The Brownian bubble tea

Abstract

We use Loewner-Kufarev evolution to decompose the Brownian loop soup into a Poisson point process on Brownian bubbles which we call the Brownian bubble tea driven by $\rho$, where $\rho$ is the driving measure for the Loewner chain. The only assumption on $\rho$ is that the evolving boundaries are locally connected. This generalizes a similar disintegration result of Lawler and Werner to essentially optimal regularity, and includes foliations with finite Loewner-Kufarev energy as well as radial and whole-plane ${\mathrm{SLE}}_{\kappa }$ for all $\kappa ⩾0$. As applications, we give Brownian loop interpretations of the conformal distortion formula for Loewner-Kufarev energy and the relationships between the Loewner energy, Loewner-Kufarev energy, and the Schwarzian action.

arXiv comment58 pages, 5 figures