Localized upper bounds of heat kernels for diffusions via a multiple Dynkin-Hunt formula
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.
Cite
@article{arxiv.1502.00213,
title = {Localized upper bounds of heat kernels for diffusions via a multiple Dynkin-Hunt formula},
author = {Alexander Grigor'yan and Naotaka Kajino},
journal= {arXiv preprint arXiv:1502.00213},
year = {2015}
}
Comments
31 pages, 1 figure; slight changes in the title and the text, an insertion of a paragraph after Theorem 3.3 explaining the origin of the name "multiple Dynkin-Hunt formula", and updates in reference information