Liouville heat kernel upper bounds at large distances

TL;DR

This paper establishes that the Liouville heat kernel exhibits rapid decay at large distances, demonstrating the semigroup's properties and contributing to the understanding of Liouville quantum gravity models.

Contribution

It provides new upper bounds for the Liouville heat kernel at large distances and proves the semigroup is $C_0$-Feller, advancing theoretical understanding of Liouville quantum gravity.

Findings

Liouville heat kernel decays rapidly at large distances

Liouville semigroup $T_t$ is $C_0$-Feller

Enhanced understanding of Liouville quantum gravity models

Abstract

We show that the Liouville heat kernel decays fast at large distances. In particular, the Liouville semigroup $T_t$ is $C_0$-Feller, where $C_0$ is the space of real-valued continuous functions on $\mathbb C$ vanishing at infinity.