Law of the SLE tip

TL;DR

This paper characterizes the distribution of the SLE tip at a fixed time as a stationary diffusion process, providing PDE-based density solutions and identifying conditions for finite negative moments, including explicit formulas for some moments.

Contribution

It introduces a novel diffusion process representation for the SLE tip law and derives explicit formulas for certain negative moments.

Findings

The SLE tip law is described as a stationary diffusion process.

The density of the SLE tip law solves a specific PDE.

Explicit formulas are provided for negative second and fourth moments.

Abstract

We analyze the law of the SLE tip at a fixed time in capacity parametrization. We describe it as the stationary law of a suitable diffusion process, and show that it has a density which is a unique solution of a certain PDE. Moreover, we identify the phases in which the even negative moments of the imaginary value are finite. For the negative second and negative fourth moments we provide closed-form expressions.