Splitting algorithm and normed convergence for drawing the random fractal Loewner curves

Abstract

In the first part of the paper we propose and study the approximation of the $SLE_\kappa$ trace via the Ninomiya-Victoir splitting algorithm. We prove the uniform convergence in probability with respect to the sup-norm to the distance between the $SLE_\kappa$ trace and the output of the Ninomiya-Victoir splitting algorithm when applied in the context of the Loewner differential equation. Further investigations on the $L^p$-norm convergence is also exhibited, shedding light on the more delicate convergence structure. In the second part we show the uniform convergence of the approximation of the $SLE_\kappa$ trace obtained using a different scheme that is based on the linear interpolation of the Brownian driving force.