Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion

Abstract

Liouville Brownian motion (LBM) is the canonical diffusion process on a Liouville quantum gravity (LQG) surface. In this work, we establish upper and lower bounds for the heat kernel for LBM when $\gamma =\sqrt{8/3}$ in terms of the $\sqrt{8/3}$-LQG metric which are sharp up to a polylogarithmic factor in the exponential.

134 pages, 8 figures; some typos have been fixed, the list of references has been updated, and a paragraph describing special features of $\sqrt{8/3}$-LQG has been added in the introduction