Boundary behavior of SLE

In my thesis, we prove the following theorem: The expected total variation of the Schwarzian derivatives for SLEkappa maps with respect to time variables has a constant asymptotic value independent of the parameter kappa.; The BMO space, or the space of functions of bounded mean oscillation, is the appropriate substitute for Linfinity in many results concerning singular integrals. This notion can be modified in the setting of continuous martingales. In a proper set-up, pre-Schwarzian derivatives of SLEkappa maps are BMO martingales. As a corollary, they satisfy the John-Nirenberg inequality.; This result may lead to an estimate on the lower bound for the Hausdorff dimension of the boundary of SLE hull. The results we obtain allow us to make a formal argument for the lower bound. The estimate for the upper bound is already established by S. Rohde and O. Schramm. While the Hausdorff dimension of the SLEkappa trace was proved by V. Beffara, it remains an open conjecture for the boundary of the hull when kappa > 4.; We prove the sharp estimates on the Holder exponents for kappa ≠ 4. In the case when kappa = 4, we consider the set of points at which the modulus of continuity of logarithmic type fails for SLE4 , and show that this exceptional set is polar for the particular logarithmic types.