Research Article | Open Access
Stochastic Current of Bifractional Brownian Motion
We study the regularity of stochastic current defined as Skorohod integral with respect to bifractional Brownian motion through Malliavin calculus. Moreover, we similarly derive some results in the case of multidimensional multiparameter. Finally, we consider stochastic current of bifractional Brownian motion as a distribution in Watanabe spaces.
The fractional Brownian motion was first introduced within a Hilbert space framework by Kolmogorov in . It was further studied by Mandelbrot and Van Ness in , who provided a stochastic integral representation of this process in terms of a standard Brownian motion in 1968. In recent years, fractional Brownian motion has become an intense object in stochastic analysis and related fields for the moment, due to its interesting properties, such as self-similarity, and its applications in various scientific areas. However, when Hurst parameters , fractional Brownian motion is neither a semimartingale nor a Markovian process. The techniques used in Brownian motion cannot be directly applied.
Nevertheless, every fractional Brownian motion has its limits in modelling certain phenomena. In order to fit better in concrete situations, several authors have recently introduced some generalized fractional Brownian motions. For instance, we mention subfractional Brownian motion (see [3, 4]) and bifractional Brownian motion (see [5, 6]).
The concept of current comes from geometric measure theory. The simplest is the functional where and is a rectifiable curve. This functional can be defined by where is a Dirac function (see ). If we want to simulate this current, we need to replace the deterministic curve with stochastic process . At the same time, the stochastic integral must be properly interpreted. Recently, people pay attentions to the research on stochastic current. Give the following map: where is a vector function on which belongs to some Banach spaces , is a stochastic process, and the integral is some version of a stochastic integral defined through regularization. Stochastic current is a continuous version of the mapping; that is, stochastic current is regarded as a stochastic element of the dual space of in .
The problem of stochastic current is motivated by the study of fluidodynamical models. In , in the study of the energy of a vortex filament naturally appear some stochastic double integrals related to Wiener process where is the kernel of the pseudodifferential operator . In the recent years, some results of stochastic currents of Gaussian processes have been obtained through different stochastic integrals in [7, 8, 10]. For example, Flandoli and Tudor  have studied the existence and regularity of stochastic currents through Malliavin calculus, where the integrals are defined as Skorohod integrals with respect to the Brownian motion and fractional Brownian motion, respectively. In  authors have shown the Sobolev regularity of the stochastic current, which is associated with the pathwise integral.
Recall that the bifractional Brownian motion is a centered Gaussian process with covariance function where parameters and . It is well known that, when , bifractional Brownian motion is a fractional Brownian motion. Since bifractional Brownian motion seems to be more flexible and more complex model than fractional Brownian motion, it seems desirable to extend the stochastic current of fractional Brownian motion to the case of bifractional Brownian motion. For this aim, motivated by [7, 11], we use Malliavin calculus and multiple integrals to discuss the stochastic current defined as divergence integral with respect to bifractional Brownian motion. Let us compare our results with the analogous ones from the case of fractional Brownian stochastic current. Note that the regularity condition of bifractional Brownian current does not depend on parameters and , while the situation is different in the case of fractional Brownian motion. On the other hand, because the problems of bifractional Brownian motion are more complex, we need some useful techniques to deal with bifractional Brownian current.
The paper is organized as follows. In Section 2, we provide some background materials from bifractional Brownian motion. In Section 3, we firstly consider the regularity of stochastic current of bifractional Brownian motion with respect to . Lastly, we regard stochastic current of bifractional Brownian motion as a distribution in Watanabe spaces.
2. Bifractional Brownian Motion
A bifractional Brownian motion is a center Gaussian process with variance where parameters and . In the case we retrieve the fractional Brownian motion, while in the case and bifractional Brownian motion corresponds to the Brownian motion.
Let be a Hilbert space. is defined as the completion of the linear space generated by the with respect to the inner product
Sometimes working with the space is not convenient, because this space also contains distributions (see ) and the norm in this space is not always tractable. We always use the subspace of , which is defined as the set of measurable functions on with We can prove that is a Banach space for the norm . At the same time, we have
Denote multiple stochastic integrals by with respect to , where . For each , has chaos expansion . Let be Ornstein-Uhlenbeck operator For each and , define Sobolev-Watanabe as the closure of the set of polynomial variables with respect to the norm where denotes the identity. Malliavin derivative operator is defined as follows: It is well known that stochastic variable belongs to if and only if
The adjoint of is always called the divergence integral (or Skorohod integral). For adapted integrands, the divergence integral coincides with the classical It integral. Hence the divergence integral is called generalized integral. If is a stochastic process, it has the following chaos expansion: where . Skorohod integral of is defined as where denotes the symmetrization of with respect to variables.
3. Stochastic Current of Bifractional Brownian Motion
3.1. Stochastic Current of One-Dimensional Case with respect to
In this section, we give stochastic current of bifractional Brownian motion as follows: where the integral is a Skorohod integral, , and .
Put where is a Gaussian kernel function of variance and is the Hermite polynomial of degree .
By Lemma 3.1 in , the following lemma is obtained. Indeed, the lemma can be regarded as a version in the case of bifractional Brownian motion.
Theorem 2. Let be a bifractional Brownian motion with Hurst parameters , satisfying and let be given by (16). Then, for each and when belongs to the negative Sobolev space .
Proof. By the chaos expansion of (see  or ), we have
where denotes multiple stochastic integrals with respect to bifractional Brownian motion .
Let us consider the Fourier transform of : By Lemma 1, we obtain Hence where denotes the symmetrization with respect to variables.
By the definition of and taking advantage of (22), we get From the following fact: we show that Firstly, we turn to estimate . Using the similar methods in [7, 11] and the following fact: we find where the last equality is established due to Taylor expansion formula of exponential function.
Since we get By , for each , we have which implies that
Use the change of variables . Furthermore, we have On the other hand, by , there exists a constant depending on and such that Putting (32)-(33) into (27), calculate When , that is, , (34) is finite.
Secondly, using the similar estimation method in the first part, consider the estimation of as follows: By Taylor expansion formula, the following equality is obvious: Recalling some results in , there exist parameters and a constant depending on and such that Thus, by (33) and (37), we obtain Calculate Use the change of variables . Hence where .
When , (40) is finite. In other words, when , (40) is finite.
From what we have said above we can draw a conclusion that when ,
From Theorem 2 we see that when , the mapping belongs to negative Sobolev space . Note that the regularity condition in Theorem 2 does not depend on and . The condition is interesting, because the condition of which belongs to negative Sobolev space is also , where is the Brownian motion (see ). In other words, they have the same regularity condition. However, the situation is different in the case of fractional Brownian motion, because the regularity condition of fractional Brownian stochastic current is , which is dependent on Hurst parameter .
3.2. Stochastic Current of d-Dimensional Case with respect to
As in , we can extend stochastic current of one-dimensional bifractional Brownian motion to the case of d-dimensional bifractional Brownian motion.
Let be the vector valued bifractional Brownian motion; that is, , where are independent one-dimensional bifractional Brownian sheet. In this part, we consider as follows: where the integrals are Skorohod integrals with respect to bifractional Brownian motion.
Theorem 3. Let be d-dimensional bifractional Brownian motion with parameters and satisfying and let be given by (42); then, for each and when , belongs to Sobolev space .
Proof. Denote by
Calculate the Fourier transform of (44) as follows: According to the definition of the normal and using Euler formula, we can prove that where and .
By the bound of and , we can get the following inequality: where and are both constants.
By the same estimation techniques of in Theorem 2, we can obtain the estimation of . Here we need to discuss the estimation of .
Applying again, we can write According to , there exists a constant depending upon , and such that Therefore Use the change of variables . Thus where .
On the other hand, by  for arbitrary , there exists a constant such that In order to be simple, here we only consider the case of .
Comparing (51) with (52), we find that When , that is, , (53) is finite. That is to say, when , there is
It is interesting to contrast Theorems 2 and 3 with Propositions 3 and 4 in . Parameters and of bifractional Brownian motion do not affect the regularity condition of bifractional Brownian current. However, the regularity condition of stochastic current of one-dimensional fractional Brownian motion is different from the case of d-dimensional setting (see ). In other words, Hurst parameters of fractional Brownian motion have influence on fractional Brownian currents in the case of different dimension.
3.3. Stochastic Current of d-Dimensional Bifractional Brownian Motion with respect to
Let be vector valued bifractional Brownian motion where vectors and ; that is, . In this part, is given by where
Using the chaos expansion of divergent integral with respect to bifractional Brownian motion, we can obtain the following expression:
Theorem 4. Let be a bifractional Brownian motion with parameters and satisfying . If is given by (55), then, for every and , is one member of Sobolev-Watanabe spaces , where .
Proof. In order to be convenient, we always replace the normal with the normal . Using the chaos expansion of and by the definition of the normal of , we verify that