Brownian Motion on Spaces with Varying Dimension

Abstract

In this thesis we introduce and study Brownian motion with or without drift on state spaces with varying dimension. Starting with a concrete such state space that is the plane with an infinite pole on it, we construct a Brownian motion on it and derive sharp two-sided global estimates on its transition density functions (also called heat kernel). These two-sided estimates are of Guassian type. However, we show that the parabolic Harnack inequality fails for such process and the measure on the underlying state space does not satisfy volume doubling property. Brownian motion on some other state spaces with varying dimensions are also studied in this thesis. For instance, we study Brownian motion on a plane with multiple lines and Brownian motion on a plane with an arc. Similar to Brownian motion with varying dimension, drifted Brownian motion with varying dimension can be characterized by infinitesimal generators and by non-symmetric Dirichlet forms. By establishing and using Duhamel's formula, we show the transition density of Brownian motion with drift on spaces with varying dimension is comparable to that of Brownian motion with varying dimension without drift. We also derive the Green function estimates for this process on bounded smooth domains and establish H"{o}lder regularity for its parabolic functions.