UC San Diego

The fractional Brownian motions are a family of stochastic processes which resemble Brownian motion in many key ways, yet lack the quality of independence of increments. This dissertation focuses on proving smoothness of densities for solutions to differential equations driven by fractional Brownian motion, provided the vector fields satisfy a particular stratification condition. This result is achieved using the methods of Malliavin calculus. Examples of such solutions include the area process for any two-dimensional projection of the fractional Brownian motion