  Roughly speaking, a space with varying dimension consists of at least two
components with different dimensions. In this paper we will concentrate on the
one, which can be treated as $\mathbb{R}^3$ tying a half line not contained by
$\mathbb{R}^3$ at the origin. The aim is twofold. On one hand, we will
introduce so-called distorted Brownian motions on this space with varying
dimension (dBMVDs in abbreviation) and study their basic properties by means of
Dirichlet forms. On the other hand, we will prove the joint continuity of the
transition density functions of these dBMVDs and derive the short-time heat
kernel estimates for them.
