Explicit heat kernels of a model of distorted Brownian motion on spaces with varying dimension

In this paper, we study a particular model of distorted Brownian motion (dBM) on state spaces with varying dimension. Roughly speaking, the state space of such a process consists of two components: a $3$-dimensional component and a $1$-dimensional component. These two parts are joined together at the origin. The restriction of dBM on the $3$- or $1$-dimensional component receives a strong "push" towards the origin. On each component, the "magnitude" of the "push" can be parametrized by a constant $\gamma >0$. In this article, using probabilistic method, we get the exact expressions for the transition density functions of dBM with varying dimension for any $0