Stochastic partial differential equations with fractal noise : two different approaches
This thesis deals with stochastic partial differential equations driven by fractional noises. In this work, problems related to this topics are tackled and solved from two fairly different points of view. On one side we prove existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems on bounded smooth domains with Dirichlet boundary conditions by means of semigroup theory and fixed point arguments. Main ingredients are the definition of a product of a function and a (not too irregular) distribution as well as a corresponding norm estimate. As an application, transport stochastic partial differential equations driven by fractional Brownian noises are considered in the pathwise sense. On the other side we deal with stochastic differential equations driven by fractal noises in Banach spaces. More precisely, we deal with abstract Cauchy problems driven by fractional Brownian processes in Banach spaces and look for weak and mild solutions. To this aim, a fractional Brownian motion in separable Banach spaces is introduced by means of cylindrical processes. The related stochastic integral is then defined as cylindrical stochastic process and its properties are investigated. When the Banach space is a function space then the equation becomes a stochastic partial differential equation driven by a fractional noise.