Stochastic partial differential equations with fractal noise
The thesis is concerned with partial differential equations under random perturbations. Our formulation is close to the random field approach, the method is pathwise but elementary, that is we do not use rough path theory. Our main tools are fractional calculus, semigroup theory and Stieltjes integration. We show existence, uniqueness and regularity results for a class of boundary initial value problems under the influence of deterministic noise terms. Solutions are defined in the mild sense, using some integral operator of Stieltjes type. Pathwise applications to stochastic problems then include for instance linear and non-linear one-dimensional heat equations under fractional Brownian perturbations with temporal Hurst index greater $1/2$ and appropriate spatial Hurst index.