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

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.

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