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Brownian motion on some spaces with varying dimension. (English) Zbl 1466.60165

Summary: 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.

MSC:

60J65 Brownian motion
60J35 Transition functions, generators and resolvents
31C25 Dirichlet forms
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J45 Probabilistic potential theory
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