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

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