## 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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## Additional Information

**Alexander Grigor’yan**- Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
- MR Author ID: 203816
- 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
- MR Author ID: 887388
- Email: nkajino@math.kobe-u.ac.jp
- 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 - © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3572263