Localized upper bounds of heat kernels for diffusions via a multiple Dynkin-Hunt formula

TL;DR

This paper establishes localized upper bounds for heat kernels of diffusions using a novel probabilistic multiple Dynkin-Hunt formula, with applications to Liouville Brownian motion in random geometry.

Contribution

It introduces a new multiple Dynkin-Hunt formula to derive global heat kernel bounds from local behavior assumptions.

Findings

Proves sub-Gaussian off-diagonal heat kernel bounds under local conditions.

Develops a probabilistic approach based on a new transition function formula.

Applies results to Liouville Brownian motion in Gaussian free field geometry.

Abstract

We prove that for a general diffusion process, certain assumptions on its behavior \emph{only within a fixed open subset} of the state space imply the existence and sub-Gaussian type off-diagonal upper bounds of the \emph{global} heat kernel on the fixed open set. The proof is mostly probabilistic and is based on a seemingly new formula, which we call a \emph{multiple Dynkin-Hunt formula}, expressing the transition function of a Hunt process in terms of that of the part process on a given open subset. This result has an application to heat kernel analysis for the \emph{Liouville Brownian motion}, the canonical diffusion in a certain random geometry of the plane induced by a (massive) Gaussian free field.