Propagation Phenomena and Transitions in Complex Systems 2014View this Special Issue
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Stochastic Fractional Heat Equations Driven by Fractional Noises
This paper is concerned with the following stochastically fractional heat equation on driven by fractional noise: , where the Hurst parameter and denotes the Skorokhod integral. A unique solution of that equation in an appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and the Hölder continuity of the solution on both space and time parameters is discussed. On the other hand, the absolute continuity of the solution is also obtained.
Recently, there has been intense interest (see, e.g., Podlubny , Samko et al. , Heydari et al. [3, 4], Cattani et al. [5, 6], Liao , and Hu ) in fractional calculus and its applications. Many mathematical problems in physics and engineering with respect to systems and processes are represented by a kind of equations, more precisely fractional order differential equations driven by fractional order noise. One of the emerging branches of this study is the theory and applications of fractional (partial) differential equations. The increasing interest in this class of equations is motivated both by their applications to fluid dynamic traffic model, viscoelasticity, heat conduction in materials with memory, electrodynamics with memory and also because they can be employed to approach nonlinear conservation laws (see  and references therein). For instance, the research discussed by Gurtin and Pipkin  and Nunziato  provides a description of heat conduction in materials with fading memory. Besides, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of great significance to import the stochastic effects into the investigation of fractional (partial) differential systems.
As known, publications treating stochastic partial differential equation (SPDE) involving fractional derivatives gain interests. In fact, these kinds of equations may be widely used in physics, fractal medium, quantum fields, risk management, statistical mechanics, and other areas (see Droniou and Imbert , Uchaikin and Zolotarev , Toma , Bakhoum and Toma , and Li et al. [16, 17]). Most of them investigate evolution type equations driven by a fractional power of the Laplacian. Mueller  and Wu  proved the existence of a solution of stochastically fractional heat and Burgers equation perturbed by a stable noise, respectively. Boulanba et al.  studied the existence, uniqueness, Hölder regularity, and absolute continuity of the solution for a class of fractional stochastic partial differential equations driven by spatially correlated noise. Other related references include Chang and Lee , Debbi and Dozzi , Sun et al. , Truman and Wu , Liu et al. , and Wu .
On the other hand, there has been an increasing interest in studying SPDEs driven by fractional noise. Recall that a fractional Brownian motion (fBm) is a centered Gaussian process with the covariance given by with , referring Biagini et al. , Mishura , and Nualart  for a comprehensive account on the fBm. The fBm has some interesting properties, such as self-similarity, Hölder continuity, and long-range dependence. It has been applied in various scientific areas including telecommunications, turbulence, image processing, and finance engineering. Moreover, there have been many papers about SPDEs driven by fractional noise. Hu  showed the existence and uniqueness of the solutions for a class of second-order stochastic heat equations, via chaos expansion. Nualart and Ouknine  explored the existence and uniqueness of mild solution to a class of second-order heat equations with additive fractional noise (fractional in time and white in space) when the Hurst parameter . In a successive paper, Tindel et al.  studied a linear stochastic evolution equation driven by an infinite-dimensional fBm in the cases of the Hurst parameter above and below , respectively. Hu and Nualart  studied the -dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of an fBm with in time. More works for the fields can be found in Balan , Balan and Tudor [35, 36], Bo et al. [37, 38], Jiang et al. [39–41], and the references therein.
Let , , be the canonical fractional Brownian fields (FBF) with the Hurst parameter on the canonical probability space , where with , . The fractional white noise is denoted by ; that is, formally . In this paper, we focus on the following stochastically fractional partial differential equation driven by fractional noise:where “” denotes the Skorokhod integral, , , , , and denotes a nonlocal fractional differential operator defined by where denotes the fractional differential derivative with respect to the th coordinate defined via its Fourier transform by where . The precise meaning of the solution of (2) will be stated in Section 2.
The structure of this paper is as follows. In Section 2, we briefly present some basic notations and preliminaries. Section 3 consists of the existence and the Lyapunov exponent estimate of the solution to (2). In Section 4, we check the Hölder continuity of the solution with both space and time parameters. In Section 5, we prove that the law of the solution of (2) is absolutely continuous with respect to the Lebesgue measure on . Finally, Section 6 concludes the paper.
In this section, we will first introduce multiple stochastic integrals with respect to fractional Brownian fields and define a solution of (2) in sense after proposing the fractional differential operator and introducing some properties. Then, we will recall the Malliavin calculus with respect to fractional noises.
2.1. Skorokhod Integral with respect to Fractional Brownian Fields
Definition 1. A multiparameter fractional Brownian field with multiparameter for and is a centered Gaussian field defined on some probability space with the covariance as follows:for all , , and .
Throughout the paper, we restrict our consideration on the multiparameter fractional Brownian field with the parameter for .
Firstly, we briefly introduce the stochastic integral with respect to the fractional Brownian field .
For any and , we write , where . Introduce the following Hilbert space: where and . Let ; one can define the following stochastic integral:(see, e.g., Hu ).
Proposition 2. Let . Then, (1).(2).
Denote by the symmetric tensor product. Let be an orthonormal basis of . Then, is an orthonormal basis of . It is easy to see that where we denote DefineWe call symmetric to (()-dimensional) variables . Now, define Denote by the symmetric tensor product. Let be the Hermite polynomial of degree . It is defined by For and , define the multiple integral of Itö type of the function byThen,by the polarization argument.
For each , we writeThen, the following isometry holds:Note that, for , there exists a sequence such that in . It follows from (15) that is Cauchy in , and the limit point of (as ) is independent of the choice of . We call the limit point the multiple integral of Itö type and denote it byIt is easy to see that, for ,
Let , where is the th chaos of (see, e.g., Hu ). For each , we introduce a Hilbert space denoted by and define In particular, if , one has .
2.2. Definition of the Solution
In order to define the solution of (2), we will introduce the nonlocal fractional differential operator defined by where , , and denotes the fractional differential derivative with respect to the th coordinate 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 .
In one space dimension, the operator is a closed, densely defined operator on . 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 if , 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, citing Debbi , Debbi and Dozzi , and Komatsu  for more details about this operator.
According to Komatsu , can be represented for byand 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 the Fourier transform, we see that is given by as follows: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 3. Let ; we have 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 .
For and any multi-index and , denote by the Green function of the deterministic equation as follows:Clearly, Now, we describe a solution of (2) in sense.
Definition 4. We say that a stochastic field is a solution of (2) in sense, if (1) is jointly measurable;(2) is well defined for all and as an element of for certain ;(3)the following equation holds in :
The following embedding proposition given by Mémin et al.  is useful for our derivations below.
Lemma 5. If and , one has where is a constant depending only on .
2.3. Malliavin Calculus
In this subsection, let us brief the basic elements of the Malliavin calculus (see, e.g., Nualart ).
Since , , is Gaussian, we might develop the Malliavin calculus with respect to fractional noises in order to prove the existence of the laws of the solutions of SPDE driven by fractional noises.
Let for and be the space of all “smooth cylindrical” random variables, where denotes the class of all bounded infinitely differentiable functions on , whose partial derivatives are also bounded. The Malliavin derivative of an element , with respect to , is defined by Let be the completion of under the norm Then, is the domain of the closed operator . For each and , define which might be extended as a closed operator on with the domain being the closure of under the normLet be an orthonormal basis of . Then, if and only if for each andIn this case,
On the other hand, the divergence operator is the adjoint of the operator and is uniquely defined by the following duality relationship: if and only ifNote that if and only if is integrable with respect to . In the literature, is called the Skorokhod integral with respect to .
The following propositions (see Wei  for the case of fractional noises), which are the Malliavin calculus with respect to fractional-colored noises, can deduce the laws for solutions to the corresponding stochastic partial differential equations.
Proposition 6. Let . If is a square integrable random variable that is measurable with respect to the -field , then
Remark 7. Let be an -adapted random field. By Proposition 6, we have , a.s., for any and .
Proposition 8. Suppose ; if , a.s., then the distribution of the random variable is absolutely continuous with respect to Lebesgue measure.
3. Lyapunov Exponent Estimate of the Solution
In this section, we will establish the existence and uniqueness of the solution of (2) and give the Lyapunov exponent estimate of the solution. We denote and . In fact, iterating (32) yields formally thatwherewhere Sym is the symmetrization with respect to (-dimensional) variables andWe compute the norm of each chaos; that is, Assume that , where, in what follows, inessential constants will be denoted generically by , even if they vary from line to line. By the isometric equality (18), we obtain Let Thus, where Now, we estimate .
Lemma 10. There is a constant such thatwhere .
Lemma 11. Let and . Then, if . Moreover,for .
Proof. Let , , , and . Then, by the Hölder inequality, one getsFollowing Hu , one can get a constant such that, for any value such that (i.e, ),Set (i.e, ); similar to the proof of (3.6) in Bo et al. , we get where is the Gamma function. Thus, we obtain We continue to use the notation introduced previously. The Stirling formula yields that, for , where the function satisfies . Hence, for , by the definition of , where and is the Mittag-Leffler function with the parameter . Note that ; then, by the asymptotic property of the Mittag-Leffler function (see, e.g., Podlubny ), one hasif . This completes the proof of the lemma.
From the above lemmas, we get the following.
Theorem 13 (existence, uniqueness, and Lyapunov exponent estimate). Let and . If , then (2) has a unique solution when . Moreover,for .
4. Hölder Regularity
Theorem 14. Assume that . If there exists some such that, for all (compact subset of ), for some large enough in , where , then the solution is Hölder continuous in and Hölder continuous in , where and .
Proof. It follows from the proof of Lemma 11. If , then when . In particular, since . On the other hand, from (43), it follows that Thus, for and , we haveNow, we will estimate , , and , respectively. Following Boulanba et al. , one getswhere and .
By Lemma 5 and (62) with , one can get