This talk will be devoted to presenting a recent joint work with Alexander Grigor yan(University of Bielefeld)proving that for a general diffusion process,certain assumptions on its behavior ONLY within a FIXED open subset of the state space imply the existence and sub-Gaussian type off-diagonal upper bounds of the heat kernel on the fixed open set.The proof is mostly probabilistic and is based on a seemingly new formula,which we call a "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 Liouville Brownian motion,the canonical diffusion in a certain random geometry of the plane induced by a(massive)Gaussian free field.