The Scientific World Journal

Volume 2014 (2014), Article ID 601327, 7 pages

http://dx.doi.org/10.1155/2014/601327

Research Article

Variational Solutions and Random Dynamical Systems to SPDEs Perturbed by Fractional Gaussian Noise

Caibin Zeng,1,2 Qigui Yang,¹ and Junfei Cao³

¹School of Sciences, South China University of Technology, Guangzhou 510640, China

²School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China

³Department of Mathematics, Guangdong University of Education, Guangzhou 510310, China

Received 22 August 2013; Accepted 27 October 2013; Published 15 January 2014

Academic Editors: T. Prieto-Rumeau and K. Saito

Copyright © 2014 Caibin Zeng 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.

This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient : , where is a nonlinear operator satisfying some monotonicity conditions. Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized -Laplacian equation.

1. Introduction

Recently, fractional Brownian motion (fBm) has been used successfully to model a variety of physical phenomena such as hydrology, turbulence, economic data, telecommunications, biology, and medicine. As a centered Gaussian process, it is characterized by the stationarity of its increments and the long-memory property. In general, the fBm represents a natural one-parameter extension (represented by the Hurst parameter ) of classical Brownian motion. It becomes the standard Brownian motion when equals to . However, it was proved in [1] that the fBm is neither Markovian nor a semimartingale when , which differs significantly from the standard Brownian motion. Thus, the well-established classical theory of stochastic analysis is not applicable to stochastic differential equations (SDEs) driven by fBm with . This situation motivates a main mathematical challenge: how to extend the results in the classical stochastic analysis to fBm? Over the last years, some new techniques have been developed in order to define stochastic integrals with respect to fBm [2–10]. These techniques have the following common points: they get harder as gets smaller; the more the paths of the stochastic process are irregular, the harder it is to integrate against them. Therefore, path regularity is a key benchmark to evaluate the mathematical tractability of any model with dependent noise.

On the other hand, the generation of a random dynamical system (or stochastic flow) from a stochastic partial differential equation (SPDE) is a fundamental problem in the study of its dynamics. It is well-known that a large class of partial differential equations (PDEs) with stationary random coefficients and Itô stochastic ordinary differential equations (ODEs) generate random dynamical systems (cf. the monograph [11]). However, stochastic equations driven by fBm do not generate a Markov process, which precludes the study of invariant measures for fBm-driven systems using classical tools. This motivates the study that fBm-driven SPDEs generate random dynamical systems. To the best of our knowledge, the generation of a random dynamical system (RDS) from a SPDE perturbed by general noise (in particular fBm) is far from fully solved. Nevertheless, some initial work has been done previously in this issue [12–14]. Under some regularity conditions, the asymptotic behavior of SPDEs driven by fBm was studied in [15, 16]. It should be pointed out that these papers were studied by using the semigroup (or mild solution) approach, for which it is necessary to have a linear operator in the drift part that has to generate a semigroup. There is also enormous research activity on nonlinear SPDEs since all kinds of dynamics with stochastic influence in nature or man-made complex systems can be modeled by such systems. In this case, variational approach has been used to investigate nonlinear SPDEs which are not necessarily of semilinear type. For more detailed examples, we refer the readers to [17, 18] and references therein. Within this framework, there seems to be only the work [19] analyzing the RDS from nonlinear SPDEs driven by infinite dimensional fBm. In [19], the existence of random attractors for a large class of SPDEs driven by general additive noise (including fBm) was established.

What is new in this paper is that we will add a suitable volatility coefficient for the infinite dimensional fBm. Based on the variational approach to SPDEs, we prove the existence and uniqueness of a variational solution to this general type of SPDEs perturbed by an infinite dimensional fBm with a suitable volatility coefficient, where the drift part is a nonlinear operator satisfying the standard monotonicity and coercivity conditions. Moreover, we show that this variational solution generates a continuous RDS.

This paper is organized as follows. In Section 2, we introduce some basic notations on RDS and stochastic integral with respect to fBm. Section 3 contains the main results and the detailed proof. In Section 4, we apply the main results to two examples of SPDE including the stochastic generalized -Laplacian equation with a suitable volatility coefficient for fBm.

2. Preliminaries

Throughout the paper, we assume that is a probability space, is a separable Hilbert space, and stands for a space of all -valued random variables such that .

2.1. Stochastic Integral for fBm

Definition 1. Let . A Gaussian stochastic process is said to be a real-valued standard fractional Brownian motion (fBm) with Hurst parameter if it satisfies that and

Assume that is a bounded linear and nonnegative symmetric operator on and has finite trace; that is, there exists a complete orthogonal of such that

Then, an infinite dimensional fBm with the incremental covariance operator is defined by

where are independent real-valued fBms and the convergence in (3) holds -a.s. as well as in . Throughout this paper, it is usually assumed that . By the Kolmogorov continuity criterion (cf. [20, Theorem ]), we have that has a continuous version since

Now, we recall stochastic integral [21] with respect to the fBms and . For some fixed , let be the space of elementary functions:

where is another separable Hilbert space. For , we define the stochastic integral

For , let be the space of measurable functionas such that

where . From [22, Proposition ], there exists some constant only depending on and such that

Then stochastic integral can be (almost surely) uniquely extended from to . Indeed it allows a continuous version due to the statement of [22, Proposition ].

Now, we turn to the stochastic integral with respect to the infinite dimensional fBm. Denote by the family of Hilbert-Schmidt linear operators from to . Let such that the function is contained in for each and . Then, we define the stochastic integral as

where the sum in right hand side of (9) converges absolutely in . Indeed, it, from [22, Lemma ], follows that the stochastic integral (9) is an -valued Gaussian process.

2.2. Random Dynamical Systems

In this subsection, we recall some basic concepts and results on random dynamical systems that will be used to describe the dynamics of systems under the influence of a noise. For more details we refer the readers to the monograph [11].

Definition 2. A metric dynamical system with time consists of a measurable flow

which is -measurable and satisfies the flow property

for all . Additionally, we assume that the measure is invariant with respect to the flow .

Let be the associated Borel--algebra. Then the operators forming the flow are given by the Wiener shift:

According to [12, Theorem 2.3], we have the following conclusion.

Remark 3. Let be the flow of the Wiener shift and be the distribution of , then the quadruple defines a metric dynamical system.

Furthermore, it holds that

We now recall the notion of random dynamical system.

Definition 4. A random dynamical system (RDS) with one-side time and phase space is a pair consisting of the metric dynamical system and a mapping

which is -measurable and satisfies the cocycle property

for all and . is said to be a continuous RDS if the mapping

is continuous for all and .

By (13) and [13, Lemma 5], we get the following property.

Remark 5. For , it follows that

3. Main Results

Let be a Gelfand triple; namely, is a reflexive Banach space such that continuously and densely, is a separable Hilbert space defined in the previous section and by the Riesz isomorphism. We consider the following stochastic partial differential equations (SPDEs):

where denotes a path of the infinite dimensional fBm with covariance function such that . Let be progressively measurable; that is, for , is -measurable, and such that the function is contained in for each and . Furthermore, we impose some conditions on and as follows.

(hemicontinuity): for all , the map

 is continuous on . (weak monotonicity): there exists such that for all

 (coercivity): there exist , , , and an -adapted process , such that for all ,  

 (boundedness): there exist and an -adapted process such that for all ,  

 :there exists a finite dimensional subspace

 such that for the corresponding orthogonal projection in , we have for all .

Remark 6. Noting that are the standard monotonicity and coercivity conditions for SPDEs (cf. [18, Chapter 4]), on the other hand, assumption looks quite abstract at first glance. But it is very useful to deduce the following property. By and [22, Corollary ], we have that and are all in , where is as in . That is,

Definition 7. A continuous -valued -adapted process is called a variational solution to (18) if and

holds for , -a.s.

Now we claim and prove the main results.

Theorem 8. Suppose hold. For any given , there exists a unique variational solution to (18).

Proof. Let us define

Then for we get a new PDE

under the transformation , where . First, we show the existence and uniqueness of solutions to (27) by checking that satisfies conditions to . Since is an -valued Gaussian process, it is obvious to check the hemicontinuity and weak monotonicity for follow from that of . For the coercivity, we also have that

where the 1st inequality holds using , the 2nd inequality holds using that , and the 3rd inequality holds using Young’s inequality with a small value . Next, by , we estimate

Together (28) and (29), we get

Let , , and let be the sum in the last parenthesis. Then we get

So we can pick to be sufficiently small such that . Moreover, it follows from Remark 6 that and are all in , and holds due to . Thus, . This completes the proof of the coercivity of .

Indeed, by we get

Let and . Then we get

It is easy to check that since and . Thus we prove the boundedness of . Therefore, according to [18, Theorem ], (27) has a unique solution

Recall the transformation , we have

By Remark 6 we further have

Consequently, (18) has a unique variational solution in the sense of Definition 7.

The next theorem shows that the unique solution of (18) generates a random dynamical system in such a way that is defined by the solution of the SPDE at time , for a noise path , with initial point .

Theorem 9. The solution of (18) defines a continuous random dynamical system given by

Proof. Trivially . Let us check then the cocycle property: for and , we have

Making the change of variable leads to

Applying Remark 5 we get

By the pathwise uniqueness of the solution to (27), we get

Now we turn to prove the measurability of . Note that the maps and are continuous, thus we only need to prove the measurability of . By the proof of the existence and uniqueness of solutions to (18), we know that is the weak limit of a subsequence of the Galerkin approximations in . Let be a Dirac sequence with supp, where is an open ball of radius centered at the point . Then is well defined for large enough. For each such and we have

when . Since is measurable, so is , . Moreover, it follows from (42) that is measurable. On the other hand, is continuous in . Therefore, and the measurability of imply the measurability of , . Since this is true for all , we get the measurability of . Consequently, we complete the proof that defines a continuous RDS.

4. Application to Examples

In this section, we give two examples as application of the general results obtained in Theorems 8 and 9.

Example 1. Let , , is open. Consider the following Gelfand triple:

and the SPDE

where is an infinite dimensional fBm with Hurst parameter , and is chosen as Section 3.

According to [18, Example ], it is well known that hold for with . Thus, by the above results, for suitable the SPDE (44) has a unique variational solution taking values in ;, moreover, this solution defines a continuous RDS.

Example 2 (stochastic generalized -Laplacian equation). Again let , , is open. Denote by the completion of with respect to the norm defined as:

Suppose is monotone, continuous and that for some strictly positive constants

for all . Then we consider the following Gelfand triple

and the SPDE

According to [22, subsection 5.4], it is well known that hold for with . Thus, by the above results, for suitable , the SPDE (48) has a unique variational solution taking values in ; moreover, this solution defines a continuous RDS.

5. Conclusion

Using the variational approach we have studied a general type of fBm-driven nonlinear SPDEs with a suitable volatility coefficient in Hilbert space. We proved the existence and uniqueness of variational solutions to such system under some monotonicity and coercivity conditions. We further proved that this variational solution generates a random dynamical system. Finally, we applied the main results to two types of SPDEs including the stochastic generalized -Laplacian equation. It is useful to note that our results for SPDEs in a Hilbert space can reduce to known results for a standard infinite dimensional Wiener process if the Hurst parameter though the techniques of proof are different. Furthermore, the conditions ()–() can be replaced by some much weaker assumptions (e.g., locally monotone) according to some recent results in [23]. When the nonlinear SPDEs are perturbed by a multiplicative fBm, it is interesting to study the random dynamical system from such more general nonlinear SPDEs by variational approach. This will be the subject for future investigation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the anonymous referees and editors for their valuable comments and suggestions. This work was partly supported by the National Natural Science Foundation of China (no. 11301090, no. 11271139, and no. 61104138), and the Fundamental Research Funds for the Central Universities (no. 2014ZB0033).

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