Pathwise and probabilistic analysis in the context of Schramm-Loewner Evolutions (SLE)

On the pathwise analysis side, this thesis contains results between Rough Path Theory and Schramm-Loewner Evolutions (SLE). In the study of Rough Differential Equations questions such as continuity of the solutions, methods of approximations of solutions and their uniqueness/non-uniqueness depending on the behavior of parameters of the equation, appear naturally. We adapt these type of questions to the study of the backward and forward Loewner differential equation in the upper half-plane, conformal welding homeomorphism and the SLE traces. On the probabilistic analysis side, we study a coordinate change of the Loewner equation in which we obtain via a random time change, a stochastic dynamics on a specific line in the upper half-plane, that is depending only on the argument of the points. In this setting, we analyze this dynamics and obtain results about related objects.