Stochastic Current of Bifractional Brownian Motion

The fractional Brownianmotionwas first introducedwithin a Hilbert space framework by Kolmogorov in [1]. It was further studied by Mandelbrot and Van Ness in [2], 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 H ̸ = 1/2, 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


Introduction
The fractional Brownian motion was first introduced within a Hilbert space framework by Kolmogorov in [1].It was further studied by Mandelbrot and Van Ness in [2], 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  ̸ = 1/2, 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 [7]).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 [8].
The problem of stochastic current is motivated by the study of fluidodynamical models.In [9], 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 (1 − Δ) − .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 [7] 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 [10] 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  ∈ (0, 1) and  ∈ (0, 1].It is well known that, when  = 1, 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.

Bifractional Brownian Motion
In this section, we briefly recall some notations and facts of bifractional Brownian motion, and for details see [5,6,11].
A bifractional Brownian motion  , is a center Gaussian process with variance where parameters  ∈ (0, 1) and  ∈ (0, 1].In the case  = 1 we retrieve the fractional Brownian motion, while in the case  = 1 and  = 1/2 bifractional Brownian motion corresponds to the Brownian motion.
Let H be a Hilbert space.H is defined as the completion of the linear space generated by the I [0,] ,  ∈ [0, ] with respect to the inner product Sometimes working with the space H is not convenient, because this space also contains distributions (see [11]) and the norm in this space is not always tractable.We always use the subspace H 1 of H, which is defined as the set of measurable functions  on [0, ] with We can prove that H 1 is a Banach space for the norm ‖ ⋅ ‖ H 1 .
At the same time, we have Denote multiple stochastic integrals by   (  ) with respect to  , , where   ∈ H ⊗ .For each  ∈  2 ([0, ]),  has chaos expansion  = ∑ ∞ =0   (  ).Let  be Ornstein-Uhlenbeck operator For each  ∈ (1, ∞) and  ∈ R, 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  ,2 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 Itô integral.Hence the divergence integral is called generalized Itô integral.If  is a stochastic process, it has the following chaos expansion: where   (⋅, ) ∈ H ⊗(+1) .Skorohod integral of  is defined as where  ()  denotes the symmetrization of   with respect to  + 1 variables.

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,  ∈ R, and  > 0. Put where   2 () is a Gaussian kernel function of variance  2 and   () is the Hermite polynomial of degree .By Lemma 3.1 in [7], the following lemma is obtained.Indeed, the lemma can be regarded as a version in the case of bifractional Brownian motion.Lemma 1. Use  x  () to denote the Fourier transform of the function  →    (); then Applying Lemma 1 and as in [7], we can obtain the stochastic current of bifractional Brownian motion.Theorem 2. Let  , be a bifractional Brownian motion with Hurst parameters  ∈ (0, 1),  ∈ (0, 1] satisfying 2 > 1 and let () be given by (16).Then, for each  ∈ Ω and when  > 1/2, () belongs to the negative Sobolev space  − (R; R).
Let us consider the Fourier transform of ( −  , ): By Lemma 1, we obtain Hence where () denotes the symmetrization with respect to  + 1 variables.
By the definition of ‖ ⋅ ‖  − (R;R) and taking advantage of (22), we get . ( From the following fact: we show that (25) Firstly, we turn to estimate Δ 1 .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 [5], for each  1 ,  1 ∈ [0, ], we have which implies that Use the change of variables  = {[( ,  1 −  ,  1 ) 2 ]} 1/2 .Furthermore, we have On the other hand, by [11], there exists a constant  3,1 (, ) depending on  and  such that Putting (32)-( 33) into (27), calculate is finite.
Secondly, using the similar estimation method in the first part, consider the estimation of Δ 2 as follows: By Taylor expansion formula, the following equality is obvious: Recalling some results in [11], there exist parameters ,  and a constant  3,2 (, ) depending on  and  such that Thus, by ( 33) and (37), we obtain Use the change of variables where  3,3 (, ) =  3,1 (, ) 3,2 (, ).
From Theorem 2 we see that when  > 1/2, the mapping () = ∫ [0,]  ( −  ,  ) ,  belongs to negative Sobolev space  − (R; R).Note that the regularity condition in Theorem 2 does not depend on  and .The condition is interesting, because the condition of () = ∫ [0,]  ( −   )  which belongs to negative Sobolev space is also  > 1/2, where   is the Brownian motion (see [7]).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  > 1/2 − 1/2, which is dependent on Hurst parameter .

Stochastic Current of d-Dimensional
Case with respect to .As in [7], we can extend stochastic current of onedimensional bifractional Brownian motion to the case of ddimensional bifractional Brownian motion.
Proof.Denote   () by where ). Calculate the Fourier transform of (44) as follows: ) . ( According to the definition of the normal ‖()‖ where By the bound of sin() and cos(), we can get the following inequality: where  3,4 and  3,5 are both constants.By the same estimation techniques of Δ 2 in Theorem 2, we can obtain the estimation of Δ 4, .Here we need to discuss the estimation of Δ 3, . Applying According to [11], there exists a constant  ) 2 ]} 1/2 .Thus where  = ( 1 , . . .,   ).
On the other hand, by [6] for arbitrary  ≥ 0, In order to be simple, here we only consider the case of  = 1.
Comparing (51) with (52), we find that When It is interesting to contrast Theorems 2 and 3 with Propositions 3 and 4 in [7].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 [7]).In other words, Hurst parameters  of fractional Brownian motion have influence on fractional Brownian currents in the case of different dimension.