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

TL;DR

This paper derives sharp two-sided bounds for the heat kernel of Liouville Brownian motion on a specific quantum gravity surface, linking it to the associated metric with near-optimal precision.

Contribution

It provides the first sharp two-sided heat kernel bounds for Liouville Brownian motion at b3=a8/3, connecting the process to the b3=a8/3-LQG metric.

Findings

Established upper and lower bounds for the heat kernel of LBM at b3=a8/3.

Bounds are sharp up to a polylogarithmic factor in the exponential.

Connected the heat kernel estimates to the b3=a8/3-LQG metric.

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.