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January 2019 Brownian motion on some spaces with varying dimension
Zhen-Qing Chen, Shuwen Lou
Ann. Probab. 47(1): 213-269 (January 2019). DOI: 10.1214/18-AOP1260


In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density function (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.


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Zhen-Qing Chen. Shuwen Lou. "Brownian motion on some spaces with varying dimension." Ann. Probab. 47 (1) 213 - 269, January 2019.


Received: 1 May 2016; Revised: 1 October 2017; Published: January 2019
First available in Project Euclid: 13 December 2018

zbMATH: 07036337
MathSciNet: MR3909969
Digital Object Identifier: 10.1214/18-AOP1260

Primary: 60J35 , 60J60
Secondary: 31C25 , 60H30 , 60J45

Keywords: Brownian motion , Green function , Heat kernel estimates , Hölder regularity , Laplacian , Space of varying dimension , transition density function

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 1 • January 2019
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