  In this paper we introduce and study Brownian motion on 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 functions (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 Holder 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.
