Continuity in κ in SLEκ theory using a constructive method and Rough Path Theory

Abstract

Questions regarding the continuity in κ of the SLE κ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of κ we use the same Brownian motion. It is very natural to assume that with probability one, SLE κ depends continuously on κ . It is rather easy to show that SLE is continuous in the Carath´eodory sense, but showing that SLE traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence κ j → κ ∈ (0 , 8 / 3), for almost every Brownian motion SLE κ traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the SLE κ traces for varying parameter κ ∈ (0 , 8 / 3). The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by √ κB t when started away from the origin are continuous in the p -variation topology in the parameter κ , for all κ ∈ R + .