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

TL;DR

This paper derives explicit heat kernels for a model of distorted Brownian motion on a space with two components of different dimensions joined at a point, revealing detailed transition densities.

Contribution

It provides the first explicit formulas for heat kernels of distorted Brownian motion on spaces with varying dimensions, using probabilistic methods.

Findings

Exact transition density functions are obtained for all positive times.

The model captures the effect of a strong push towards the junction point.

Results enhance understanding of diffusion processes on complex geometric structures.

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" 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 $0<t<\infty$.