  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.
