An Asymptotic Radius of Convergence for the Loewner Equation and Simulation of $$SLE_{\kappa }$$ Traces via Splitting

In this paper, we shall study the convergence of Taylor approximations for backward Loewner differential equation (driven by Brownian motion) near origin. More concretely, whenever initial condition (which lies in upper half plane) is small and has form $Z_{0} = \varepsilon i$, show these exhibit an $O(\varepsilon)$ error provided time horizon $\varepsilon^{2+\delta}$ $\delta > 0$. Statements theorem will be given using both rough path $L^{2}(\mathbb{P})$ estimates. Furthermore, over $\varepsilon^{2-\delta}$, see that "higher degree" terms within expansion become larger than "lower $\varepsilon$. sense, on which are accurate scales like $\varepsilon^{2}$. This scaling comes naturally from when growing vector field derivatives balanced against decaying iterated integrals motion. As well as being theoretical interest, may used a guiding principle developing adaptive step size strategies perform efficiently addition, result highlights limitations stochastic methods (such Euler-Maruyama Milstein methods) approximating $SLE_{\kappa}$ traces. Due to analytically tractable fields equation, Ninomiya-Victoir (or Strang) splitting particularly suited SLE simulation. singularity at origin can lead large numerical errors, employ proposed Tom Kennedy discretize traces splitting. We believe scheme first high order method been successfully applied