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

Abstract

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” toward the origin. On each component, the “magnitude” of a “push” can be parametrized by a constant $\mathit{\gamma }>0$. In this article, using the probabilistic method, we get the exact expressions for the transition density functions of dBM with varying dimension for any $0<t<\mathrm{\infty }$.