Escape probability and transience for SLE

TL;DR

This paper provides uniform and sharp estimates for the probability that SLE curves of various types retreat far from their endpoints after approaching them, focusing on the case where   .

Contribution

It introduces uniform, sharp probability estimates for SLE curves' retreat behavior near their endpoints, extending understanding of SLE transience.

Findings

Estimates are uniform over all initial segments.

Results are sharp up to a universal constant.

Applicable to chordal, radial, and two-sided radial SLE_.

Abstract

We give estimates for the probability that a chordal, radial or two-sided radial SLE$_\kappa$ curve retreats far from its terminal point after coming close to it, for $\kappa \leq 4$. The estimates are uniform over all initial segments of the curve, and are sharp up to a universal constant.