Some topics on the fractional Brownian motion and stochastic partial differential equations

Abstract

In this dissertation, we investigate some problems in fractional Brownian motion and stochastic partial differential partial differential equations driven by fractional Brownian motion and Hilbert space valued Wiener process. This dissertation is organized as follows. In Chapter 1, we introduce some preliminaries on fractional Brownian motion and Malliavin calculus, used in this research. Some main original results are also stated also in this chapter. In Chapter 2, the notion of fractional martingale as the fractional derivative of order &alpha of a continuous local martingale, where -1/2 <α< 1/2, is introduced. Then we show that it has a nonzero finite variation of order 2/(1+2α), under some integrability assumptions on the quadratic variation of the local martingale. As an application, we achieve our objective, an extension of Lévy's characterization theorem to fractional Brownian motion. Chapter 3 is concerned with the problem of exponential moments of the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter 0<H<1. We first apply Clark-Ocone formula to deduce an explicit integral representation for this random variable and then derive the existence of some exponential moments. In Chapter 4, we establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula we establish the smoothness of the density of the solution, and the Holder regularity of the solution in the space and time variables. We also derive a Feynman-Kac formula for the stochastic heat equation in the Skorohod sense and we obtain Feynman-Kac formula to each Wiener chaos of the solution. In Chapter 5, A version of the Feynman-Kac formula for the multidimensional stochastic heat equation with spacially correlated noise is established. For a class of stochastic heat equations, we study the Holder continuity of the solutions, and get an explicit expression for the Malliavin derivatives of the solutions by using the Feynman-Kac formula. Based on the above results and the result from the Malliavin calculus, we show that the law of the solution of the stochastic heat equation has smooth density.