The Annals of Probability

Brownian motion on some spaces with varying dimension

Zhen-Qing Chen and Shuwen Lou

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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.

Article information

Ann. Probab., Volume 47, Number 1 (2019), 213-269.

Received: May 2016
Revised: October 2017
First available in Project Euclid: 13 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 31C25: Dirichlet spaces 60H30: Applications of stochastic analysis (to PDE, etc.) 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Space of varying dimension Brownian motion Laplacian transition density function heat kernel estimates Hölder regularity Green function


Chen, Zhen-Qing; Lou, Shuwen. Brownian motion on some spaces with varying dimension. Ann. Probab. 47 (2019), no. 1, 213--269. doi:10.1214/18-AOP1260.

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