UC Berkeley

In this thesis, the well-posedness of stochastic differential equations (SDEs) with singular coefficients is discussed. First, it is proved that SDEs with elliptic diffusion possess a unique solution when drift vector fields belong to the Orlicz-critical space. Then, it is shown that SDEs with degenerate and hypoelliptic diffusion are well-posed for a large class of singular drifts. A basic theory on Lorentz spaces and the analysis on the homogeneous Carnot group will also be introduced.