Planar Brownian motion and Gaussian multiplicative chaos

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 (Ann. Probab. 22 (1994) 566–625), 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 (Acta Math. 186 (2001) 239–270).