Fractional Brownian motion and an application to fluids

Abstract

In this chapter, we review the fractional Brownian motion (fBm) and some of its properties. It is a Gaussian process that is characterized by its covariance function and its hurst parameter H ∈ (0, 1). When H > 1/2, we introduce the stochastic integral with respect to a fBm by using fractional integrals. This is a pathwise approach based on the Riemann Stieltjes construction using the Hölder continuity of the process. An application related to fluids is provided. This is an integrodifferential equation representing the dynamic of a vortex filament associated to an inviscid, incompressible, homogeneous fluid in R3. We prove existence and uniqueness of solutions in a functional space of Sobolev type.