that for fixed , ,

Note that

Using (A.5) in Appendix with , then , one can get

Now, we deal with , together with

Recalling is defined by (11), one can get

Using Gronwall’s inequality, we can get

The proof of the theorem is completed.

We now prove the Freidin-Wentizell inequality as follows:

Suppose is a Brownian sheet. For and , we define

and

Using Girsanov’s theorem, the process is a Brownian sheet under . Suppose is a solution of (1) under . Then,

Now, one can prove (36). Note that, under , then,

So under , by Gronwall’s Lemma, one can get

Now, one can change (34) the proof to the following theorem.

Theorem 5. Suppose and . For each , , and , there exists a constant satisfying

In the following, we give a key Lemma to prove Theorem 1, which is similar to Candon-Weber [2], and the proof is omitted.

Lemma 3. Suppose , and satisfying

for any and . Suppose which is an almost surely continuous, -adapted process satisfying , and for , suppose

Then, for any , there exist a positive constant C, Ç(), and such that for allÇ(),

Proof of Theorem 5. Suppose

Then, there exists satisfying

If and , by Lemma 2, we can get

The proof of Theorem 3 is completed.

In the following, we first give Garsis-Rodemich-Rumser’s Lemma in Bally et al. [29].

Lemma 4. Assume and are a family of real-valued stochastic processes. Suppose the following is true.

Assumption 3. One has

as .

Assumption 4. For any , there exists satisfying

Then, for any ,

Now we can prove Theorem 2.

Proof of Theorem 2. Set . We will prove that

To this end, we need to prove that Assumptions 3 and 4 are satisfied for . Note that

where

and

Using Taylor’s formula, there exists a such that

Note that is Lipschitz continuous and ; one can get

So

Using Hölder’s inequality, for and , one can get

where . Together with (5) of Lemma A.1 and Proposition 1, there exists a constant only depending on satisfying

Since , for , together with Hölder’s inequality and (5) of Lemma A.1, we can get

where . Together with (52), (56), and (57), we can get

Using Gronwall’s inequality, one can get

which implies Assumption 3 in Lemma 4.
Now we prove Assumption 4 in Lemma 4. We will prove and satisfy Assumption 4 in Lemma 4. Using Hölder’s inequality and (A.9), one can get

where and . Similarly,

where . Together with (89) and (90), we can get

Similarly, one can get

Together with (91) and (92), we can get

Then, for any and such that and , using Lemma 4,

The proof of Theorem 2 is completed.

Proof of Theorem 3. By Theorem 1, obeys large deviation principles on , with the rate function I given by (30) and the speed function . Using Dembo and Zeitouni ([30] Theorem 4.2.13), to prove the large deviation principles of is -exponentially equivalent to , that is,

holds for any .
Since

To prove (67), we only need to prove

Note the decomposition

For any , , and , by Hölder’s inequality, (32), and (A.9), we can get

where . Similarly,

where . Together with (99) and (100), we can get

Note that , we have

Thus, for , we have

Applying Gronwall’s Lemma to , we have

By (64) to prove (67), we only need to prove that for any ,

Note that

where

By the same method in the proof of (59), we have

Similar to the proof of (53), we have

Together with (40) and (A.11), for any , we can get that

where

By Gronwall’s Lemma, one can get

Applying Lemma 4 with

for any fixed , one can get that for any ,

We can get

Together with (73), (78), (82), (83), and (85), we have

This completes the proof.

Appendix

Green Function

The nonlocal factional differential operator is defined by

where is called the index of stability and improperly referred to as the skewness which satisfy and when .

The operator is a closed, densely defined operator on , and it is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction. This operator is a generalization of various well-known operators, such as the Laplacian operator (when ), the inverse of the generalized Riesz-Feller potential (when ), and the Riemann-Liouville differential operator (when or ). It is self-adjoint only when , and in this case, it coincides with the fractional power of the Laplacian. We refer the readers to Debbi and Dozzi [31] for more details about this operator.

According to Komatsu [32], can be represented for , by

and for , by

where are two nonnegative constants satisfying , is a smooth function for which the integral exists, and is its derivative. This representation identifies it as the infinitesimal generator for a nonsymmetric α-stable Lévy process.

Suppose is the fundamental solution to the following equation:

where is the Dirac distribution. Using Fourier transform, one can get which is given by

Let us list some known facts on which will be used later on (see, e.g., Debbi and Dozzi [31]).

Lemma A.1. Suppose one can get the following:

(1) is not symmetric with respect to .(2)For any and ,

or equivalently

(3)For any , (4)For , there exist constants and satisfying

(5) if and only if .

Lemma A.2. For and , we have

where is a constant.

Proof. For any and ,

By the mean-value theorem, for , one can get that

and

Therefore, if , that is, , then,

Similarly, we can check

Hence, the inequality (A.9) holds. As for the inequality (A.10), for any and ,

By mean-value theorem, it holds that

Note that

Hence, one can attain

So

Therefore, if i.e. , then, for ,

Similarly,

So (A.10) holds. Now let us prove that (A.11) holds. For any and ,

The proof of the lemma is completed.

Lemma A.3. Suppose , , and satisfying

Suppose is the Green kernel, or with . We define by

with . Then, is a bounded linear operator which satisfies the following:

(1)When , then, for all ,

(2)When , then, for all ,

Proof. Together with Minkowski’s inequality, Young inequality and (4) of Lemma A.1. We can get

where one has used the result that, for ,

The proof of (2) is omitted since it is similar to case (1). The proof of this lemma is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Acknowledgments

Xichao Sun is partially supported by the Natural Science Foundation of Anhui Province (1808085MA02) and the Natural Science Foundation of Bengbu University (2017GJPY04, 2016KYTD02, 2017jxtd2, and 2017ZR08). Ming Li is partially supported by National Natural Science Foundation of China under the project Grant nos. 61672238, 61272402, and 61070214.

References

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, NY, USA, 1984.

C. Cardon-Weber, “Large deviations for a Burgers’-type SPDE,” Stochastic Processes and their Applications, vol. 84, no. 1, pp. 53–70, 1999.

D. Marquez-Carreras and M. Sarra, “Large deviation principle for a stochastic heat equation with spatially correlated noise,” Electronic Journal of Probability, vol. 8, no. 12, 2003.

T. El Mellali and M. Mellouk, “Large deviations for a fractional stochastic heat equation in spatial dimension driven by a spatially correlated noise,” Stochastics and Dynamics, vol. 16, no. 1, article 1650001, 2016.

Y. M. Jiang, K. H. Shi, and Y. J. Wang, “Large deviation principle for the fourth-order stochastic heat equations with fractional noises,” Acta Mathematica Sinica, English Series, vol. 26, no. 1, pp. 89–106, 2010.

A. Budhiraja, P. Dupuis, and M. Fischer, “Large deviation properties of weakly interacting processes via weak convergence methods,” The Annals of Probability, vol. 40, no. 1, pp. 74–102, 2012.

A. Budhiraja, W. T. Fan, and R. Wu, “Large deviations for Brownian particle systems with killing,” Journal of Theoretical Probability, vol. 6, pp. 1–40, 2016.

W. Liming, “Moderate deviations of dependent random variables related to CLT,” The Annals of Probability, vol. 23, no. 1, pp. 420–445, 1995.

A. Guillin and R. Liptser, “Examples of moderate deviation principle for diffusion processes,” Discrete and Continuous Dynamical Systems - Series B, vol. 6, no. 4, pp. 803–828, 2006.

C. Cattani and A. Ciancio, “On the fractal distribution of primes and prime-indexed primes by the binary image analysis,” Physica A: Statistical Mechanics and its Applications, vol. 460, pp. 222–229, 2016.

A. Budhiraja, P. Dupuis, and A. Ganguly, “Moderate deviation principles for stochastic differential equations with jumps,” The Annals of Probability, vol. 44, no. 3, pp. 1723–1775, 2016.

R. Wang and T. Zhang, “Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise,” Potential Analysis, vol. 42, no. 1, pp. 99–113, 2015.

Y. Li, R. Wang, and S. Zhang, “Moderate deviations for a stochastic heat equation with spatially correlated noise,” Acta Applicandae Mathematicae, vol. 139, no. 1, pp. 59–80, 2015.

J. Yang and Y. Jiang, “Moderate deviations for fourth-order stochastic heat equations with fractional noises,” Stochastics and Dynamics, vol. 16, no. 6, article 1650022, 2016.

J. Droniou and C. Imbert, “Fractal first-order partial differential equations,” Archive for Rational Mechanics and Analysis, vol. 182, no. 2, pp. 299–331, 2006.

E. G. Bakhoum and C. Toma, “Alternating coordinate-momentum representation for quantum states based on bopp operators for modelling long-distance coherence aspects,” Mathematical Problems in Engineering, vol. 2015, Article ID 818019, 7 pages, 2015.

C. Levy and M. Pinchas, “Maximum likelihood estimation of clock skew in IEEE 1588 with fractional Gaussian noise,” Mathematical Problems in Engineering, vol. 2015, Article ID 174289, 24 pages, 2015.

A. Mardani, M. R. Hooshmandasl, M. H. Heydari, and C. Cattani, “A meshless method for solving the time fractional advection–diffusion equation with variable coefficients,” Computers & Mathematics with Applications, vol. 75, no. 1, pp. 122–133, 2018.

O. Niculescu, D. G. Dimitriu, V. P. Paun, P. D. Matasaru, D. Scurtu, and M. Agop, “Experimental and theoretical investigations of a plasma fireball dynamics,” Physics of Plasmas, vol. 17, no. 4, article 042305, 2010.

V. Paun, “Fractal surface analysis of Zircaloy-4 SEM micrographs using the time-series method,” Central European Journal of Physics, vol. 7, no. 2, 2009.

M. Pinchas, “Symbol error rate for nonblind adaptive equalizers applicable for the SIMO and FGn case,” Mathematical Problems in Engineering, vol. 2014, Article ID 606843, 11 pages, 2014.

C. Mueller, “The heat equation with Lévy noise,” Stochastic Processes and their Applications, vol. 74, no. 1, pp. 67–82, 1998.

J. Wu, “Fractal Burgers equation with stable Lévy noise,” in International Conference SPDE and Applications-VII, pp. 4–10, 2004.

J. Liu, L. Yan, and Y. Cang, “On a jump-type stochastic fractional partial differential equation with fractional noises,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 16, pp. 6060–6070, 2012.

D. Wu, “On the solution process for a stochastic fractional partial differential equation driven by space–time white noise,” Statistics & Probability Letters, vol. 81, no. 8, pp. 1161–1172, 2011.

J. B. Walsh, “An introduction to stochastic partial differential equations,” in École d'Été de Probabilités de Saint Flour XIV – 1984, vol. 1180 of Lecture Notes in Mathematics, pp. 265–439, Springer, Berlin, Heidelberg, 1986.

D. Nualart and Y. Ouknine, “Regularization of quasilinear heat equations by a fractional noise,” Stochastics and Dynamics, vol. 4, no. 2, pp. 201–221, 2004.

H. Doss and P. Priouret, “Petites perturbations de systemes dynamiques avec reflexion,” in Séminaire de Probabilités XVII 1981/82, vol. 986 of Lecture Notes in Mathematics, pp. 353–370, Springer, Berlin, Heidelberg, 1983.

V. Bally, A. Millet, and M. Sanz-Sole, “Approximation and support theorem in Holder norm for parabolic stochastic partial differential equations,” The Annals of Probability, vol. 23, no. 1, pp. 178–222, 1995.

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Second edition, 1998.

L. Debbi and M. Dozzi, “On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension,” Stochastic Processes and their Applications, vol. 115, no. 11, pp. 1764–1781, 2005.

T. Komatsu, “On the martingale problem for generators of stable processes with perturbations,” Osaka Journal of Mathematics, vol. 21, pp. 113–132, 1984.