Onsager-Machlup functional for $\text{SLE}_{\kappa}$ loop measures

TL;DR

This paper connects two renormalization methods for Brownian loop measures on the Riemann sphere and interprets Loewner energy as an Onsager--Machlup functional for SLE$_ppa$ loop measures, extending to a broader class of loop measures.

Contribution

It establishes a relationship between normalized Brownian loop measure and Werner's measure, enabling the interpretation of Loewner energy as an Onsager--Machlup functional for SLE and related loop measures.

Findings

Unified the two renormalization approaches for Brownian loop measures.

Provided an interpretation of Loewner energy as an Onsager--Machlup functional.

Extended the framework to Malliavin--Kontsevich--Suhov loop measures.

Abstract

We relate two ways to renormalize the Brownian loop measure on the Riemann sphere. One by considering the Brownian loop measure on the sphere minus a small disk, known as the normalized Brownian loop measure; the other by taking the measure on simple loops induced by the outer boundary of the Brownian loops, known as Werner's measure. This result allows us to interpret the Loewner energy as an Onsager--Machlup functional for SLE$_\kappa$ loop measure for any fixed $\kappa \in (0, 4]$, and more generally, for any Malliavin--Kontsevich--Suhov loop measure of the same central charge.