The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$

TL;DR

This paper explores the energy of deterministic Loewner chains, linking it to large deviation principles for SLE$_ ext{0+}$ and demonstrating the reversibility of the energy under time-reversal of the curve.

Contribution

It establishes the reversibility of the energy of deterministic Loewner chains using interpretations from SLE$_ ext{0+}$ and large deviation theory.

Findings

Energy equals the large deviation rate function for SLE$_ ext{0+}$.

Reversibility of the energy under curve time-reversal.

Demonstrates energy invariance between forward and reverse curves.

Abstract

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_\kappa$ as $\kappa$ tends to 0 and the known reversibility of the SLE$_\kappa$ curves for small $\kappa$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.