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

Grigor’yan, Alexander; Kajino, Naotaka

doi:10.1090/tran/6784

Localized upper bounds of heat kernels for diffusions via a multiple Dynkin-Hunt formula
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by Alexander Grigor’yan and Naotaka Kajino PDF

Trans. Amer. Math. Soc. 369 (2017), 1025-1060 Request permission

Abstract:

We prove 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 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 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.

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