In this paper we statistically analyze the Fokker-Planck (FP) equation of Schramm-Loewner evolution (SLE) and its variant $\mathrm{SLE}(\kappa ,{\rho }_c)$. After exploring the derivation and the properties of the Langevin equation of the tip of the SLE trace, we obtain the long- and short-time behaviors of the chordal SLE traces. We analyze the solutions of the FP and the corresponding Langevin equations and connect it to the conformal field theory (CFT) and present some exact results. We find the perturbative FP equation of the $\mathrm{SLE}(\kappa ,{\rho }_c)$ traces and show that it is related to the higher-order correlation functions. Using the Langevin equation we find the long-time behaviors in this case. The CFT correspondence of this case is established and some exact results are presented.

• Received 1 January 2015
• Revised 30 April 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022113