Local Times And Stochastic Currents Of Gaussian Processes

In recent years, fractional Brownian motion (fBm) has become an intense object in stochastic analysis and related fields for the moment, due to its interesting proper-ties, such as self-similarity, and its applications in various scientific areas. However, when Hurst parameter H≠2/1, fBm is neither a semimartingale nor a Markovian pro-cess. FBm can not be directly dealt by many methods and results in stochastic analysis. On the other hand, fBm may be restrictive as a model. In order to simulate the real situation precisely, it is urgent for us to introduce other Gaussian processes. There-fore, it is interesting and challenge work to study local times and stochastic currents of these Gaussian processes.In this paper, we use white noise analysis approach and Malliavin calculus method to study the local times and stochastic currents of these Gaussian processes. The main innovative results of this paper are as follows.In section 3, we discuss the local times of Gaussian processes through white noise analysis approach. Firstly, prove that the generalized local time of the Wiener integral with respect to Brownian motion is a Hida distribution. Secondly, for a given point, certify that the local time of fBm is a Hida distribution, and give the chaos expansion of the local time in terms of multiple Ito integral. Similar results of d-dimension fBm with N-parameter are researched. We obtain the chaos expansion of the local time in terms of Hermite polynomial. Thirdly, the multiple intersection local times of fBm are considered. Under the mild conditions, the multiple intersection local times of fBm are regarded as Hida distributions. Fourthly, the collision local times of two inde-pendent fractional Brownian motions are considered as Hida distributions. Under the mild conditions, get the chaos expansions and the kernel functions of two independent fractional Brownian motions. Finally, the results of two independent fractional Brow-nian motions can be extended to the case of two independent multifractional Brownian motions through the similar method and local nondeterministic properties.In section 4, mainly study the stochastic currents of Gaussian processes. We firstly define Brownian stochastic current and fractional Brownian stochastic current in the sense of Wick integral, respectively. We prove that these stochastic currents are both Hida distributions via white noise analysis method. Next, the conditions of regularity of bifractional Brownian stochastic current are obtained through Malliavin calculus method. Finally, using similar approach, the conditions of regularity of subfractional Brownian stochastic current are obtained.