Abstract. The development of Schramm-Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde [MR05], is to sample Brownian motion at discrete times, interpolate appropriately in between and solve explicitly the Loewner equation with this approximation. This algorithm always produces piecewise smooth non self-intersecting curves whereas SLE κ has been proven to be simple for κ ∈ [0, 4], self-touching for κ ∈ (4, 8) and space-filling for κ ≥ 8. In this paper we show that this sequence of curves converges to SLE κ for all κ = 8 by giving a condition on deterministic driving functions to ensure the sup-norm convergence of simulated curves when we use this algorithm.

The regularity of Loewner curves Huy V. Tran Chair of the Supervisory Committee:Professor Steffen Rohde MathematicsThe Loewner differential equation, a classical tool that has attracted recent attention due to Schramm-Loewner evolution (SLE), provides a unique way of encoding a simple 2-dimensional curve into a continuous 1-dimensional driving function. In this thesis we study the curve in three cases according to the regularity of driving function: weakly Hölder-1/2, Hölder-1/2 with norm less than 4 and C α with α ∈ (1/2, ∞). In the first case, given the existence of the curve we show that the standard algorithm simulating the curve converges in a strong sense. One direct application is for simulating SLE. In the second case, wegive another proof of Marshall, Rohde [26] and Lind [22] in which the curve exists and is a quasi-arc. A sufficient condition for the rectifiability of the curve is also given. In the final case, we show that the Loewner curve is in C α+1/2 . The thesis is a combination of three projects [39], [32] and [21] which are joint work with

We revisit regularity of SLE trace, for all κ = 8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia-Rodemich-Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1 + κ/8, 2), improving on previous works of Werness, and also (optimal) Hölder regularityà la Johansson Viklund and Lawler.

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm-Loewner evolution SLE, Hölder continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic setting and obtain a new and elementary proof that Hölder-1/2 driving functions with norm less than 4 generate simple arcs. We also give a sufficient condition for driving functions to generate curves that are graphs of Lipschitz functions.

To explore the relation between properties of Loewner chains and properties of their driving functions, we study Loewner chains driven by functions U of finite total variation. Under a slow point condition, we show the existence of a simple trace γ and establish the continuity of the map from U to γ with respect to the uniform topology on γ and to the total variation topology on U . In the spirit of the work of Wong [19] and Lind-Tran [9], we also obtain conditions on the driving function that ensures the trace to be continuously differentiable.