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Advances in Mathematical Physics
Volume 2014 (2014), Article ID 479873, 10 pages
Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise
1Department of Mathematics and Physics, Bengbu College, 1866 Caoshan Road, Bengbu, Anhui 233030, China
2School of Mathematics and Statistics, Nanjing Audit University, 86 West Yu Shan Road, Pukou, Nanjing 211815, China
Received 21 September 2013; Revised 27 December 2013; Accepted 27 December 2013; Published 12 February 2014
Academic Editor: Ming Li
Copyright © 2014 Xichao Sun and Junfeng Liu. 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.
We consider a class of stochastic fractional equations driven by fractional noise on , with Dirichlet boundary conditions. We formally replace the random perturbation by a family of sequences based on Kac-Stroock processes in the plane, which approximate the fractional noise in some sense. Under some conditions, we show that the real-valued mild solution of the stochastic fractional heat equation perturbed by this family of noises converges in law, in the space of continuous functions, to the solution of the stochastic fractional heat equation driven by fractional noise.
In recent years, there has been considerable interest in studying fractional equations due to interesting properties and applications in various scientific areas including image analysis, risk management, and statistical mechanics (see Droniou and Imbert  and Uchaikin and Zolotarev  for a survey of applications). Much effort has been devoted to apply the fractional calculus to mathematical problems in science and engineering. For example, Chen et al.  and Li et al.  studied the fractional-order networks and Li  investigated fractal time series. More works on the fields can be found in [6–12] and the references therein. Stochastic partial differential equation involving fractional Laplacian operator (which is an integrodifferential operator) has been studied by many authors. For example, Mueller  and Wu  proved the existence of a solution of stochastic fractional heat and Burgers equation perturbed by a stable noise, respectively. Other related references are Chang and Lee , Truman and Wu , Liu et al. , Wu , and the references therein.
On the other hand, weak convergence to Brownian motion, fractional Brownian motion, and related stochastic processes have been considered extensively since the work of Taqqu  and Delgado and Jolis . Recently, many researchers are interested in studying weak convergence of stochastic differential equation. Some surveys could be found in Bardina et al. , Boufoussi and Hajji , and Mellall and Ouknine . Bardina et al.  studied the convergence in law, in the space of continuous functions, of the solution of with vanishing initial data and Dirichlet boundary conditions, towards the solution of where and is a noisy input which converges to white noise . Mellall and Ouknine  considered the quasilinear stochastic heat equation on with Dirichlet boundary conditions and initial condition , where is a fractional noise with Hurst parameter .
Motivated by these works, we consider the weak convergence for the following stochastic fractional heat equation driven by fractional noise on : where is the fractional Laplacian operator with respect to the spatial variable, to be defined in Section 2 which was recently introduced by Debbi  and Debbi and Dozzi , and is a fractional noise on with Hurst index defined on a complete probability space . Actually, we understand (5) in the sense of Walsh , and so one can present a mild formulation of (5) as follows: where denotes the Green function associated with (5).
The rest of this paper is organized as follows. In Section 2, we begin by making some notation and by recalling some basic preliminaries which will be needed later. In Section 3, we will prove weak limit theorems for (5) in space . Most of the estimates of this paper contain unspecified constants. An unspecified positive and finite constant will be denoted by , which may not be the same in each occurrence. Sometimes we will emphasize the dependence of these constants upon parameters.
In this section, we briefly recall some basic definitions of fractional noise and Green function.
2.1. Fractional Noise
For each , let be the -field generated by the random variables and the sets of probability zero, and denote by the -field of progressively measurable subsets of .
We denote by the set of step functions on . Let be the Hilbert space defined as the closure of with respect to the scalar product where covariance kernel and denotes the Lebesgue measure of the set .
According to Nualart and Ouknine , 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 the square integrable kernel given by with , and one can get Moreover, the kernel satisfies the following property: being the covariance kernel of the fractional Brownian motion. Then, for any pair of step functions and in we have because As a consequence, the operator provides an isometry between the Hilbert space and . Hence, the Gaussian family defined by is a space-time white noise, and the process has an integral representation of the form Now, we can present a mild formulation of (5) as follows: That is, the last term of (6) is equal to
2.2. Green Function
In this subsection, we will introduce the nonlocal factional differential operator defined via its Fourier transform by In this paper, we will assume that , , is the largest even integer less or equal to (even part of ), and .
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 , Debbi and Dozzi , and Komatsu  for more details about this operator.
According to Komatsu , can be represented for by and for by where and are two nonnegative constants satisfying and is a smooth function for which the integral exists, and is its derivative. This representation identifies it as the infinitesimal generator for a nonsymmetric -stable Lévy process.
Let be the fundamental solution of the following Cauchy problem: where is the Dirac distribution. By Fourier transform, we see that is given by The relevant parameters , called the index of stability, and (related to the asymmetry), improperly referred to as the skewness, are real numbers satisfying , and when .
Lemma 1. Let ; one has the following:(1)the function is not in general symmetric relatively to and it is not everywhere positive;(2)for any and , or equivalently, (3) for any ;(4)For , there exist some constants and such that, for all , (5) if and only if .
3. Main Results and Its Proof
Our aim is to prove that the mild solution of (5), given by (17), can be approximated in law in the space by the processes where is a weak approximation of a Brownian sheet; that is, , is a family of Kac-Stroock processes in the plane which is square integral a.s., defined by and is a standard Poisson process in plane.
Theorem 2. Let , , be the Kac-Stroock processes in the plane. Assume that is a continuous function and satisfies the following linear growth conditions: and uniformly Lipschitz conditions Then, the family of stochastic processes defined by (27) converges in law, as tends to infinity, in the space , to the mild solution of (5), given by (17).
In order to prove Theorem 2, we will focus on the linear problem, which is amount to establish the convergence in law, in the space , of the solutions of with vanishing initial data and Dirichlet boundary conditions, toward the solution of where the solutions of (31) and (32) are, respectively, given by
In the following, we need two results which can be found in Bardina et al. . The first one leads to the tightness in of a family.
Lemma 3. Let be a family of random variables taking values in . The family of the laws of is tight, if there exist , and a constant such that and, for every and ,
The second one is a technical lemma.
Lemma 4. Denote by the Kac-Stroock kernels; for any even , there exists a constant , such that, for any and satisfying and , one can get for any .
Proposition 5. The family of processes is tight in .
Proof. We first estimate the moment of order of the quantity
which is equal to
By Lemma 4, one can get
Using the continuous embedding established in 
Let . Thanks to the mean-value theorem, one can get Therefore, if , that is, , then Similarly, one can get Hence Now we are in the position to deal with . Consider By mean-value theorem, it holds that Noting that one can get Then Therefore, if , we have Similarly, Thus, we have Now we deal with ; similar to the proof of , we get Together with (40)–(55), one can get By Lemma 3, the proof can be completed.
Proof. We claim that, for any and , the law of linear combination
converges weakly to the law of a random variable defined by
This will be done by proving the convergence of the corresponding characteristic functions; that is,
Since for any fixed , , then for any , , there exists a sequence of simple functions such that converges to in as .
To simplify notation, we define Then where We will proceed to prove (59) in three steps.
Step 1. By the mean-value theorem, there exists a constant such that Using the same method as the proof of Lemma 4, by Hölder inequality we can get So uniformly converges to with respect to .
Step 2. We proceed to deal with . Using the mean-value theorem again, Thanks to Donsker’s theorem, as , the laws of processes converge weakly to the law of since is a simple function. So we can get , as .
Step 3. Finally we deal with . Using the mean-value theorem again, one can get Then by the Hölder inequality and the variance for a stochastic integral, we get So, , for all , as . Our proof is completed.
As a consequence of the last two properties, we can state the following.
Now, we can give the proof of Theorem 2.
Proof of Theorem 2. Let us recall first the mild solution of (5), which is given by
and the approximation sequence toward the mild solution of (5), which fulfils
where , , stand for the Kac-Stroock process which is square integrable a.s.
Moreover, the approximating sequence has continuous paths a.s., for all which can be obtained by using the properties of the Green function, the fact that a.s., together with a Gronwall-type argument.
On the other hand, consider the following function: where is a continuous function, and Then it can be proved that this last function admits a unique continuous solution. Now, according to Theorem 3.5 in Bardina et al. , the function is continuous. Considering one can get that converges in law in to , as goes to infinity. On the other hand, we have that and , and hence the continuity of implies the convergence in law of to in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their sincere gratitude to the editor and the anonymous referees for their valuable comments and error corrections. Xichao Sun is partially supported by Natural Science Foundation of Anhui Province: Stochastic differential equation driven by fractional noise and its application in finance (no.1408085QA10), Natural Science Foundation of Anhui Province (no.1408085QA09 ), and Key Natural Science Foundation of Anhui Education Commission (no.KJ2013A183). Junfeng Liu is partially supported by Mathematical Tianyuan Foundation of China (no.11226198).
- J. Droniou and C. Imbert, “Fractal first order partial differential equations,” Archive for Rational Mechanics and Analysis, vol. 182, pp. 229–261, 2004.
- V. V. Uchaikin and V. M. Zolotarev, Chance and Stability, Stable Distributions and Their Applications, Modern Probability and Statistics, 1999.
- L. Chen, Y. Chai, R. Wu, J. Sun, and T. Ma, “Cluster synchronization in fractional-order complex dynamical networks,” Physics Letters A, vol. 376, no. 35, pp. 2381–2388, 2010.
- M. Li, W. Zhao, and C. Cattani, “Delay bound: fractal traffic passes through network servers,” Mathematical Problems in Engineering, vol. 2013, Article ID 157636, 15 pages, 2013.
- M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010.
- S. Cohen and A. Lindner, “A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process,” Journal of Statistical Planning and Inference, vol. 143, no. 8, pp. 1295–1306, 2013.
- K. Jaczak-Borkowska, “Generalized BSDEs driven by fractional Brownian motion,” Statistics and Probability Letters, vol. 83, no. 3, pp. 805–811, 2013.
- M. Besal and C. Rovira, “Stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter ,” Stochastics and Dynamics, vol. 12, no. 4, pp. 1–24, 2012.
- D. T. Nguyen, “Mackey-Glass equation driven by fractional Brownian motion,” Physica A, vol. 391, no. 22, pp. 5465–5472, 2012.
- Y. S. Mishura and S. V. Posashkova, “Stochastic differential equations driven by a Wiener process and fractional Brownian motion: convergence in Besov space with respect to a parameter,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1166–1180, 2011.
- M. Li and W. Zhao, “Quantitatively investigating locally weak stationarity of modified multifractional Gaussian noise,” Physica A, vol. 391, no. 24, pp. 6268–6278, 2012.
- M. Li, Y. Q. Chen, J. Y. Li, and W. Zhao, “Hölder scales of sea level,” Mathematical Problems in Engineering, vol. 2012, Article ID 863707, 22 pages, 2012.
- C. Mueller, “The heat equation with Lévy noise 1,” Stochastic Processes and Their Applications, vol. 74, no. 1, pp. 67–82, 1998.
- J. Wu, “Fractal Burgers equation with stable Lévy noise,” in Proceedings of the 7th International Conference SPDE and Applications, January 2004.
- T. Chang and K. Lee, “On a stochastic partial differential equation with a fractional Laplacian operator,” Stochastic Processes and Their Applications, vol. 122, no. 9, pp. 3288–3311, 2012.
- A. Truman and J. Wu, “Fractal Burgers equation driven by Lévy noises,” in Stochastic Partial Differential Equations and Applications, G. da Prato and L. Tubaro, Eds., vol. 245 of Lecture Notes in Pure and Applied Mathematics, pp. 295–310, Chapman Hall/CRC Taylor, 2006.
- 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 and Probability Letters, vol. 81, no. 8, pp. 1161–1172, 2011.
- M. S. Taqqu, “Weak convergence to fractional brownian motion and to the rosenblatt process,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 31, no. 4, pp. 287–302, 1975.
- R. Delgado and M. Jolis, “Weak approximation for a class of Gaussian processes,” Journal of Applied Probability, vol. 37, no. 2, pp. 400–407, 2000.
- X. Bardina, M. Jolis, and L. Quer-Sardanyons, “Weak convergence for the stochastic heat equation driven by Gaussian white noise,” Electronic Journal of Probability, vol. 15, pp. 1267–1295, 2010.
- B. Boufoussi and S. Hajji, “An approximation result for a quasi-linear stochastic heat equation,” Statistics and Probability Letters, vol. 80, no. 17-18, pp. 1369–1377, 2010.
- T. E. Mellall and Y. Ouknine, “Weak convergence for qualinear stochastic heat equation driven by a fractional noise with Hurst parameter ,” Stochastics and Dynamics, vol. 13, no. 3, pp. 1–13, 2013.
- L. Debbi, “On some properties of a high order. Fractional differential operator which is not in general selfadjoint,” Applied Mathematical Sciences, vol. 1, pp. 1325–1339, 2007.
- 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.
- 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. 266–439, Springer, Berlin, Germany, 1986.
- D. Nualart and Y. Ouknine, “Regularization of quasilinear heat equation by a fractional noise,” Stochastics and Dynamics, vol. 4, no. 2, pp. 201–221, 2004.
- T. Komatsu, “On the martingale problem for generators of stable processes with perturbations,” Osaka Journal of Mathematics, vol. 21, pp. 113–132, 1984.