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

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 $\kappa \to 0$ 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 $D$.