A level line of the Gaussian free field with measure-valued boundary conditions

TL;DR

This paper extends the concept of level lines of the Gaussian free field to measure-valued boundary conditions by constructing SLE-like curves from CLE and Brownian excursions, demonstrating continuous dependence on the measure's intensity.

Contribution

It introduces a novel construction of GFF level lines with measure-valued boundary conditions, expanding the understanding of GFF boundary behaviors.

Findings

Constructed SLE-like curves from CLE and Brownian excursions.

Proved continuous dependence of curve law on Brownian excursion intensity.

Extended GFF level line theory to measure-valued boundary conditions.

Abstract

In this article, we construct samples of SLE-like curves out of samples of CLE and Poisson point process of Brownian excursions. We show that the law of these curves depends continuously on the intensity measure of the Brownian excursions. Using such construction of curves, we extend the notion of level lines of GFF to the case when the boundary condition is measure-valued.