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

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