Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise

Abstract

We consider a class of stochastic fractional equations driven by fractional noise on t , x ∈ 0 , T × 0,1 ∂ u / ∂ t = [superscript] D δ α [/superscript] u + f t , x , u + [superscript] ∂ 2 [/superscript] [superscript] B H [/superscript] t , x / ∂ t ∂ x , 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 ...9E; 0 , T × 0,1 of continuous functions, to the solution of the stochastic fractional heat equation driven by fractional noise.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Xichao Sun 1 and Junfeng Liu 2

Academic Editor:Ming Li

1, Department of Mathematics and Physics, Bengbu College, 1866 Caoshan Road, Bengbu, Anhui 233030, China
2, School 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; 12 February 2014

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.

1. Introduction

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 [1] and Uchaikin and Zolotarev [2] 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. [3] and Li et al. [4] studied the fractional-order networks and Li [5] 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 [13] and Wu [14] 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 [15], Truman and Wu [16], Liu et al. [17], Wu [18], 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 [19] and Delgado and Jolis [20]. Recently, many researchers are interested in studying weak convergence of stochastic differential equation. Some surveys could be found in Bardina et al. [21], Boufoussi and Hajji [22], and Mellall and Ouknine [23]. Bardina et al. [21] studied the convergence in law, in the space ...9E; ( [ 0 , t ] × [ 0,1 ] ) of continuous functions, of the solution of [figure omitted; refer to PDF] with vanishing initial data and Dirichlet boundary conditions, towards the solution of [figure omitted; refer to PDF] where ( t , x ) ∈ [ 0 , T ] × [ 0,1 ] and _{θ n} is a noisy input which converges to white noise W . Mellall and Ouknine [23] considered the quasilinear stochastic heat equation on [ 0,1 ] [figure omitted; refer to PDF] with Dirichlet boundary conditions [figure omitted; refer to PDF] and initial condition u ( 0 , x ) = _{u 0} ( x ) , x ∈ [ 0,1 ] , where ^{∂ 2 B H} ( t , x ) / ∂ t ∂ x is a fractional noise with Hurst parameter H ∈ ( 1 / 2,1 ) .

Motivated by these works, we consider the weak convergence for the following stochastic fractional heat equation driven by fractional noise on ( t , x ) ∈ [ 0 , T ] × [ 0,1 ] : [figure omitted; refer to PDF] where ^{D δ α} is the fractional Laplacian operator with respect to the spatial variable, to be defined in Section 2 which was recently introduced by Debbi [24] and Debbi and Dozzi [25], and ^{B H} ( t , x ) is a fractional noise on [ 0 , T ] × [ 0,1 ] with Hurst index H > 1 / 2 defined on a complete probability space { Ω , F , { _{F t } t ...5; 0} , P } . Actually, we understand (5) in the sense of Walsh [26], and so one can present a mild formulation of (5) as follows: [figure omitted; refer to PDF] where _{G α , δ} ( · , * ) 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 ...9E; ( [ 0 , t ] × [ 0,1 ] ) . Most of the estimates of this paper contain unspecified constants. An unspecified positive and finite constant will be denoted by C , which may not be the same in each occurrence. Sometimes we will emphasize the dependence of these constants upon parameters.

2. Preliminaries

In this section, we briefly recall some basic definitions of fractional noise and Green function.

2.1. Fractional Noise

For each t ∈ [ 0 , T ] , let ^{F t H} be the σ -field generated by the random variables { ^{B H} ( t , A ) , t ∈ [ 0 , T ] , A ∈ [Bernoulli] [ 0,1 ] } and the sets of probability zero, and denote by ...AB; the σ -field of progressively measurable subsets of [ 0 , T ] × Ω .

We denote by ... the set of step functions on [ 0 , T ] × [ 0,1 ] . Let [Hamiltonian (script capital H)] be the Hilbert space defined as the closure of ... with respect to the scalar product [figure omitted; refer to PDF] where covariance kernel _{R H} ( t , s ) = ( 1 / 2 ) [ ^{t 2 H} + ^{s 2 H} - | t - s ^{| 2 H} ] and | A | denotes the Lebesgue measure of the set A .

According to Nualart and Ouknine [27], the mapping _{1 [ 0 , t ] × A} [arrow right] ^{B H} ( t , A ) can be extended to an isometry between [Hamiltonian (script capital H)] and the Gaussian space _{H 1} ( ^{B H} ) associated with ^{B H} and denoted by [figure omitted; refer to PDF]

Define the linear operator ^{K H *} : ... ... ^{L 2} ( [ 0 , T ] ) by [figure omitted; refer to PDF] where _{K H} is the square integrable kernel given by [figure omitted; refer to PDF] with _{c H} = ( 2 H Γ ( 3 / 2 - H ) / Γ ( H + 1 / 2 ) Γ ( 2 - 2 H ) ^{) 1 / 2} , and one can get [figure omitted; refer to PDF] Moreover, the kernel _{K H} satisfies the following property: [figure omitted; refer to PDF] _{R H} ( t , s ) being the covariance kernel of the fractional Brownian motion. Then, for any pair of step functions [straight phi] and ψ in ... we have [figure omitted; refer to PDF] because [figure omitted; refer to PDF] As a consequence, the operator ^{K H *} provides an isometry between the Hilbert space [Hamiltonian (script capital H)] and ^{L 2} ( [ 0 , T ] × [ 0,1 ] ) . Hence, the Gaussian family { W ( t , A ) , t ∈ [ 0 , T ] , A ∈ [Bernoulli] [ 0,1 ] } defined by [figure omitted; refer to PDF] is a space-time white noise, and the process ^{B H} has an integral representation of the form [figure omitted; refer to PDF] Now, we can present a mild formulation of (5) as follows: [figure omitted; refer to PDF] That is, the last term of (6) is equal to [figure omitted; refer to PDF]

2.2. Green Function

In this subsection, we will introduce the nonlocal factional differential operator ^{D δ α} defined via its Fourier transform ... by [figure omitted; refer to PDF] In this paper, we will assume that | δ | ...4; min ... { α - [ α _{] 2} , 2 + [ α _{] 2} - α } , i = 1 , ... , d , [ α _{] 2} , is the largest even integer less or equal to α (even part of α ), and δ ∈ 2 ... + 1 .

The operator ^{D δ α} is a closed, densely defined operator on ^{L 2} ( ... ) 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 α = 2 ), the inverse of the generalized Riesz-Feller potential (when α > 2 ), and the Riemann-Liouville differential operator (when | δ | = 2 + [ α _{] 2} or | δ | = α - [ α ] ). It is self-adjoint only when δ = 0 and, in this case, it coincides with the fractional power of the Laplacian. We refer the readers to Debbi [24], Debbi and Dozzi [25], and Komatsu [28] for more details about this operator.

According to Komatsu [28], ^{D δ α} can be represented for 1 < α < 2 by [figure omitted; refer to PDF] and for 0 < α < 1 by [figure omitted; refer to PDF] where ^{κ - δ} and ^{κ + δ} are two nonnegative constants satisfying ^{κ - δ} + ^{κ + δ} > 0 and [straight phi] is a smooth function for which the integral exists, and ^{[straight phi] [variant prime]} is its derivative. This representation identifies it as the infinitesimal generator for a nonsymmetric α -stable Lévy process.

Let _{G α , δ} ( t , x ) be the fundamental solution of the following Cauchy problem: [figure omitted; refer to PDF] where _{δ 0} ( · ) is the Dirac distribution. By Fourier transform, we see that _{G α , δ} ( t , x ) is given by [figure omitted; refer to PDF] The relevant parameters α , called the index of stability, and δ (related to the asymmetry), improperly referred to as the skewness, are real numbers satisfying | δ | ...4; min ... { α - [ α _{] 2} , 2 + [ α _{] 2} - α } , and δ = 0 when δ ∈ 2 ... + 1 .

Let us list some known facts on _{G α , δ} ( t , x ) which will be used later on (see, e.g., Debbi [24] and Debbi and Dozzi [25]).

Lemma 1.

Let α ∈ ( 0 , ∞ ) / { ... } ; one has the following:

(1) the function _{G α , δ} ( t , · ) is not in general symmetric relatively to x and it is not everywhere positive;

(2) for any s , t ∈ ( 0 , ∞ ) and x ∈ ... , [figure omitted; refer to PDF] or equivalently, [figure omitted; refer to PDF]

(3) _{G α , δ} ( s , · ) * _{G α , δ} ( t , · ) = _{G α , δ} ( s + t , · ) for any s , t ∈ ( 0 , ∞ ) ;

(4) For n ...5; 1 , there exist some constants C and _{C n} > 0 such that, for all x ∈ ... , [figure omitted; refer to PDF]

(5) ^{∫ 0 T} ... _{∫ ...} ... | _{G α , δ} ( t , x ) ^{| λ} d t d x < ∞ if and only if 1 / α < λ < α .

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 ...9E; ( [ 0 , t ] × [ 0,1 ] ) by the processes [figure omitted; refer to PDF] where { _{θ n} ( t , x ) _{} n ∈ ...} is a weak approximation of a Brownian sheet; that is, { _{θ n} ( t , x ) _{} n ∈ ...} , ( t , x ) ∈ [ 0 , T ] × [ 0,1 ] is a family of Kac-Stroock processes in the plane which is square integral a.s., defined by [figure omitted; refer to PDF] _{N n} ( t , x ) = N ( n t , n x ) and { N ( t , x ) , ( t , x ) ∈ [ 0 , T ] × [ 0,1 ] } is a standard Poisson process in plane.

Theorem 2.

Let { _{θ n} ( t , x ) , ( t , x ) ∈ [ 0 , T ] × [ 0,1 ] } , n ∈ ... , be the Kac-Stroock processes in the plane. Assume that _{u 0} : [ 0,1 ] [arrow right] ... is a continuous function and f satisfies the following linear growth conditions: [figure omitted; refer to PDF] and uniformly Lipschitz conditions [figure omitted; refer to PDF] Then, the family of stochastic processes { _{u n} , n ∈ ... } defined by (27) converges in law, as n tends to infinity, in the space ...9E; ( [ 0 , T ] × [ 0,1 ] ) , to the mild solution u 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 ...9E; ( [ 0 , T ] × [ 0,1 ] ) , of the solutions of [figure omitted; refer to PDF] with vanishing initial data and Dirichlet boundary conditions, toward the solution of [figure omitted; refer to PDF] where the solutions of (31) and (32) are, respectively, given by [figure omitted; refer to PDF]

In the following, we need two results which can be found in Bardina et al. [21]. The first one leads to the tightness in ...9E; ( [ 0 , T ] × [ 0,1 ] ) of a family.

Lemma 3.

Let { _{X n} , n ∈ ... } be a family of random variables taking values in ...9E; ( [ 0 , T ] × [ 0,1 ] ) . The family of the laws of { _{X n} , n ∈ ... } is tight, if there exist ^{p [variant prime]} , p > 0 , δ > 2 , and a constant C such that [figure omitted; refer to PDF] and, for every t , ^{t [variant prime]} ∈ [ 0 , T ] and x , ^{x [variant prime]} ∈ [ 0,1 ] , [figure omitted; refer to PDF]

The second one is a technical lemma.

Lemma 4.

Denote by _{θ n} ( t , x ) the Kac-Stroock kernels; for any even n ∈ ... , there exists a constant _{C n} , such that, for any _{t 1} , _{t 2} ∈ [ 0 , T ] and _{x 1} , _{x 2} ∈ [ 0,1 ] satisfying 0 < _{t 1} < _{t 2} < 2 _{t 1} and 0 < _{x 1} < _{x 2} < 2 _{x 1} , one can get [figure omitted; refer to PDF] for any f ( s , x ) ∈ ^{L 2} ( [ 0 , T ] × [ 0,1 ] ) .

Proposition 5.

The family of processes { _{X n} , n ∈ ... } is tight in ...9E; ( [ 0 , T ] × [ 0,1 ] ) .

Proof.

We first estimate the moment of order m of the quantity [figure omitted; refer to PDF] which is equal to [figure omitted; refer to PDF] By Lemma 4, one can get [figure omitted; refer to PDF] Using the continuous embedding established in [27] [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF]

Let θ ∈ ( 0 , min ... { 1 , ( α + 1 ) H - 1 } ) . Thanks to the mean-value theorem, one can get [figure omitted; refer to PDF] Therefore, if 1 / α - ( 1 + θ ) / α H + 1 > 0 , that is, θ < ( α + 1 ) H - 1 , then [figure omitted; refer to PDF] Similarly, one can get [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] Now we are in the position to deal with I I . Consider [figure omitted; refer to PDF] By mean-value theorem, it holds that [figure omitted; refer to PDF] Noting that [figure omitted; refer to PDF] one can get [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Therefore, if μ ∈ ( 0 , ( ( α + 1 ) H - 1 ) / α ) , we have [figure omitted; refer to PDF] Similarly, [figure omitted; refer to PDF] Thus, we have [figure omitted; refer to PDF] Now we deal with I I I ; similar to the proof of I , we get [figure omitted; refer to PDF] Together with (40)-(55), one can get [figure omitted; refer to PDF] By Lemma 3, the proof can be completed.

Proposition 6.

The family of processes { _{X n} , n ∈ ... } defined by (33) converges to the process X given by (34), in the sense of finite-dimensional distributions, as n tends to infinity, in the space ...9E; ( [ 0 , T ] × [ 0,1 ] ) .

Proof.

We claim that, for any _{a 1} , ... , _{a m} ∈ ... and ( _{t 1} , _{x 1} ) , ... , ( _{t m} , _{x m} ) ∈ [ 0 , T ] × [ 0,1 ] , the law of linear combination [figure omitted; refer to PDF] converges weakly to the law of a random variable defined by [figure omitted; refer to PDF] This will be done by proving the convergence of the corresponding characteristic functions; that is, [figure omitted; refer to PDF] Since for any fixed ( t , x ) ∈ [ 0 , T ] × [ 0,1 ] , ^{K H *} _{G α , δ} ( t - s , x - y ) ∈ ^{L 2} ( [ 0 , T ] × [ 0,1 ] ) , then for any ( _{t j} , _{x j} ) ∈ [ 0 , T ] × [ 0,1 ] , j ∈ { 1,2 , ... , m } , there exists a sequence ( ^{K H *} _{G α , δ} ( _{t j} - s , _{x j} - y ) _{) k} of simple functions such that ( ^{K H *} _{G α , δ} ( _{t j} - s , _{x j} - y ) _{) k} converges to ^{K H *} _{G α , δ} ( _{t j} - s , _{x j} - y ) in ^{L 2} ( [ 0 , T ] × [ 0,1 ] ) as k [arrow right] ∞ .

To simplify notation, we define [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] We will proceed to prove (59) in three steps.

Step 1. By the mean-value theorem, there exists a constant C > 0 such that [figure omitted; refer to PDF] Using the same method as the proof of Lemma 4, by Hölder inequality we can get [figure omitted; refer to PDF] So V uniformly converges to 0 with respect to n .

Step 2. We proceed to deal with V I . Using the mean-value theorem again, [figure omitted; refer to PDF] Thanks to Donsker's theorem, as n [arrow right] ∞ , the laws of processes [figure omitted; refer to PDF] converge weakly to the law of [figure omitted; refer to PDF] since ( ^{K H *} _{G α , δ} ( _{t j} - s , _{x j} - y ) _{) k} is a simple function. So we can get V I [arrow right] 0 , as n [arrow right] ∞ .

Step 3. Finally we deal with V I I . Using the mean-value theorem again, one can get [figure omitted; refer to PDF] Then by the Hölder inequality and the variance for a stochastic integral, we get [figure omitted; refer to PDF] So, V I I [arrow right] 0 , for all 1 ...4; j ...4; m , as n [arrow right] ∞ . Our proof is completed.

As a consequence of the last two properties, we can state the following.

Theorem 7.

The family of processes { _{X n} , n ∈ ... } defined by (33) converges in law, as n tends to infinity, in the space ...9E; ( [ 0 , T ] × [ 0,1 ] ) , to the process X defined by (34).

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 [figure omitted; refer to PDF] and the approximation sequence toward the mild solution of (5), which fulfils [figure omitted; refer to PDF] where { _{θ n} ( t , x ) , ( t , x ) ∈ [ 0 , T ] × [ 0,1 ] } , n ∈ ... , stand for the Kac-Stroock process which is square integrable a.s.

Moreover, the approximating sequence _{u n} has continuous paths a.s., for all n ∈ ... which can be obtained by using the properties of the Green function, the fact that _{θ n} ∈ ^{L 2} ( [ 0 , T ] × [ 0,1 ] ) a.s., together with a Gronwall-type argument.

On the other hand, consider the following function: [figure omitted; refer to PDF] where η : [ 0 , T ] × [ 0,1 ] [arrow right] ... is a continuous function, and [figure omitted; refer to PDF] Then it can be proved that this last function admits a unique continuous solution. Now, according to Theorem 3.5 in Bardina et al. [21], the function ψ is continuous. Considering [figure omitted; refer to PDF] one can get that _{X n} converges in law in ...9E; ( [ 0 , T ] × [ 0,1 ] ) to X , as n goes to infinity. On the other hand, we have that _{u n} = ψ ( _{X n} ) and u = ψ ( X ) , and hence the continuity of ψ implies the convergence in law of _{u n} to u in ...9E; ( [ 0 , T ] × [ 0,1 ] ) .

Acknowledgments

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).

Conflict of Interests

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

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