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


Authors: Alexander Grigor’yan and Naotaka Kajino
Journal: Trans. Amer. Math. Soc. 369 (2017), 1025-1060
MSC (2010): Primary 35K08, 60J35, 60J60; Secondary 28A80, 31C25, 60J45
DOI: https://doi.org/10.1090/tran/6784
Published electronically: April 14, 2016
MathSciNet review: 3572263
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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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Additional Information

Alexander Grigor’yan
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
Email: grigor@math.uni-bielefeld.de

Naotaka Kajino
Affiliation: Department of Mathematics, Graduate School of Science, Kobe University, Rokkodai-cho 1-1, Nada-ku, 657-8501 Kobe, Japan
Email: nkajino@math.kobe-u.ac.jp

DOI: https://doi.org/10.1090/tran/6784
Keywords: Hunt process, multiple Dynkin-Hunt formula, diffusion, heat kernel, sub-Gaussian upper bound, exit probability estimate
Received by editor(s): February 7, 2015
Published electronically: April 14, 2016
Additional Notes: The first author was supported by SFB 701 of the German Research Council (DFG)
The second author was supported by SFB 701 of the German Research Council (DFG) and by JSPS KAKENHI Grant Number 26287017
Article copyright: © Copyright 2016 American Mathematical Society

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