# Planar Brownian motion and Gaussian multiplicative chaos

## TL;DR

This paper constructs Gaussian multiplicative chaos measures for planar Brownian motion local times, extending previous work and providing refined estimates for the size of thick points in the subcritical regime.

## Contribution

It introduces a new construction of chaos measures for Brownian local times and relates them to flat measures, extending prior results to the entire subcritical phase.

## Key findings

- Established a nondegenerate limit for the size of thick points.
- Extended Gaussian chaos construction to all subcritical regimes.
- Refined estimates for thick point sizes in planar Brownian motion.

## Abstract

We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereby considerably refining estimates of Dembo, Peres, Rosen and Zeitouni.